Kvs Pb1 Maths Papers

PM SHRI KENDRIYA VIDYALAYA, MALKAPURAM
Collection of PB 1 Papers
MATHEMMATICS-041
CLASS XII
R. S. N. ACHARYULU, PGT(MATHS)
KENDRIYA VIDYALAYA SANGATHAN, AGRA REGION
MATHEMATICS PRE-BOARD EXAMINATION 2023-24
Class – XII
Mathematics (Code-041)
Time: 3 hours Maximum marks: 80
General Instructions:
1. The Question paper contains - five sections A, B, C, D and E. Each section is
compulsory. However, there are internal choices in some questions.
2. Section A has 18 MCQ’s and 02 Assertion-Reason based questions of 1 mark

each.
3. Section B has 8 Response based MCQ’s of 2 marks each.
4. Section C has 4 Short Answer (SA) type questions of 3 marks each.
5. Section D has 4 Long Answer (LA) type questions of 5 marks each.
6. Section C has 3 case based/source based/integrated/other type questions

of 4 marks each.
SECTION-A
(Multiple Choice Question)
Each question carries 1 mark
Q1. Let f: R→R be defined by f(x)= x4. Then, the correct option is
(a) f is one-one and onto
(b) f is many- one and onto
(c) f is one-one but not onto
(d) f is neither one-one nor onto
1 2
Q2. Let A = [ ]. Then the value of A–1 =
3 4
1 1 2 1 −4 2 1 4 −2 1 −4 −2
(a) 2 [ ] (b) 2 [ ] (c) 2 [ ] (d) 2 [ ]
3 4 3 −1 −3 1 −3 1
Page 1 of 7
Q3. If A is 2 × 3 matrix such that AB and AB′ both are defined, then find the order of
the matrix B
(a) 2 × 3 (b) 3 × 3 (c)2 × 2 (d) Not defined
2 4 2x 4
Q4. If | |=| |, then the possible value(s) of x is/are
5 1 6 x
(a) 1 (b) √3 (c) −√3 (d) ±√3
Q5. If |A| = |kA|, where A is a square matrix of order 2, then sum of all possible
values of k is
(a) 1 (b) -1 (c) 2 (d) 0
Q6. The area of triangle with vertices (-3,0), (3,0) and (0,k) is 9 sq units. Then the
value of k will be
(a) 9 (b) 3 (c) -9 (d) 6
1−cos 4𝑥
𝑥≠0
Q7. The value of 𝑘 for which the function, 𝑓(𝑥 ) = { 8𝑥 2 is continuous at 𝑥 =
𝑘 𝑥=0
0,
(a) 0 (b) -1 (c) 1 (d) 2
Q8. Let the function f : R →R be defined by f(x)=2x + cos x. Then f(x)

(a) has a maximum at x = π (b) has a maximum at x=0
( c) is a decreasing function (d) is an increasing function
cos 2x−cos 2θ
Q9. ∫ dx is equal to
cos x−cos θ
(a) 2(sin x+ x cos θ) +c (b) 2(sin x- x cos θ) + c

(c) 2(sin x +2x cos θ) +c (d) 2 (sin x- sin θ) +c
dy y
Q10. The Integrating Factor of the given differential equation − x = 2x 2 is
dx
1 1
(a) x2 (b) x (c) - (d)
x x
dy dy
Q11. The degree and order of the differential equation + sin( ) = 0. is given by
dx dx
( a ) 1 and 1 ( b ) 1 and not defined

( c ) degree not defined and 1 ( d ) None of these
Page 2 of 7
 
Q12. For given vectors, a = 3 î – ˆj + 2 k̂ and b = – 2 î + 3 ˆj – k̂ , the unit vector in the
 
direction of the vector a + b is
(a)
5

1 ˆ ˆ ˆ
i  jk  (b)
1
6

î  ĵ  k̂  (c)
6

1 ˆ ˆ ˆ
i  jk  (d)
6

1 ˆ ˆ
i  j  2kˆ 
     
Q13. The value of λ for which vectors 2i  j  3k and i  j  4k are orthogonal is
(a)12 (b) 14 (c) 16 (d) none of these


Q14. If| a | = 5,|𝑏̅| = 13 and |𝑎⃗ × 𝑏⃗⃗| = 25, then 𝑎̅·𝑏̅ is equal to
(a) 12 (b) 5 (c) 13 (d) 60

Q15. The vector equation for the line passing through the points (–1, 0, 2)
and (3, 4, 6) is:
(a). i + 2k + λ(4i + 4j + 4k) (b). i – 2k + λ(4i + 4j + 4k)
( c). -i+2k+ λ(4i + 4j + 4k) (d). -i+2k+ λ(4i – 4j – 4k)
Q16. The linear programming problem minimize Z= 3x+2y, subject to constraints
x + y ≤ 8, 3x+5y ≤ 15, x, y ≥0, has
(a) One solution (b) No feasible solution
(c) Two solutions (d) Infinitely many solutions
Q17. Points within and on the boundary of the feasible region represent ……
(a) Infeasible solution (b) Feasible solution

(c) Objective solution (d) None of these
Q18. .The value of P(E|F), where E: no tail appears, F: no head appears,

when two coins are tossed in the air.
(a). 0 (b). ½ (c). 1 (d) None of the above
Page 3 of 7
ASSERTION-REASON BASED QUESTIONS
In the following questions, a statement of assertion(A) is followed by
A statement of
Reason(R). Choose the correct answer out of the following choices.
(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true but R is not the correct explanation of A.
(c) A is true but R are false.
(d) A is false but R is true
Q19. Assertion: The function f(x) = x2 is differentiable function ∀ x ∈ R

Reason: The function f(x) = |x| is a differentiable function ∀ x ∈ R
Q20.Assertion (A): The equation of line which is parallel to 2î + ĵ + 3k̂ and which
𝑥−5 𝑦+2 𝑧−4
pass through the point (5, - 2, 4) is = =
2 1 3
Reason (R): Equation of a line passing through a point A (𝑥1 , 𝑦1 , 𝑧1 ) and parallel to
x−x1 y−y1 z−z1
the given vector (a î + b ĵ + c k̂) is given by: = = .
a b c
SECTION-B
(This section comprises of very short answer type-question (VSA)of 2 marks each)
13𝜋
Q21. Write the principal value of cos −1 (cos ) OR
6
2𝑥−1
Show that the function f: R→ R defined by f(x)= , x∈R is one-one and onto
3
Q22. The amount of pollution content added in air in a city due to x-diesel vehicles is
given by P(x) = 0.005x3 + 0.02x2 + 30x. Find the marginal increase in pollution content
when 3 diesel vehicles are added.
4 sinθ
Q23. Prove that y= – θ is an increasing function of θ in [0, π/2].
2+cosθ
Q24. A ladder 5 m long is leaning against a wall. The bottom of the ladder is pulled
along the ground away from the wall at the rate of 2cm/sec. How fast is its height on
the wall decreasing when the foot of the ladder is 4m away from the wall?
sin2 x−cos2 x 3−5 sinx

Q25. Find ∫ dx OR Find ∫ cos2x
dx
sinx cosx
Page 4 of 7
SECTION-C
(This section comprises of short answer type-question (SA)of 3 marks each)
1 1 dy y
Q26. If x  asin t
, y  a cos t
, show that  OR
dx x
𝑑𝑦
If y=(𝑠𝑖𝑛𝑥)𝑥 +(𝑥)𝑠𝑖𝑛𝑥 , then find .
𝑑𝑥
dx 𝜋
Q27. Evaluate: ∫ √3x2+6x+12 OR Evaluate:∫0 log(1 + 𝑡𝑎𝑛𝑥 ) 𝑑𝑥
4
π xtanx
Q28. Evaluate∫0 dx
tanx+secx
dy
Q29. Solve cos 2 x + y = tanx.
dx
Q30. Solve the following linear programming problem graphically:
Maximize Z = 4x + y
subject to the constraints: x + y ≤ 50 , 3x + y ≤ 90 , x ≥ 0, y ≥ 0
Q31. Bag I contain 3 red and 4 black balls and Bag II contains 4 red and 5 black
balls. One ball is transferred from Bag I to Bag II and then a ball is drawn from bag
II at random. The ball so drawn is found to be red in colour. Find the probability that
the transferred ball is black. OR
A problem in mathematics is given to 3 students whose chances of solving it are 1/2,

1/3 and 1/4, what is the probability that the (i) problem is solved (ii) exactly one of
them will solve
SECTION-D
(This section comprises of long answer type-questions (LA)of 5 marks each)
Q32. Show that the relation R on the set Z of all integers defined by (x,y) ∈R(x-y)
is divisible by 3 is an equivalence relation. OR
𝑥
Show that the function f:R→ 𝑅 defined by f(x) =𝑥 2+1 ∀ x∈ 𝑅 neither one-one nor onto.
Page 5 of 7
2 2 −4 1 −1 0
Q33. Given A= [−4 2 −4] , B = [2 3 4] , find BA and use this to solve the
2 −1 5 0 1 2
system of equations y + 2z = 7, x – y = 3, 2x + 3y + 4z = 17.
Q34. Find the area of the region bounded by the curve x 2=y, y=x+2 and x-axis, using
integration.
x 1 y  3 z  5 x2 y4 z6

Q35. Show that the lines   and   intersect. Find their
3 5 7 1 3 5
point of intersection
OR
Find the shortest distance between lines
𝑥−8 𝑦+9 𝑧−10 𝑥−15 𝑦−29 𝑧−5
= = and = =
3 −16 7 3 8 −5
SECTION-E
[This section comprises of 3 case- study/passage based questions of 4 marks each
with sub parts. The first two case study questions have three sub parts (i), (ii),
(iii) of marks 1,1,2 respectively. The third case study question has two sub parts
of 2 marks each.]
Q36. In an elliptical sport field, the authority wants to design a rectangular soccer
𝑥2 𝑦2
field with maximum possible area. The sport field is given by the graph of 2
+ = 1.
𝑎 𝑏2
(i) If the length and the breadth of the rectangular field be 2x and 2y respectively,
the find the area function in terms of x.
(ii) Find the critical point of the function.
(iii) use first derivative test to find the length 2x and width 2y of the soccer field ( in
terms of a and b) that maximize its area.
Page 6 of 7
Q37. Ginni purchased an air plant holder which is in the shape of a tetrahedron. Let
A, B, C and D are the coordinates of the air plant holder where A (1, 1, 1),B(2, 1, 3),
C (3,2,2) and D(3,3,4).
Based on the above information, answer the following questions.

(i) Find the position vector of →
𝐴𝐵
(ii) Find the unit vector along →
𝐴𝐷
(iii) Find the area of ΔABC using vector method.
Q38. The distribution of a random variable X is given as under
𝑘𝑥 2 , 𝑓𝑜𝑟 𝑥 = 1, 2, 3
P (X = x) = {2𝑘𝑥, 𝑓𝑜𝑟 𝑥 = 4, 5 , 6
0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
Where k is a constant,
Based on the above information, answer the following questions:
(i) What is the value of k?
(ii) Find the value of P (X > 4) and Mean Also.
END OF PAPER
Page 7 of 7
SET – 1 (A)
KENDRIYA VIDYALAYA SANGATHAN, AHMEDABAD REGION 2023 – 24
1ST PRE – BOARD EXAMINATION - 2024
SUBJECT: MATHEMATICS MAX. MARKS: 80 MARKS
CLASS: XII TIME: 3 HOURS
1. This Question paper contains – five sections A, B, C, D and E.
Each section is compulsory. However, there are internal choices in
some questions.
2. Section – A has 20 MCQ’s, 18 Very Short Answer (Type – 1)
and 02 Assertion – Reason based questions of 1 mark each.
3. Section – B has 5 Very Short Answer (VSA) – type questions of
2 marks each.
4. Section – C has 6 Short Answer (SA) – type questions of 3
marks each.
5. Section – D has 4 Long Answer (LA) – type questions of 5
marks each.
6. Section – E has 3 source based/ case based / passage
based/integrated units of assessment (4 marks each) with sub
parts.
SECTION – A
Q. 1: The matrix 𝐵 ′ 𝐴𝐵 is symmetric, if:
(a) 𝐴 is symmetric (b) 𝐴 is skew – symmetric
(c) 𝐵 is symmetric (d) 𝐵 is skew – symmetric
Q. 2: Let R be the relation in the set N given by
𝑅 = {(𝑎, 𝑏): 𝑎 = 𝑏 – 2, 𝑏 > 6}. Choose the correct answer.
(a) (2, 4) ∈ 𝑅 (b) (3, 8) ∈ 𝑅
(c) (6, 8) ∈ 𝑅 (d) (8, 7) ∈ 𝑅
Q. 3: Let A be a square matrix of order 2. If all its entries are 0, 1 and
2, then number of such matrices are:
Page 1 of 9
(a) 9 (b) 12
(c) 27 (d) 81
Q. 4: 1 2 𝑥
If (2𝑥 3) ( ) ( ) = 𝑂 , find the value of x.
−3 0 8
23 23
(a) 0, (b) 0 ,−
2 2
41 41
(c) 0 ,− (d) 0,
2 2
Q. 5: If A and B are invertible matrix, then which of the following is not
true?
(a) 𝑎𝑑𝑗 𝐴 = |𝐴|. 𝐴−1 (b) 𝑑𝑒𝑡 (𝐴−1 ) = {𝑑𝑒𝑡 (𝐴)}−1
(c) (𝐴𝐵)−1 = 𝐵−1 𝐴−1 (d) (𝐴 + 𝐵)−1 = 𝐵−1 + 𝐴−1
Q. 6: 2 −2 𝑥
If 𝐴=( 2 2 −2) is a singular matrix, then 𝑥 = :
−2 2 2
(a) 1 (b) 2
(c) −1 (d) −2
Q. 7: The value of
𝜋⁄4
∫0 √1 + 𝑠𝑖𝑛 2𝑥 𝑑𝑥 is:
(a) 0 (b) 1
1
(c) −1 (d)
√2
Q. 8: The integrating factor of the differential equation

𝑑𝑦
(𝑥 𝑙𝑜𝑔 𝑥) + 𝑦 = 2 𝑙𝑜𝑔 𝑥 is:
𝑑𝑥
(a) 𝑒 𝑥 (b) 𝑙𝑜𝑔 𝑥
(c) 𝑙𝑜𝑔 (𝑙𝑜𝑔 𝑥) (d) 𝑥
Q. 9: The degree of the differential equation
𝑑2𝑦 𝑑𝑦 2 𝑑2𝑦
+ 3 ( ) = 𝑥 2 𝑙𝑜𝑔 ( 2 ) is:
𝑑𝑥 2 𝑑𝑥 𝑑𝑥
(a) 1 (b) 2
(c) 0 (d) Not defined
Q. 10: Let A and B be two events such that 𝑃 (𝐴) = 0.6 , 𝑃 (𝐵) = 0.2 ,
𝐴′
and 𝑃 (𝐴 | 𝐵) = 0.5 . Then 𝑃 ( ′ ) equals?
𝐵
(a) 1⁄ (b) 3⁄
10 10
(c) 3⁄ (d) 6⁄
8 7
Page 2 of 9
Q. 11: If a line makes angles 𝛼 , 𝛽 , 𝛾 with the positive directions of
the coordinate axes, then the value of 𝑠𝑖𝑛2 𝛼 + 𝑠𝑖𝑛2 𝛽 + 𝑠𝑖𝑛2 𝛾
is:
(a) 1 (b) 2 (c) 3 (d) 4
Q. 12: The value of 𝜆 for which the two vectors 2 𝑖̂ − 𝑗̂ + 2 𝑘
̂ and
3 𝑖̂ + 𝜆 𝑗̂ + 𝑘̂ are perpendicular is:

(a) 𝜆=2 (b) 𝜆=4
(c) 𝜆=6 (d) 𝜆=8
Q. 13: The angle between the vectors ̂
𝑖̂ + 𝑗̂ and 𝐽̂ − 𝑘 is:
𝜋 2𝜋 𝜋 5𝜋
(a) (b) (c) − (d)
3 3 3 6
Q. 14: The position vector of the point which divides the join of points
⃗⃗
with position vectors 𝑎⃗ + 𝑏 ⃗⃗ − ⃗𝑏⃗
and 2 𝑎 in the ratio 1 ∶ 2
internally, is:
⃗⃗
3 𝑎⃗⃗ + 2 𝑏
(a) (b) 𝑎⃗
3
⃗⃗
5 𝑎⃗⃗ − 𝑏 ⃗⃗
4 𝑎⃗⃗ + 𝑏
(c) (d)
3 3
Q. 15: Which of the following is true for the function 𝑥 + 1 ?

𝑥
(a) local maximum value = 2

(b) local minimum value = – 2
(c) neither maximum nor minimum value exists
(d) local minimum value is greater than local maximum value
Q. 16: For the curve 𝑦 = 5 𝑥 – 2 𝑥3 , if x increases at the rate of
𝑢𝑛𝑖𝑡𝑠
2 𝑠𝑒𝑐
, then how fast is the slope of curve changing when
𝑥 = 3?
(a) decreasing at the rate of 72 units/sec
(b) increasing at the rate of 72 units/sec
(c) decreasing at the rate of 24 units/sec
(d) increasing at the rate of 24 units/sec
Page 3 of 9
Q. 17: The corner points of the
bounded feasible region are
shown in the figure. The
maximum value of the
objective function
𝑍 = 250 𝑥 + 75 𝑦 occurs
at:
(a) (20 , 0)
(b) (0 , 60)
(c) (10 , 50)
(d) (0 , 0)
Q. 18: The corner points of
the bounded feasible
region are shown in
the figure. The
maximum value of
the objective
function
𝑍 =𝑝𝑥+𝑞𝑦
occurs at (15 , 15) and (0 , 20) . Then the relation between
𝑝 𝑎𝑛𝑑 𝑞 is:
(a) 𝑝=3𝑞
(b) 3𝑝=𝑞
(c) 𝑝=2𝑞
(d) 𝟐𝑝=𝑞
ASSERTION – REASON QUESTIONS
Each of these questions contains two statements, Assertion and
Reason. Each of these questions also has four alternative
choices, only one of which is the correct answer. You have to
select one of the codes (a), (b), (c) and (d) given below.
(a) Assertion is correct, reason is correct; reason is a correct
explanation for assertion.
Page 4 of 9
(b) Assertion is correct, reason is correct; reason is not a
correct explanation for assertion.
(c) Assertion is correct, reason is incorrect.
(d) Assertion is incorrect, reason is correct.
Q. 19: Assertion (A): The angle made by vector 2 𝑖̂ + 𝑗̂ − 2 𝑘̂ with
2
𝑥 – 𝑎𝑥𝑖𝑠 is 𝑐𝑜𝑠−1 (3) .
Reason (R): The scalar components of a unit vector represent
direction cosines of the vector.
Q. 20: Assertion (A): The differential coefficient of
𝑥
𝑠𝑒𝑐 (𝑡𝑎𝑛−1 𝑥) with respect to 𝑥 is .
√1 + 𝑥 2
Reason (R): The Differential coefficient of the function with

respect to 𝑥 is the first order derivative of the function.
SECTION – B
Q. 21: Find the point(s) on the line 𝑥 + 2
=
𝑦 + 1
=
𝑧 − 3
at a distance
3 2 2
5 units from the point (1 , 3 , 3) .

Q. 22: If 𝑐𝑜𝑠−1 𝛼 + 𝑐𝑜𝑠−1 𝛽 + 𝑐𝑜𝑠−1 𝛾 = 3 𝜋 , then find the value of
𝛼(𝛽 + 𝛾) − 𝛽(𝛾 + 𝛼) + 𝛾(𝛼 + 𝛽) .
OR
√3
Find the value of 𝑡𝑎𝑛−1 [2 𝑠𝑖𝑛 (2 𝑐𝑜𝑠 −1 )] .
2
Q. 23: Find the area of the region bounded by the curves 𝑥2 = 4 𝑦 ,
𝑦 = 3 and 𝑥 − 𝑎𝑥𝑖𝑠 .
Q. 24: A kite moving horizontally at the height of 151.5 m. If the speed
of the kite is 10 m/sec, how fast is the string being let out when
the kite is 250 m away from the boy who is flying the kite? The
height of the boy is 1.5 m.
OR
The volume of the cube increases at a constant rate. Prove that
the increase in its surface area varies inversely as the length of
the side.
Page 5 of 9
Q. 25: If 𝑎⃗ + 𝑏
⃗⃗ + 𝑐⃗ = 0 and |𝑎⃗| = 3 , |𝑏⃗⃗| = 5 and |𝑐⃗| = 7 , find
angle between 𝑎 ⃗⃗ .
⃗⃗⃗ and 𝑏
SECTION – C
Q. 26: Draw a rough sketch of the curve
𝑦 = |𝑥 − 1| + 2 , 𝑥 = −2 , 𝑥 = 3 , 𝑦 = 0
and find the area of the region bounded by them using
integration.
Q. 27:
A candidate has to reach the examination centre in time.
Probability of him going by bus or scooter or by other means of
3 1 3
transport are 10 , 10 , 5 respectively. The probability that he will
1 1
be late is 𝑎𝑛𝑑 respectively, if he travels by bus or scooter.
4 3
But he reaches in time if he uses any other mode of transport.

He reached late at the centre. Find the probability that he
travelled by bus.
OR
Two dice are thrown simultaneously. Let X denote the number of
sixes, find the probability distribution of X. Also, find the expected
value of X.
Q. 28: 1
Integrate: ∫ {𝑙𝑜𝑔(𝑙𝑜𝑔 𝑥) + } 𝑑𝑥
(𝑙𝑜𝑔 𝑥)2
OR
𝑒𝑥
Find: ∫ 𝑑𝑥
√5 − 4 𝑒𝑥 − 𝑒2𝑥
Q. 29: Solve the following Linear Programming Problem graphically:

Minimize 𝑍 = 60 𝑥 − 30 𝑦 + 1800
subject to the constraints:
𝑥 + 𝑦 ≤ 30
𝑥 + 𝑦 ≥ 15
𝑥 ≤ 15
𝑦 ≤ 20
Page 6 of 9
𝑥 ≥ 0, 𝑦 ≥ 0.
Q. 30: If 𝑥 = 𝑡𝑎𝑛 (1 𝑙𝑜𝑔 𝑦 ) , prove that
𝑎
𝑑2𝑦 𝑑𝑦
(1 + 𝑥 2 ) + ( 2𝑥 – 𝑎) = 0
𝑑𝑥 2 𝑑𝑥
𝑥 𝑥
Q. 31:
Solve the differential equation: 2 𝑦 𝑒 𝑑𝑥 + (𝑦 − 2 𝑥 𝑒 ) 𝑑𝑦 = 0
𝑦 𝑦
OR
𝑑𝑦
Solve the differential equation: − 3𝑦 𝑐𝑜𝑡 𝑥 = 𝑠𝑖𝑛 2𝑥
𝑑𝑥
SECTION – D
Q. 32: 2 3 1
If 𝐴 = [−3 2 1 ] , find A-1. Using A-1 solve the following
5 −4 −2
system of linear equations:
2𝑥 − 3𝑦 + 5𝑧 = 11, 3𝑥 + 2𝑦 − 4𝑧 = −5, 𝑥 + 𝑦 − 2𝑧 = −3
Q. 33: Let 𝑓 ∶ 𝑁 → 𝑅 be a function defined as
𝑓(𝑥) = 4 𝑥2 + 12 𝑥 + 15. Show that 𝑓 ∶ 𝑁 → 𝑆 where, S is
the range of 𝑓 , is a bijection.
Q. 34: Find the foot of the perpendicular from the point (0, 2, 3) on
𝑥 + 3 𝑦 − 1 𝑧 + 4
the line = = . Also find the length of the
5 2 3
perpendicular.
OR
Show that the lines
𝑟⃗ = (3 𝑖̂ + 2 𝑗̂ − 4 𝑘̂) + λ (𝑖̂ + 2 𝑗̂ + 2 𝑘̂)

and 𝑟⃗ = (5 𝑖̂ − 2 𝑗̂) + 𝜇 (3 𝑖̂ + 2 𝑗̂ + 6 𝑘̂)
are intersecting. Hence, find their point of intersection.
Q. 35: 𝜋⁄2 𝑐𝑜𝑠2 𝑥
Evaluate: ∫ 𝑑𝑥
0 𝑐𝑜𝑠2 𝑥 + 4 𝑠𝑖𝑛2 𝑥
OR
(𝑥 2 + 2) (𝑥 2 + 3)
Evaluate: ∫ (𝑥 2 + 1) (𝑥2 + 𝑑𝑥
4)
SECTION – E
Page 7 of 9
Q. 36: CASE – STUDY 1: Read the following passage and answer
the following questions: (2 + 2)
Mayank, Rajendra and Dinesh throw a dice alternately till one of
them gets a 6 and wins the game. Rajendra starts first then
Mayank throws and then Dinesh throws the die.
(i) Find the probability of winning of Dinesh?

(ii) Find the probability of winning of Mayank?
Q. 37: CASE – STUDY 2: Read the following passage and answer
the following questions: (1 + 1 + 2)
A tank with rectangular base and
rectangular sides, open at the top
is to be constructed so that its
depth is 2𝑚 and volume is
8 𝑚3 .
(i) Assuming dimension of the
base as 𝑥 𝑎𝑛𝑑 𝑦 , write the
relationship between 𝑥 𝑎𝑛𝑑 𝑦 .
(ii) Write the surface area of the tank in terms of 𝑥 .
(iii) Find the dimension of the tank for minimum expense?
OR
Page 8 of 9
If building of tank costs ₹ 70 𝑝𝑒𝑟 𝑚2 for the base and
₹ 45 𝑝𝑒𝑟 𝑚2 for sides, what is the cost of the least

expensive tank?
Q. 38: CASE – STUDY 3: (1 + 1 + 2)
The use of electric
vehicles will curb air
pollution in the long run.
The use of electric
vehicles is increasing
every year and estimated
number of electric
vehicles in use at any time t is given by the function
𝑉(𝑡) = 𝑡3 − 3 𝑡2 + 3 𝑡 − 100 , where t represents time and
𝑡 = 1, 2, 3, … … corresponds to year 2021, 2022, 2023, …….
respectively.
Based on the above information answer the following:
(i) Can the above function be used to estimate number of
vehicles in the year 2020? Justify.
(ii) Find the estimated number of vehicles in the year 2040.
(iii) Prove that the function 𝑉(𝑡) is an increasing function.
*********************************
Page 9 of 9
SET – 1 (A)
केंद्रीय विद्यालय संगठन, अहमदाबाद संभाग 2023-24
KENDRIYA VIDYALAYA SANGATHAN, AHMEDABAD REGION 2023-24
कक्षा बारहिीं के ललए संचयी परीक्षा
CUMULATIVE TEST FOR CLASS-XII

SUBJECT: MATHEMATICS M.M.: 80
CLASS: XII TIME: 3 Hrs.
MARKING SCHEME
Q.NO MARKING POINTS MARKS

ALLOTED
1. (a) 1
2. (c) 1
3. (d) 1
4. (b) 1
5. (d) 1
6. (d) 1
7. (b) 1
8. (b) 1
9. (d) 1
10. (c) 1
11. (b) 1
12. (d) 1
13. (a) 1
14. (d) 1
15. (d) 1
16. (a) 1
17. (c) 1
Page 1 of 8
18. (b) 1
19. (a) 1
20. (a) 1
21. Any point on the line 𝑥 + 2

=𝑦 + 1
=𝑧 − 3
= 𝑟 be 𝑃 (3 𝑟 − 2 , 2 𝑟 − 1 , 2 𝑟 + 3) 1
3 2 2
such that 𝑃𝑄 = 5 where 𝑄 (1 , 3 , 3) .
⇒ 𝑟 = 0 ,2
Points are (−2 , −1 , 3) and (4 , 3 , 7) . 1
22. 𝑐𝑜𝑠 −1 𝛼 + 𝑐𝑜𝑠 −1 𝛽 + 𝑐𝑜𝑠 −1 𝛾 = 3 𝜋 1
⇒ 𝑐𝑜𝑠 −1 𝛼 = 𝜋, 𝑐𝑜𝑠 −1 𝛽 = 𝜋 , 𝑐𝑜𝑠 −1 𝛾 = 𝜋

⇒ 𝛼 = 𝛽 = 𝛾 = −1
𝛼(𝛽 + 𝛾) − 𝛽(𝛾 + 𝛼) + 𝛾(𝛼 + 𝛽) = 2 1
OR √3 𝜋 1
𝑡𝑎𝑛−1 [2 𝑠𝑖𝑛 (2 𝑐𝑜𝑠 −1 )] = 𝑡𝑎𝑛−1 [2 𝑠𝑖𝑛 ( )]
2 3
𝜋
= 𝑡𝑎𝑛−1 [√3] = 3 1
23. Correct figure 1
3
Required area = 4 ∫0 √𝑦 𝑑𝑦 = 8√3 𝑠𝑞 𝑢𝑛𝑖𝑡𝑠 1
24. Let length of string be l meter. 1

∴ 𝑥 2 + 22500 = 𝑙 2
𝑑𝑥 𝑑𝑙
⇒𝑥 =𝑙 … … (1)
𝑑𝑡 𝑑𝑡
At 𝑙 = 250 𝑚 , 𝑥 = 200 𝑚 1
𝑑𝑙 𝑑𝑙 𝑚
200 × 10 = 250 ⇒ =8
𝑑𝑡 𝑑𝑡 𝑠
𝑚
∴ the string is being let out by 8 𝑠 .
OR Let the side of a cube be x unit. 1
Page 2 of 8
Volume of cube 𝑉 = 𝑥 3
𝑑𝑉 𝑑𝑥
= 3 𝑥2 = 𝑘 (𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡)
𝑑𝑡 𝑑𝑡
𝑑𝑥 𝑘
=
𝑑𝑡 3 𝑥2
Surface area 𝑆 = 6 𝑥 2 1
𝑑𝑆 𝑑𝑥
= 12 𝑥
𝑑𝑡 𝑑𝑡
𝑑𝑆 𝑘 𝑘
= 12 𝑥 =4 ( )
𝑑𝑡 3 𝑥2 𝑥
Hence, the surface area of the cube varies inversely as length of side
25. 𝑎⃗ + 𝑏⃗⃗ + 𝑐⃗ = 0 1
𝑎⃗ + 𝑏⃗⃗ = −𝑐⃗
(𝑎⃗ + 𝑏⃗⃗). (𝑎⃗ + 𝑏⃗⃗) = (−𝑐⃗). (−𝑐⃗)
2 2 2
⃗⃗⃗| + |⃗𝑏⃗| + 2|𝑎
|𝑎 ⃗⃗| |⃗𝑏⃗| 𝑐𝑜𝑠 𝜃 = |𝑐
⃗⃗|
⇒ 𝑐𝑜𝑠 𝜃 =
15
=
1
⇒𝜃=
𝜋 1
30 2 3

Required area
1 3
1
= ∫ (−𝑥 + 3) 𝑑𝑥 + ∫ (𝑥 + 1) 𝑑𝑥
−2 1
𝑥2
1
𝑥2
3
33 1
= (− + 3𝑥) + ( 2 + 𝑥) = 𝑠𝑞 𝑢𝑛𝑖𝑡
2 −2 1 2
27. Let 𝐸1 = 𝐻𝑒 𝑤𝑖𝑙𝑙 𝑔𝑜 𝑏𝑦 𝑏𝑢𝑠 1

𝐸2 = 𝐻𝑒 𝑤𝑖𝑙𝑙 𝑔𝑜 𝑏𝑦 𝑠𝑐𝑜𝑜𝑡𝑒𝑟
𝐸3 = 𝐻𝑒 𝑤𝑖𝑙𝑙 𝑔𝑜 𝑏𝑦 𝑜𝑡ℎ𝑒𝑟 𝑚𝑒𝑎𝑛𝑠
3 1 3
∴ 𝑃(𝐸1 ) = , 𝑃(𝐸2 ) = , 𝑃(𝐸3 ) =
10 10 5
Also, let 𝐴 = 𝐻𝑒 𝑟𝑒𝑎𝑐ℎ𝑒𝑑 𝑙𝑎𝑡𝑒 𝑎𝑡 𝑡ℎ𝑒 𝑐𝑒𝑛𝑡𝑟𝑒. 1

𝐴 1 𝐴 1 𝐴
∴ 𝑃( ) = , 𝑃( ) = , 𝑃( ) = 1
𝐸 4 𝐸
1 3 𝐸 2 3
𝐸1
𝐴
𝑃(𝐸1 ) . 𝑃( ) 9
1
𝐸1
By Baye’s theorem 𝑃 ( ) = 𝐴 𝐴 𝐴 =
𝐴 𝑃(𝐸1 ) . 𝑃( )+𝑃(𝐸2 ) . 𝑃( )+𝑃(𝐸3 ) . 𝑃( ) 85
𝐸1 𝐸2 𝐸3
Page 3 of 8
OR Let 𝑋 = 𝑁𝑜 𝑜𝑓 𝑠𝑖𝑥𝑒𝑠 = 0 , 1 , 2 1
1
2
5 5 25 5 1 1 5 10 1 1 1
𝑃(𝑋 = 0) = × = , 𝑃(𝑋 = 1) = × + × = , 𝑃(𝑋 = 2) = × =
6 6 36 6 6 6 6 36 6 6 36
𝑿 0 1 2 1
2
𝑷(𝑿) 25 10 1
36 36 36
10
𝑬(𝑿) = ∑ 𝒙𝒊 𝒑𝒊 = 𝟎 + 36 + 36 = 3
2 1
1
28. 𝐼 = ∫ {𝑙𝑜𝑔(𝑙𝑜𝑔 𝑥) + (𝑙𝑜𝑔

1
} 𝑑𝑥
1
𝑥)2 2
Put 𝑙𝑜𝑔 𝑥 = 𝑡 ⇒ 𝑑𝑥 = 𝑒𝑡 𝑑𝑡
1 1 1 1 1
𝐼 = ∫ {𝑙𝑜𝑔 𝑡 + 2 } 𝑒 𝑡 𝑑𝑡 = ∫ {(𝑙𝑜𝑔 𝑡 − ) + ( + 2 )} 𝑒 𝑡 𝑑𝑡 1
2
𝑡 𝑡 𝑡 𝑡
1 ′ 1 1
Here 𝑓(𝑡) = 𝑙𝑜𝑔 𝑡 − 𝑡 ⇒ 𝑓 (𝑡) = 𝑡 +
𝑡2
1 1
= 𝑙𝑜𝑔 𝑡 − + 𝐶; 𝑢𝑠𝑖𝑛𝑔 𝑡ℎ𝑒𝑜𝑟𝑒𝑚 ∫{𝑓(𝑥) + 𝑓 ′ (𝑥)} 𝑒 𝑥 𝑑𝑥 = 𝑓(𝑥) + 𝐶
𝑡
OR 𝑒𝑥 1
𝐼=∫ 2𝑥
𝑑𝑥 2
√ 5 − 4 𝑒𝑥 − 𝑒
Put 𝑒𝑥 = 𝑡 ⇒ 𝑒𝑥 𝑑𝑥 = 𝑑𝑡
𝑑𝑡 𝑑𝑡 1
𝐼=∫ =∫ 1
√5 − 4 𝑡 − 𝑡 2 √32 − (𝑡+2)2 2
𝑡 + 2 𝑒𝑥 + 2 1
= 𝑠𝑖𝑛−1 ( ) + 𝐶 = 𝑠𝑖𝑛−1 ( )+𝐶
3 3
Page 4 of 8
1
Point 𝒁 = 𝟔𝟎 𝒙 − 𝟑𝟎 𝒚 + 𝟏𝟖𝟎𝟎
(15 , 0) 2700
(15 , 15) 2250
(10 , 20) 1800
(0 , 20) 1200
(0 , 15) 1350
Minimum 𝑍 = 1200 at 𝑥 = 0 𝑎𝑛𝑑 𝑦 = 20 .
30. 1 −1
1
1
𝑥 = 𝑡𝑎𝑛 ( 𝑙𝑜𝑔 𝑦 ) ⇒ 𝑦 = 𝑒 𝑎 𝑡𝑎𝑛 𝑥 2
𝑎
−1 𝑥
𝑑𝑦 𝑎 𝑒 𝑎 𝑡𝑎𝑛
= (1 + 𝑥 2 )
𝑑𝑥
(1 + 𝑥 2 )
𝑑𝑦
= 𝑎 𝑦 … … … (1) 1
𝑑𝑥
𝑑2𝑦 𝑑𝑦 𝑑𝑦
(1 + 𝑥 2 ) + 2𝑥 =𝑎
𝑑𝑥 2 𝑑𝑥 𝑑𝑥
𝑑2𝑦 𝑑𝑦 1
(1 + 𝑥 2) (
+ 2𝑥 – 𝑎) = 0 2
𝑑𝑥 2 𝑑𝑥
𝑥 𝑥
31. 𝑦 𝑦
1
2𝑦 𝑒 𝑑𝑥 + (𝑦 − 2 𝑥 𝑒 ) 𝑑𝑦 = 0
𝑥
𝑥
𝑑𝑥 2 𝑒𝑦 − 1
𝑦
= 𝑥
𝑑𝑦
2 𝑒𝑦
𝑑𝑥 𝑑𝑣
Put 𝑥 = 𝑣𝑦 ⇒ 𝑑𝑦 = 𝑣 + 𝑦 𝑑𝑦
𝑑𝑣 2 𝑣 𝑒𝑣 − 1 𝑑𝑣 1 1
𝑣+𝑦 = ⇒𝑦 =−
𝑑𝑦 2 𝑒𝑣 𝑑𝑦 2 𝑒𝑣
𝑑𝑦 𝑥 1
2 ∫ 𝑒𝑣 𝑑𝑣 = − ∫ ⇒ 2 𝑒𝑣 = −𝑙𝑜𝑔 𝑦 + 𝐶 ⇒ 2 𝑒𝑦 + 𝑙𝑜𝑔 𝑦 = 𝐶
𝑦
OR 𝑑𝑦 1
− 3𝑦 𝑐𝑜𝑡 𝑥 = 𝑠𝑖𝑛 2𝑥 2
𝑑𝑥
𝑑𝑦
Comparing with + 𝑃 𝑦 = Q , we have 𝑃 = − 3 𝑐𝑜𝑡 𝑥 𝑎𝑛𝑑 𝑄 = 𝑠𝑖𝑛 2𝑥
𝑑𝑥
Integrating factor = 𝑒 −3 ∫ 𝑐𝑜𝑡 𝑥 𝑑𝑥 = 𝑒 −3 𝑙𝑜𝑔 𝑠𝑖𝑛 𝑥 = 𝑠𝑖𝑛3 𝑥

1
1
Solution 𝑦. (𝐼𝐹) = ∫ 𝑄 (𝐼𝐹)𝑑𝑥 1
Page 5 of 8
1 1
𝑦. (𝑠𝑖𝑛3 𝑥) = ∫ 𝑠𝑖𝑛 2𝑥 (𝑠𝑖𝑛3 𝑥) 𝑑𝑥 = ∫ 2 𝑐𝑜𝑡 𝑥 𝑐𝑜𝑠𝑒𝑐 𝑥 𝑑𝑥
𝑦 1
= −2 𝑐𝑜𝑠𝑒𝑐 𝑥 + 𝐶 ⇒ 𝑦 = −2 𝑠𝑖𝑛2 𝑥 + 𝐶 𝑠𝑖𝑛3 𝑥
𝑠𝑖𝑛3 𝑥 2
32. 2 3 1 1
𝐴 = [−3 2 1] |𝐴| = −1 2
5 −4 −2
0 2 1 2
𝑎𝑑𝑗 𝐴 = [−1 −9 −5]
2 23 13
0 −2 −1 1
𝐴−1 = [ 1 9 5 ] 2
−2 −23 −13
0 1 −2 11 1 1
1
−1 )′
𝑿= (𝐴 𝐵 = [−2 9 −23] [−5] = [2] 2
−1 5 −13 −3 3
𝑥 = 1 ,𝑦 = 2 ,𝑧 = 3 1
2
33. One – one: Let 𝑥1 , 𝑥2 ∈ 𝑁 such that 𝑓(𝑥1 ) = 𝑓(𝑥2 )
⇒ 4 𝑥1 2 + 12 𝑥1 + 15 = 4 𝑥2 2 + 12 𝑥2 + 15 1
2
⇒ (𝑥1 − 𝑥2 ) (𝑥1 + 𝑥2 + 3) = 0 1
2
⇒ (𝑥1 − 𝑥2 ) = 0 𝑎𝑠 (𝑥1 + 𝑥2 + 3) ≠ 0 1
2
⇒ 𝑥1 = 𝑥2 1
2
Onto: 𝒚 = 4 𝑥 2 + 12𝑥 + 15
− 3 ± √𝑦 − 6 1
⇒ 4 𝑥2 + 12𝑥 + 15 − 𝑦 = 0 ⇒ 𝑥= 2
2
− 3 − √𝑦 − 6 1
Neglecting 𝑥 = ∉𝑵 2
2 2
√𝑦 − 6 − 3
∴ 𝑭𝒐𝒓 𝒂𝒍𝒍 𝒚 ∈ 𝑺, 𝒕𝒉𝒆𝒓𝒆 𝒆𝒙𝒊𝒔𝒕𝒔 𝒙 = ∈ 𝑵 𝒔𝒖𝒄𝒉 𝒕𝒉𝒂𝒕
2
2
√𝑦 − 6 − 3 √𝑦 − 6 − 3 √𝑦 − 6 − 3
𝑓( )= 4 ( ) + 12 ( ) + 15
2 2 2
= 𝑦 − 6 + 9 − 2 √𝑦 − 6 + 2 √𝑦 − 6 − 18 + 15 = 𝑦
34. Any point on the line 1
𝑥 + 3 𝑦 − 1 𝑧 + 4
𝐿1 : 5
= 2
= 3
=𝑟
Be 𝑄 (5 𝑟 − 3, 2 𝑟 + 1 , 3 𝑟 − 4)
Page 6 of 8
DR of PQ is 5𝑟 − 3, 2𝑟 − 1, 3𝑟 − 7 and DR of the line 𝐿1 is 5, 2, 3 1
𝐿1 ⊥ 𝑃𝑄 1
∴ 𝑎1 𝑎2 + 𝑏1 𝑏2 + 𝑐1 𝑐2 = 0 ⇒ 5 (5𝑟 − 3) + 2 (2𝑟 − 1) + 3( 3𝑟 − 7) = 0
⇒𝑟=1
∴ Foot of perpendicular is 𝑄 (2 , 3 , −1) 1
𝑃𝑄 = √21 1
OR If lines are intersecting, then 1

(3 𝑖̂ + 2 𝑗̂ − 4 𝑘̂ ) + λ (𝑖̂ + 2 𝑗̂ + 2 𝑘̂ ) = (5 𝑖̂ − 2 𝑗̂) + 𝜇 (3 𝑖̂ + 2 𝑗̂ + 6 𝑘̂ )
⇒ 3 + λ = 5 + 3𝜇 ⇒ λ − 3𝜇 = 2 … … (1) 1
1
2
⇒ 2 + 2λ = −2 + 2𝜇 ⇒ λ − 𝜇 = −2 … … (2)
⇒ −4 + 2λ = 6𝜇 ⇒ λ − 3𝜇 = 2 … … (3)
Solving (1) and (2), we get λ = −4 , 𝜇 = −2 1
1
2
These values satisfy the 3rd equation
∴ lines are intersecting.
̂.
Point of intersection is −𝑖̂ − 6 𝑗̂ − 12 𝑘 1
35. 𝜋 ⁄2 𝑐𝑜𝑠 2 𝑥 1
𝐼 = ∫0 𝑑𝑥 1
2
𝑐𝑜𝑠 2 𝑥 + 4 𝑠𝑖𝑛2 𝑥
𝜋 ⁄2 𝑐𝑜𝑠 2 𝑥 1 𝜋⁄2 4 − (4 − 3 𝑐𝑜𝑠 2 𝑥)
= ∫0 𝑑𝑥 = ∫0 𝑑𝑥
4 − 3 𝑐𝑜𝑠 2 𝑥 3 4 − 3 𝑐𝑜𝑠 2 𝑥
1
= 3 ∫0
𝜋⁄2
{
4
− 1} 𝑑𝑥 = 3 ∫0
1 𝜋⁄2
{
4
} 𝑑𝑥 − ∫0
1 𝜋⁄2
𝑑𝑥 1
4 − 3 𝑐𝑜𝑠2 𝑥 4 − 3 𝑐𝑜𝑠2 𝑥 3
1 𝜋⁄2 4 𝑠𝑒𝑐 2 𝑥 ⁄ 1 1 𝜋⁄2 4 𝑠𝑒𝑐 2 𝑥 𝜋 1
= 3 ∫0 { } 𝑑𝑥 − (𝑥)𝜋0 2 = ∫0 { } 𝑑𝑥 −
4 𝑠𝑒𝑐 2 𝑥 −3 3 3 1 + 4 𝑡𝑎𝑛2 𝑥 6 2
1 𝜋⁄2 𝑠𝑒𝑐 2 𝑥 𝜋
1
= 3 ∫0 { 2 } 𝑑𝑥 − 6 Put 𝑡𝑎𝑛 𝑥 = 𝑡 ⇒ 𝑠𝑒𝑐 2 𝑥 𝑑𝑥 = 𝑑𝑡 2
(1⁄2) + 𝑡𝑎𝑛2 𝑥
1 ∞ 𝑑𝑡 𝜋 2 𝜋 𝜋 𝜋 𝜋
1
= ∫0 { }− == (𝑡𝑎𝑛−1 2𝑡)∞ 1
2 0 − = − = 2
3 (1⁄2) + 𝑡 2 6 3 6 3 6 6
OR (𝑥 2 + 2) (𝑥 2 + 3)
𝐼 = ∫ (𝑥 2 + 1) (𝑥 2 + 𝑑𝑥
4)
(𝑥 2 + 2) (𝑥 2 + 3)
= (𝑦
(𝑦 + 2) (𝑦 + 3)
= 1 + (𝑦
2
= 1 + 3 ((𝑦
2 1
−
1
) 3
(𝑥 2 + 1) (𝑥 2 + 4) + 1) (𝑦 + 4) + 1) (𝑦 + 4) + 1) (𝑦 + 4)
2 1 1
= 1 + 3 (𝑥 2 − )
+ 1 𝑥2 + 4
𝐼 = ∫ {1 + 3 (𝑥 2
2 1
−
1 2
)} 𝑑𝑥 = 𝑥 + 3 (𝑡𝑎𝑛−1 𝑥 − 2 𝑡𝑎𝑛−1 2) + 𝐶
1 𝑥 2
+ 1 𝑥2 + 4
Page 7 of 8
36. (i) Let 𝐴 = 𝑅𝑎𝑗𝑒𝑛𝑑𝑟𝑎 𝑔𝑒𝑡𝑠 𝑠𝑖𝑥 , 𝐵 = 𝑀𝑎𝑦𝑎𝑛𝑘 𝑔𝑒𝑡𝑠 𝑠𝑖𝑥 , 𝐶 = 𝐷𝑖𝑛𝑒𝑠ℎ 𝑔𝑒𝑡𝑠 𝑠𝑖𝑥 2
𝑃(𝐷𝑖𝑛𝑒𝑠ℎ 𝑤𝑖𝑛𝑠) = 𝑃(𝐴′ 𝐵 ′ 𝐶) + 𝑃(𝐴′ 𝐵 ′ 𝐶 ′ 𝐴′ 𝐵 ′ 𝐶) + 𝑃(𝐴′ 𝐵 ′ 𝐶 ′ 𝐴′ 𝐵 ′ 𝐶 ′ 𝐴′ 𝐵 ′ 𝐶) + ⋯
5 5 1 5 5 5 5 5 1 5 5 5 5 5 5 5 5 1
= . . + . . . . . + + . . . . . . . . + ⋯ ….
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
5 5 1
. . 25
6 6 6
= 5 5 5 =
1− . . 91
6 6 6
(ii) 𝑃(𝑀𝑎𝑦𝑎𝑛𝑘 𝑤𝑖𝑛𝑠) = 𝑃(𝐴′ 𝐵) + 𝑃(𝐴′ 𝐵 ′ 𝐶 ′ 𝐴′ 𝐵) + 𝑃(𝐴′ 𝐵 ′ 𝐶 ′ 𝐴′ 𝐵 ′ 𝐶 ′ 𝐴′ 𝐵) + ⋯ 2

5 1 5 5 5 5 1 5 5 5 5 5 5 5 5 1
= . + . . . . + + . . . . . . . . + ⋯ ….
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
5 1
. 30
6 6
= 5 5 5 =
1− . . 91
6 6 6
37. (i) 𝑥𝑦 = 4 1
(ii) 𝑆 = 4 + 4 (𝑥 + 𝑥)
4 1
𝑑𝑆 4 𝑑2𝑆 48 1
(iii) = 4 (1 − ) 𝑎𝑛𝑑 =
𝑑𝑥 𝑥2 𝑑𝑥 2 𝑥3
𝑑𝑆 1
= 0 ⇒ 𝑥 = ±2 Neglecting negative sign 𝑥 = 2
𝑑𝑥 2
𝑑2𝑆 1
) = 6 > 0 So, surface area is minimum at 𝑥 = 2. 2
𝑑𝑥 2 𝑥=2
OR Cost 𝐶 = 280 + 180 (𝑥 + 𝑥)

4 1
𝑑𝐶 4 𝑑2𝑆 180 × 12
= 180 (1 − ) 𝑎𝑛𝑑 =
𝑑𝑥 𝑥2 𝑑𝑥 2 𝑥3
𝑑𝑆 1
= 0 ⇒ 𝑥 = ±2 Neglecting negative sign 𝑥 = 2
𝑑𝑥 2
𝑑2𝑆 1
) = 6 > 0 So, cost is minimum at 𝑥 = 2 and minimum cost 2
𝑑𝑥 2 𝑥=2
= ₹ 1000
38. (i) No, the above function cannot be used to estimate number of 1
vehicles in the year 2020 because for 2020 we have t = 0 and
𝑉 (0) = 0 − 0 + 0 − 100 = −100, which is not possible.
3
(ii) 𝑉 (20) = 20 − 3 (20)2 + 3 (20) − 100 = 6760 1
(iii) 𝑉 ′ (𝑡) = 3 𝑡2 − 6 𝑡 + 3 = 3 (𝑡2 − 2 𝑡 + 1) = 3 (𝑡 − 1)2 ≥ 0 2
Hence 𝑉(𝑡) is always increasing function.
*******************************
Page 8 of 8
KENDRIYA VIDYALAYA SANGATHAN,BHUBANESWAR REGION
FIRST PRE-BOARD EXAMINATION SESSION 2023-2024
Subject :Mathematics(041) Max.Marks: 80
Class : XII Time : 3 hours
General Instructions :
1. This Question paper contains - five sections A, B, C, D and E.
2. Each section is compulsory. However, there are internal choices in some questions.
3. Section A has 18 MCQ’s and 02 Assertion-Reason based questions of 1 mark each.
4. Section B has 5 Very Short Answer (VSA)-type questions of 2 marks each.
5. Section C has 6 Short Answer (SA)-type questions of 3 marks each.
6. Section D has 4 Long Answer (LA)-type questions of 5 marks each.
7. Section E has 3 source based/case based/passage based/integrated units of assessment (4
marks each) with sub parts.
SECTION - A
1 𝜋 -1 −1 1
sin [ −sin ( )] is equal to:
3 2
1 1
a) b) c) -1 d) 1
2 3
2 If A =[aij] is a skew-symmetric matrix of order n, then 1
1
(a)aij= ∀𝑖,𝑗 (b) 𝑎ij ≠ 0 ∀𝑖,𝑗 (c)𝑎ij = 0, 𝑤ℎ𝑒𝑟𝑒𝑖 = 𝑗 (d) 𝑎ij ≠ 0 𝑤ℎ𝑒𝑟𝑒𝑖 = 𝑗
aji
3 1 , 𝑤ℎ𝑒𝑛 𝑖 ≠ 𝑗 1
If A is a square matrix of order 2 such that aij = { A2 is equal to
0, 𝑤ℎ𝑒𝑛 𝑖 = 𝑗
1 0 1 1 1 1 1 0
(a) [ ] (b)[ ] (c)[ ]
(d) [ ]
1 0 2×2 0 0 2×2 1 0 2×2 0 1 2×2
4 𝑘 8 1
Value of 𝑘, for which A =[ ] is a singular matrix is:
4 2𝑘
a) 8 b) ±𝟖 c) ±4 d) 0
5 If A is a square matrix of order 3, |𝐴′| = −3, then |𝐴𝐴′| = 1
(a) 9 (b) -9 (c) 3 (d) -3
6 If the area of the triangle with vertices (0, k) ,3 ,0) ,(- 3,0) be 9 square units the value/s of k will be 1
(a) 9 (b) ±3 (c) −9 (d)3
𝑥
7 𝑤ℎ𝑒𝑛𝑥 < 0
The point(s), at which the function f given by 𝑓(𝑥) = { |𝑥| , is continuous is/are: 1
−1 𝑤ℎ𝑒𝑛𝑥 ≥ 0
a) 𝑥𝜖R b) 𝑥 = 0 c) 𝑥𝜖 R –{0} d) 𝑥 = −1 and 1
8 For what value of k is the function 1
sin 2𝑥
, 𝑖𝑓 𝑥 ≠ 0
𝑓(𝑥) = { 8𝑥 is continuous at 𝑥 = 0?
𝑘 , 𝑖𝑓 𝑥 = 0
1 1
(a) 4 (b) 1 (c) (d)
4 8
9 For an objective function 𝑍 = 𝑎𝑥 + 𝑏𝑦, where 𝑎, 𝑏> 0; the corner points of the feasible region 1
determined by a set of constraints (linear inequalities) are (0, 20), (10, 10), (30, 30) and (0, 40). The
condition on a and b such that the maximum Z occurs at both the points (30, 30) and (0, 40) is:
a) 𝑏 − 3𝑎 = 0 b) 𝑎 = 3𝑏 c) 𝑎 + 2𝑏 = 0 d) 2𝑎 − 𝑏 = 0
10 The feasible region corresponding to the linear constraints of a Linear Programming Problem is given 1
below
Which of the following is not a constraint to the given Linear Programming Problem?
(a) 𝑥 + 𝑦 ≥ 2 (b) 𝑥 + 2𝑦 ≤ 10 (c)𝑥 − 𝑦 ≤ 1 (d)𝑥 − 𝑦 ≥ 1
11 The value of 𝑏 for which the function 𝑓(𝑥) = 𝑥 + 𝑐𝑜𝑠𝑥 + 𝑏 is strictly decreasing over R is: 1
a) 𝑏< 1 b) No value of b exists c) 𝑏 ≤ 1 d) 𝑏 ≥ 1
12 The least value of the function 𝑓(𝑥) = 2𝑐𝑜𝑠𝑥 + 𝑥 in the closed interval [0, 𝜋/2 ] is: 1
a) 2 b) 𝜋/ 6 + √3 c) 𝜋/ 2 d) The least value does not exist
13 Which of these is equal to ∫𝑒 𝑥 log 5 𝑒 𝑥 𝑑𝑥, where C is the constant of integration? 1
(5𝑒) 𝑥 𝑥 𝑥 𝑥 𝑥
(a) +𝑐 (b)log 5 + 𝑥 + 𝑐 (c) 5 𝑒 + 𝑐 (d)(5𝑒) log 𝑥 + 𝑐
log 5𝑒
𝜋 2
14 For any integer n, the value of ∫–𝜋 𝑒 𝑐𝑜𝑠 𝑥 𝑠𝑖𝑛 3 (2𝑛 + 1)𝑥𝑑𝑥𝑖𝑠 1
(a) -1 (b) 0 (c) 1 (d) 2
15 The general solution of the differential equation 𝑦𝑑𝑥 − 𝑥𝑑𝑦 = 0 𝑖𝑠 1
(a) 𝑥𝑦 = 𝐶 (b) 𝑥 = 𝐶𝑦2 (c) 𝑦 = 𝐶𝑥 (d) 𝑦 = 𝐶𝑥2
16 If two vectors 𝑎 𝑎𝑛𝑑𝑏⃗ are such that |𝑎 | = 2 ,|𝑏⃗ | = 3 𝑎𝑛𝑑⃗⃗⃗⃗𝑎. 𝑏⃗ = 4,then|𝑎 − 2𝑏⃗ | is equal to 1
(a) √2 (b) 2√6 (c) 24 (d) 2√2
17 ABCD is a rhombus whose diagonals intersect at E. Then ⃗⃗⃗⃗⃗ 𝐸𝐴 + ⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗ + ⃗⃗⃗⃗⃗
𝐸𝐵 + 𝐸𝐶 𝐸𝐷 equals to 1
⃗
(a) 0 ⃗⃗⃗⃗⃗
(b)𝐴𝐷 (c)2 𝐵𝐷 ⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗
(d)2𝐴𝐷
18 M and N are two events such that P(M|N) = 0.3, P(M)= 0.2 and P(N) = 0.4. Which of the following is 1
the value of P(M ∩ N')?
(a) 0.8 (b) 0.12 (c) 0.1 (d) 0.08
In the following questions, a statement of assertion (A) is followed by a statement of Reason (R).
Choose the correct answer out of the following choices.
(c) A is true but R is false.
(d) A is false but R is true
19 X = {0, 2, 4, 6, 8}. 1
P is a relation on X defined by P = {(0, 2), (4, 2), (4, 6), (8, 6), (2, 4), (0, 4)}.
Assertion (A): The relation P on set X is a transitive relation.
Reason (R): The relation P has a subset of the form {(a, b), (b, c), (a, c)}, where a, b, c ∈ X.
20 Assertion (A): The function f(x) = |x − 6|(cos x) is differentiable in 𝑅 − {6}. 1
Reason (R): If a function f is continuous at a point c, then it is also differentiable at that point.
SECTION - B
sin 𝑥+cos 𝑥 −𝜋 𝜋
21 Express sin−1( ), where < 𝑥 < in the simplest form. 2
√2 4 4
OR
i) Find the domain of the function below.
1
f(x) = sec−1(5𝑥 − 3)
2
ii) Find the range (principal value branch) of the function below.
1
𝑓(𝑥) = 3 cos−1 ( )−2
2𝑥 − 1
Show your work.
22 A man 1.6 m tall walks at the rate of 0.3 m/sec away from a street light that is 4 m above the ground. 2
At what rate is the tip of his shadow moving? At what rate is his shadow lengthening?
23 1 2−𝑥 2
Evaluate :∫−1 log ( ) 𝑑𝑥
2+𝑥
24 If 𝑎 + 𝑏⃗ + 𝑐 = 0
⃗ 𝑎𝑛𝑑 |𝑎| = 3 , |𝑏⃗ | = 5 , |𝑐| = 7 , then find the value of 𝑎. 𝑏⃗ + 𝑏⃗ . 𝑐 + 𝑐. 𝑎. 2
OR
If 𝑎 = i− 𝚥̂+ 7𝑘̂𝑎𝑛𝑑𝑏⃗ = 5𝚤̂− 𝚥̂+ 𝜆𝑘̂, then find the value of 𝜆 so that the vectors 𝑎 +𝑏⃗𝑎𝑛𝑑𝑎 −𝑏⃗ are
orthogonal.
25 A refrigerator box contains 2 milk chocolates and 4 dark chocolates. Two chocolates are drawn at 2
random. Find the probability distribution of the number of milk chocolates. What is the most likely
outcome?
OR
A problem in mathematics is given to 4 students A, B, C, D. Their chances of solving the problem,
respectively, are 1/3, 1/4, 1/5 and 2/3. What is the probability that (i) the problem will be solved? (ii) at
most one of them will solve the problem?
SECTION - C
26 2𝑥 2 +3 3
Find the anti derivative of
𝑥 2 (𝑥 2 +9)
OR
𝑑𝑥
Evaluate :∫ √3−2𝑥−𝑥 2
27 3
OR
28 Solve the following Linear Programming Problem graphically: 3

Minimize: z=𝑥 + 2𝑦
subject to the constraints: 𝑥 + 2𝑦 ≥ 100 ,2𝑥 − 𝑦 ≤ 0 ,2𝑥 + 𝑦 ≤ 2, 𝑥, 𝑦 ≥ 0
29 1 3
Determine for what values of x, the function f(x) = x3 + 3( x ≠ 0) is strictly increasing or strictly
𝑋
decreasing .
30 3
OR
31 3
SECTION - D
32 5
3 2 1
If A =[4 −1 2 ] ,then find 𝐴−1 𝑎𝑛𝑑 ℎ𝑒𝑛𝑐𝑒 𝑠𝑜𝑙𝑣𝑒 𝑡ℎ𝑒 𝑓𝑜𝑙𝑙𝑜𝑤𝑖𝑛𝑔 𝑠𝑦𝑠𝑡𝑒𝑚 𝑜𝑓 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛𝑠 :
7 3 −3
3𝑥 + 4𝑦 + 7𝑧 = 14 ,2𝑥 − 𝑦 + 3𝑧 = 4, 𝑥 + 2𝑦 − 3𝑧 = 0
33 5
34 5
OR
35 Let N be the set of all natural numbers and R be a relation on N X N defined as (a,b) R (c ,d)  ad = 5
bc for all (a,b) ,(c,d)∈ 𝑁𝑋𝑁 . Show that R is an equivalence relation on N X N .Also ,find the
equivalence class of (2,6) .
OR
𝑥
Show that the function f : R→ {𝑥𝜖𝑅: −1 < 𝑥 < 1} defined by f(x) = , 𝑥 ∈ 𝑅 is an one-one and onto
1+|𝑥|
function.
SECTION - E
[This section comprises of 3 case- study/passage based questions of 4 marks each with sub parts.
The first two case study questions have three sub parts (i), (ii), (iii) of marks 1,1,2 respectively. The
third case study question has two sub parts of 2 marks each.)
36 The reliability of a COVID PCR test is specified as follows: Of people having COVID, 90% of the test
detects the disease but 10% goes undetected. Of people free of COVID, 99% of the test is judged
COVID negative but 1% are diagnosed as showing COVID positive. From a large population of which
only 0.1% have COVID, one person is selected at random, given the COVID PCR test, and the
pathologist reports him/her as COVID positive.
Based on the above information, answer the following

i) What is the probability of the ‘person to be tested as COVID positive’ given that ‘he is 1
actually having COVID?
ii) What is the probability of the ‘person to be tested as COVID positive’ given that ‘he is 1
actually not having COVID’?
iii) What is the probability that the ‘person is actually not having COVID? 2
Or
What is the probability that the ‘person is actually having COVID given that ‘he is tested as COVID 2
positive’?
37 The equations of motion of a rocket are 𝑥 = 3𝑡, 𝑦 = −4𝑡, 𝑧 = 𝑡 , where the time “t” is given in seconds
and the distance is measured in kilometers.
i) Find the position of the missile at t=2 second ? 1
ii) If the position of the rocket at a certain instant of time is (5,-8,10) then what will be 1
the height of the rocket from the ground?
iii) At what distance will the rocket be from the starting point (0,0,0) in 5 seconds? 2
OR
iii) What is the path of the missile?Write the equation of the path of the missile. 2
38
In an elliptical sport field the authority wants to design a regular soccer field with the maximum
𝑥2 𝑦2
possible area . The sport field is given by the graph of 2 + 2 = 1.
𝑎 𝑏
(i) If the length and the breadth of the rectangular field be 2x and 2y respectively, the find the
area function in terms of x. 1
(ii) Find the critical point of the function. 1
(iii) Use the first derivative Test to find the length 2x and width 2y of the soccer field that 2
maximize its area.
OR
(iii) Use the second derivative Test to find the length 2x and width 2y of the soccer field that 2
maximize its area
KENDRIYA VIDYALAYA SANGTHAN, CHANDIGARH REGION
PREBOARD-1 (2023-24)
CLASS XII
MATHEMATICS (041)
Time Allowed: 3 Hours Maximum Marks: 80
1. This Question paper contains - five sections A, B, C, D and E. Each section is compulsory.
However, there are internal choices in some questions.
6. Section E has 3 source based/case based/passage based/integrated units of assessment of 4
marks each with sub-parts.
_____________________________________________________________________________
SECTION A (Multiple Choice Questions)

4 𝜆 −3
Q1. If 𝐴 = [0 2 5 ], and 𝑖𝑓 𝐴−1 exists, then
1 1 3
(a) 𝜆 = 2 (b) 𝜆 ≠ 2 (c) 𝜆 = −2 (d) 𝜆 ≠ −2
(−𝑖 +2𝑗)2
Q2. If A = [aij] is a 2 × 3 matrix, such that aij = . then a23 is
5
1 2 9 16
(a) ) (b) ) (c) (d) )
5 5 5 5
Q3. If A is a square matrix of order 3 and |𝐴| = 5, then |𝐴 𝑎𝑑𝑗𝐴| =
(a) 125 (b) 25 (c) 625 (d) 5
6 0
Q4. If for any square matrix A, A(adjA) = [ ] then value of |𝐴|
0 6
(a) 3 (b) 6 (c) 8 (d) 1
3𝑥 5 9 5
Q5. If | |=| |, then find x.
8 𝑥 8 3
(a) 3 (b) 6 (c) ±3 (d) 1
Q6. If the function f(𝑥) = |𝑥 + 1 | + |𝑥 + 2| is not differentiable at p and q then the value of
p + q is
(a) 3 (b) -3 (c) ±3 (d) 0
1
𝑥𝑠𝑖𝑛 𝑥 , 𝑖𝑓 𝑥 ≠ 0
Q7. The value of k which makes the function (𝑥) = { , is continuous at x = 0
𝑘 − 1 , 𝑖𝑓 𝑥 = 0
(a) 1 (b) 0 (c) ±1 (d) -1
𝑠𝑖𝑛𝑥
Q8. If ∫ 𝑠𝑖𝑛(𝑥−𝑎) 𝑑𝑥 = 𝐴𝑥 + 𝐵 𝑙𝑜𝑔|sin (𝑥 − 𝑎)| + 𝐶 then the value of A2+ B2 is
(a) 2x (b) 0 (c) 1 (d) π
Q9. The direction cosine of a line equally inclined to the axes is?
1 1 1
(a) 1, 1, 1 (b) ±1, ±1,±1 (c) ± ,± ,± (d) 1, 0, 0
√3 √3 √3
Q10. Let A and B be two events. If P (A) = 0.2, P (B) = 0.4, P (A∪B) = 0.6, then P (A|B) is
equal to
(a) 0.5 (b) 0.8 (c) 0.3 (d) 0
Q11. If m is the order and n is the degree of given differential equation,

3
𝑑2 𝑦 𝑑𝑦 5
(𝑑𝑥 2 ) + (𝑑𝑥 ) + 𝑥 4 = 0 then what is the value of m + n
(a) 5 (b) 9 (c) 4 (d) 7
Q12. If 𝑎⃗ and 𝑏⃗⃗ be two-unit vectors and 𝜃 is the angle between them. Then 𝑎⃗ + 𝑏⃗⃗ is a unit
Vector, if 𝜃 =
𝜋 𝜋 𝜋 2𝜋
(a) (b) (c) (d)
4 3 2 3
Q13. If magnitude of vector 𝑥(𝑖̂ + 𝑗̂ + 𝑘̂) is ‘3 units’ then the value of 𝑥 is

1 1
(a) ) (b) ) √3 (c) ± (d) ) ± √3
3 √3
Q14. The projection of 𝑎⃗ = 2𝑖̂ − 𝑗̂ + 𝑘̂ along 𝑏⃗⃗ = 𝑖̂ + 2𝑗̂ + 2𝑘̂ is

2 2
(a) 3 (b) - 3 (c) 2 (d) √6
3−2𝑥 𝑦+ 5 6− 𝑧
Q15. The direction ratio of the line = = is
4 3 6
(a) 4, 3, 6 (b) 2, 3, - 6 (c) 2, -3, 6 (d) – 2, -3, -6
Q16. The objective function Z = 4 x + 3 y can be maximized subjected to the constraints

3x + 4y ≤24, 4x + 3y ≤ 24 , x, y ≥ 0
(a) at only one point ( b) does not exist
(c) at two points only (d) at an infinite number of points
Q17.The corner points of feasible solution region determined by the system of linear
constraints are (0, 10), (5, 5), (15,15), (0,20). Let 𝑍 = 𝑝𝑥 + 𝑞𝑦, where p, q> 0.
Condition on p and q so that the maximum of Z occurs both the point (5,5) and (0,10) is
(a) p = 2q (b) p = q (c) q = 2p (d) q = 3p
Q18. A couple has 2 children. Then probability that both are boys, if it is known that one of
the children is a boy is?
1 1 2
(a) (b) ) (c) (d) ) 1
3 2 3
ASSERTION- REASON BASED QUESTIONS

In the following questions a statement of Assertion (A) is followed by a statement of
Reason (R).
(a) Both A and R are true, and R is the correct explanation of A.
(b) Both A and R are true, but R is not the correct explanation of A.
(c) A is true, R is false.
(d) A is false, R is true.
Q19. Assertion (A): sin−1 (−𝑥) = − sin−1 𝑥 ; 𝑥 ∈ [−1,1]

𝜋
Reason (R): sin−1: [−1,1] → [ 0, ] is a bijection function.
2
𝟐
𝒙 + 𝟏 ,𝒙 ≤ 𝟏
Q20. Assertion (A): 𝒇(𝒙) = { is continuous at x=1
𝟐𝒙 , 𝒙 > 𝟏
Reason (R): LHL and RHL both are equal and it is equal to f(1).
SECTION-B
This section comprises of very short answer type question (VSA) of 2 marks each.
𝜋 1
Q21.If 0 < α < 2 and sinα = 2 ; Evaluate: tan−1{2cos (3𝜋 − 2𝛼)}
OR
f : R → R defined by 𝑓(𝑥) = 𝑥|𝑥|. Is function one-one and onto?
𝑑𝑦
Q22. If 𝑥 𝑚 𝑦 𝑛 = (𝑥 + 𝑦 )𝑚 + 𝑛 , then find .
𝑑𝑥
Q23. A man 1.6 m tall walks at a rate of 0.3 m/sec away from a street light that is 4m above
the ground. At what rate is the tip of the shadow moving?
1
Q24. Evaluate ∫ 𝑥(𝑥+1) 𝑑𝑥
1 2 1
Q25. For two events A and B let 𝑃(𝐴) = 2 𝑃(𝐴𝑈𝐵) = 3 𝑎𝑛𝑑 𝑃(𝐴 ∩ 𝐵) = 6 , then find the
value of 𝑃(𝐴 ∩ 𝐵).
SECTION C
(This section comprises of short answer type question (SA) of 3 marks each)
(𝑥 2 +1)𝑒 𝑥
Q26 Evaluate ∫ (𝑥+1)2
𝑑𝑥
Q27. Solve the differential equation: y (1 + ex) dy = (y + 1) ex dx.
1
Q28. Find ∫ 𝑠𝑖𝑛(𝑥−𝑎).𝑐𝑜𝑠(𝑥−𝑏) 𝑑𝑥
Or
2
Evaluate ∫−1|𝑥 3 − 𝑥| 𝑑𝑥
𝑥+ 3
Q29.Find ∫ √5 𝑑𝑥
−4𝑥 − 𝑥 2
OR
𝑥2
Find ∫ (𝑥 2 + 1)(𝑥 2 + 4) 𝑑𝑥
Q30. Solve the following linear programming problem graphically:

Minimise Z = 200x + 500y
Subject to constraints: 𝑥 + 2𝑦 ≥ 10 , 3𝑥 + 4𝑦 ≤ 24, 𝑥 ≥ 0, 𝑦 ≥ 0;
Q31. 𝑆ℎ𝑜𝑤 𝑡ℎ𝑎𝑡 𝑓(𝑥) = |𝑥 − 1| 𝑖𝑠 𝑛𝑜𝑡 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡𝑖𝑎𝑏𝑙𝑒 𝑎𝑡 𝑥 = 1

SECTION D
(This section comprises of long answer-type question (LA) of 5 marks each)
2 3 1
Q32. If A = [−3 2 1 ] , find A-1 and use it to solve the system of equations:
5 −4 −2
2x – 3y + 5z = 11, 3x + 2y – 4z = - 5, x + y – 2z = - 3
Q33. Consider a function f : R+ → [-5, ∞) defined as f(x) = 9x2 + 6x – 5. Show that f is

one- one and onto function, Where R+ is the set of all non-negative real numbers.
OR
Let N be the set of all natural numbers & R be the relation on N × N defined by
{ (a,b) R (c,d) iff a + d = b + c}. Show that R is an equivalence relation.
Q34. Using integration find the area of the region bounded by the curve y = cos x
between x = 0 and x = 2π.
Q35. Find the equation of the lines through origin which intersect the line
𝑥−3 𝑦−3 𝑧 𝜋
= = 1 at an angle of
2 1 3
OR
Find equation of line which passes through (1, 1, 1) and intersects the lines
𝑥−1 𝑦−2 𝑧−3 𝑥+2 𝑦−3 𝑧+1
= = and = = .
2 3 4 1 2 4
SECTION E
(This section comprises of 3 case study/ passage- based questions of 4 marks each with
two sub- parts. First two case study questions have three sub-parts of marks 1, 1, 2
respectively. The third case study question has two sub-parts of 2 marks each.)
Q36. If Sunil and his friend walking through along two sides of parallelogram shaped
agricultural field. Vector representation of sides of field are 𝑎⃗ = 𝑖̂ + 4𝑗̂ + 2𝑘̂ and
𝑏⃗⃗ = 3𝑖̂ − 2𝑗̂ + 7𝑘̂
( i) find the both diagonals of field
(ii) find area of the field by using sides
(iii) find the area of the field by using diagonals .
Q37. Case-Study II- A m a n has an expensive square shape piece of golden board of
size 24 cm is to be made into a box without top by cutting square of side x
from each corner a nd folding the flaps to form a box.
(i) What is the Volume of the open box formed by folding up the flap:
dy
(ii) In the first derivative test, if changes its sign from positive to negative as x increases
dx
through c1, then which value is attained by the function at x = c1.
(iii) What should be the side of the square piece to be cut from each corner of the board to
be hold the maximum volume?
OR
(iii) Find the maximum volume of the box?
Q38. Case-Study III - Read the text carefully and answer the questions:
A doctor is to visit a patient. From the past experience, it is known that the
probabilities that he will come by cab, metro, bike or by other means of transport are
respectively 0.3, 0.2, 0.1, and 0.4. The probabilities that he will be late are 0.25, 0.3, 0.35,
and 0.1 if he comes by cab, metro, bike and other means of transport respectively.
i. When the doctor arrives late, what is the probability that he comes by
metro?
ii. When the doctor arrives late, what is the probability that he comes by
other means of transport?
************************************END OF PAPER*************************
12MAT02 QP
KENDRIYA VIDYALAYA SANGATHAN, CHENNAI REGION
FIRST PRE-BOARD EXAMINATION (2023-24)
CLASS XII MATHEMATICS TIME: 3 hrs
Max Marks : 80
1. This Question paper contains five sections A, B, C, D and E. Each section is
2. Section A has 18 MCQ’s and 02 Assertion-Reason based questions of 1mark each.
3. Section B has 5 Very Short Answer (VSA)-type questions of 2marks each.
4. Section C has 6 Short Answer (SA)-type questions of 3marks each.
5. Section D has 4 Long Answer (LA)-type questions of 5marks each.
6. Section E has 3 source based/case based/passage based/integrated units of
assessment (4marks each) with sub-parts.
SECTION A
(Multiple Choice Questions)
Each question carries 1mark
𝑑𝑦 2 𝑑𝑦 2
1). Sum of the order and degree of the differential equation (𝑥 + ) = (𝑑𝑥 ) + 1 is
𝑑𝑥
(a) 1 (b) 2 (c) 3 (d) 4
2). The corner points of the feasible region determined by the system of linear constraints are (0,3), (1,1)
and (3,0). Let Z = px + qy, where p, q > 0. Condition on p and q so that the minimum of Z occurs at both
the points (3,0) and (1,1) is
(a) 2p = q (b) p =2q (c) q = p (d) q = 3p
3) . If A is a square matrix such that A2 = A, then the value of 7A – (I+A)3 , where I is an identity matrix,
is
(a) – I (b) I (c) A (d) A – I
𝑑𝑦 𝑦
4). The general solution of the differential equation 𝑑𝑥 = 𝑥 , is
(a) log y = kx (b) xy = k (c) y = kx (d) y = k log x
5). The region represented by the inequation system 𝑥, 𝑦 ≥ 0 , 𝑦 ≤ 6, 𝑥 + 𝑦 ≤ 3 is

(a) unbounded in the first quadrant (b) unbounded in first and second quadrants
(c) bounded in first quadrant containing the origin
(d) bounded in first quadrant not containing the origin
1 1 1
6). A problem in mathematics is given to three students whose chances of solving it are , ,
2 3 4
respectively. If the event of their solving the problem are independent, then the probability that the
problem will be solved, is
1 1 1 3
(a) 4 (b) 3 (c) 2 (d) 4
Page 1 of 6
7). If the area of the triangle with vertices (-3,0) ,(3,0) and (0,k) is 9 sq units, then the value(s) of k will be
(a) ± 9 (b) ± 3 (c) ± 8 (𝑑) ± 4
8). Let A be a 3×3 matrix such that |𝑎𝑑𝑗 𝐴| = 64. Then |𝐴| is equal to
(a) 8 only (b) -8 only (c) 64 (d) 8 or -8
3 4
9). If A = [ ] and 2A + B is a null matrix, then B is equal to
5 2
6 8 −6 −8 3 4 −3 −4
(a) [ ] (b) [ ] (c) [ ] (d) [ ]
10 4 −10 −4 5 2 −5 −2
2 0
10). If [ ] = P + Q , where P is a symmetric and Q is a skew symmetric matrix, then Q is equal to
5 4
−5 5 −5 5
0 0 2 2
2 2 2 2
(a) [ 5 ] (b) [−5 ] (c) [ 5 ] (d) [ 5 ]
0 0 4 4
2 2 2 2
4𝑥 2 + 3𝑏𝑥, 𝑥 ≠ 1
11). If f(x) is continuous at x=1 and f(x) = { then the value of ‘b’ is
5𝑥 − 4, 𝑥=1
(a) 3 (b) - 1 (c) 1 (d) - 3
𝑑2 𝑦
12). If y = 𝑒 −𝑥 , then 𝑑𝑥 2 is equal to
(a) y (b) - y (c) x (d) – x.
𝑑
13). If 𝑑𝑥 [𝑓(𝑥)] = 𝑎𝑥 + 𝑏 𝑎𝑛𝑑 𝑓(0) = 0 , then f(x) is equal to
𝑎𝑥 2 𝑎𝑥 2
(a) a + b (b) b (c) + 𝑏𝑥 (d) + 𝑏𝑥 + 𝑐
2 2
14). If 𝑎⃗ and 𝑏⃗⃗ are such that |𝑎⃗. 𝑏⃗⃗|= |𝑎⃗ × 𝑏⃗⃗|, then the angle between 𝑎⃗ and 𝑏⃗⃗ is
𝜋 𝜋 𝜋 𝜋
(a) 6 (b) 3 (c) 4 (d) 2
15). In a triangle OAC, if B is the midpoint of side AC and ⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗ = 𝑏⃗⃗, 𝑡ℎ𝑒𝑛 𝑂𝐶
𝑂𝐴 = 𝑎⃗, 𝑂𝐵 ⃗⃗⃗⃗⃗⃗ 𝑖𝑠 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 .
⃗⃗
𝑎⃗⃗+ 𝑏
(a) 𝑎⃗ + 𝑏⃗⃗ (b) (c) 2𝑎⃗ + 𝑏⃗⃗ (d) 2 𝑏⃗⃗ − 𝑎⃗
2
16). The value of (𝑖̂ × 𝑗̂). 𝑘̂ + 𝑖.̂ 𝑗̂

(a) - 1 (b) 0 (c) 2 (d) 1
17). The two lines 𝑟⃗ = (𝑖̂ + ̂𝑗 − ̂𝑘)+ 𝜇 (2𝑖̂ + 3̂𝑗 − 6 ̂𝑘) and 𝑟⃗ = (2𝑖̂ − ̂𝑗 − ̂𝑘)+ α (6𝑖̂ + 9̂𝑗 − 18 ̂𝑘 ) are
(a) parallel (b) intersecting (c) perpendicular (d) skew
18). The direction cosines of a vector equally inclined to the axes OX, OY and OZ are
1 1 1 1 1 1
(a) (1,1,1) (b) ( , , ) (c) (3 , 3 , 3) (d) (1,2,3)
√3 √3 √3
Page 2 of 6
ASSERTION-REASON BASED QUESTIONS:
In the following questions, a statement of Assertion (A) is followed by a statement of
Reason (R). Choose the correct answer out of the following choices.
(d) A is false but R is true.
19). Assertion (A) : Let A = {2,4,6} and B = {3,5,7,9} and defined a function f = {(2,3),(4,5),(6,7)}
from A to B. Then, f is not onto.
Reason(R): A function f : A → B is onto, if every element of B is the image of some element of A
under f.
3𝜋
20). Assertion (A) : If f(x) = log(sin x), x > 0 is strictly decreasing in (𝜋, )
2
′
Reason(R): If 𝑓 (𝑥) > 0 , then f(x) is strictly increasing function.
SECTION B
This section comprises of very short answer type-questions (VSA) of 2marks each
21). Find the principal value of tan-1(√3 ) – cot-1 (-√3).
(OR)
3𝜋
Find the range of the function f(x) = 2 sin -1 x +
2
22). The total revenue in rupees received from the sale of ‘x’ units of a product is given by
R(x) = 13x2 + 26x + 15. Find the marginal revenue when x = 7.
23). Find the interval(s) in which the function f : R → R defined by f(x) = x2 𝑒 −𝑥 , is increasing
24). If f(x) = 9 x2 + 12 x + 2 , then find the minimum value of f(x).

(OR)
Find the maximum profit that a company can make , if the profit function is given by
P(x) = 72 + 42 x – x2 , where x is the number of units and P is the profit in rupees.
25). Evaluate :
𝜋
2
∫−𝜋 ( 𝑥 3 + 𝑥 cos 𝑥 + 𝑡𝑎𝑛5 𝑥) 𝑑𝑥
2
SECTION C
This section comprises of Short Answer type questions (SA) of 3marks each
𝜋 𝑑2 𝑦
26). If x = a (cos t + t sin t) and y = a (sin t – t cos t) , 0 < t < 2 , find 𝑑𝑥 2 .
2𝜋 1
27). Evaluate ∶ ∫0 dx.
1+𝑒 𝑠𝑖𝑛 𝑥
(OR)
𝜋
Evaluate : ∫0 log(1 + tan 𝑥) 𝑑𝑥.
4
Page 3 of 6
6𝑥+7
28). Evaluate ∶ ∫ dx.
√(𝑥−5)(𝑥−4)
dx
29). Solve the differential equation + x cot 𝑦 = 2 y + y 2 cot 𝑦 where y ≠ 0
dy
(OR)
𝑦
Solve the differential equation [𝑥 𝑠𝑖𝑛2 (𝑥 ) − 𝑦] dx + x dy =0
30) Solve the following linear programming problem graphically.

Minimize Z = 3 x + 9 y
Subject to
x + 3y ≤60, x + y ≥ 10 , x ≤ y , x, y ≥ 0
(OR)
Solve the following linear programming problem graphically.
Minimize Z = 50 x + 70 y
Subject to
2x + y ≥ 8 , x + 2y ≥ 10 , x, y ≥ 0
31). A random variable X has the following probability distribution, where k is some real number
X 0 1 2 otherwise
P(X) k 2k 3k 0
(1) Determine the value of k

(2) Find P(X<2)
(3) Find P(X>2)
SECTION D
This section comprises of Long Answer-type questions (LA) of 5 marks each
32). Prove that the relation R on the set N×N defined by (a,b) R (c,d) ⇒ a + d = b + c, for every
(a,b), (c,d) ∈ 𝑁 × 𝑁 is an equivalence relation.
(OR)
If 𝑅1 𝑎𝑛𝑑 𝑅2 are equivalence relations in a set A , show that 𝑅1 ∩ 𝑅2 is also an equivalence relation.
𝑥2 𝑦2 𝑥 𝑦
33). Find the area of the smaller region bounded by the ellipse + = 1 and the line + =1
9 4 3 2
Page 4 of 6
34). Using matrix method, solve
2 3 10 4 6 5 6 9 20
+ + =4, − + =1, + − =2
𝑥 𝑦 𝑧 𝑥 𝑦 𝑧 𝑥 𝑦 𝑧
𝑥+3 𝑦−1 𝑧+4

35). Find the coordinates of the image of the point P(0,2,3) with respect to the line = =
5 2 3
(OR)
Find the shortest distance between the lines whose equations are
𝑟1 = 𝑖̂ + 𝑗̂ + 𝜆(2𝑖̂ − 𝑗̂ + 𝑘̂)
⃗⃗⃗⃗ 𝑟2 = 2𝑖̂ + 𝑗̂ − 𝑘̂ + 𝜇(3𝑖̂ − 5𝑗̂ + 2 𝑘̂ )
and ⃗⃗⃗⃗
SECTION E
This section comprises of 3 case-study/passage-based questions of 4 marks each. First two questions
have three sub-parts (1),(2),(3) of marks 1,1,2 respectively. The third case study question has two
sub-parts of 2 marks each.
36). Read the following passage and answer the questions given below.
A house is being constructed and a lot of planning is put into it. Now a person is confused about the
window. He wants the window in the form of a rectangle surmounted by a semicircle such that the
perimeter of the window is to be 10 metres. If radius of the semicircular portion is ‘r’ metres and
height of the rectangular portion is ‘x’ metres, then
(1) Write a relation between x and r.

(2) Represent the area in terms of ‘r’.
(3) Find the critical point, with respect to area, in terms of ‘r’.
(OR)
(3) What are dimensions so that maximum light may enter the room?
(Note: Internal choice is for option 3)
Page 5 of 6
37). A student of class XII wants to find the displacement of an object using the formula
1 𝐹⃗
𝑠⃗ = 𝑢
⃗⃗𝑡 + 𝑎⃗ 𝑡 2 and 𝑎⃗ = , 𝑤ℎ𝑒𝑟𝑒 𝑢
⃗⃗ = 2𝑖̂ 𝑚/𝑠 and mass of the object is 2 kg. Force on
2 𝑚
the object are as (Newton unit)

𝐹1 = 2𝑖̂ + 3𝑗̂ − 𝑘̂ , ⃗⃗⃗⃗
⃗⃗⃗⃗ 𝐹2 = 2𝑖̂ + 2𝑗̂ − 3𝑘̂
From the above information answer the following

(1) Find the net force on the object.
(2) Find the acceleration of the object.
(3) Find the unit vector perpendicular to both ⃗⃗⃗⃗ ⃗⃗⃗⃗2 .
𝐹1 and 𝐹
(OR)
(3) Find the displacement in 2 seconds.
(Note: Internal choice is for option 3)
38). A company has two plants to manufacture TVs. The first plant manufactures 70% of the TVs
and the rest are manufactured by the second plant. 80% of the TVs manufactured by the first
plant are rated of standard quality, while that of the second plant only 60% are of standard quality.
One TV is selected at random.
Based on the above information answer the following :

(i) Find the probability that the selected TV is of standard quality.
(ii) Find the probability that the TV is of standard quality, given that it was manufactured by the first
plant.
***************
Page 6 of 6
Pre-Board Exam(2023-24) Set-II
Class-XII
Time:3h Subject-Mathematics MM-80
Instructions:
1. This Question paper contains - five sections A, B, C, D and E. Each section is compulsory. However,
there are internal choices in some questions.
2. Section A has 18 MCQ's and 02 Assertion-Reason based questions of 1 mark each.
6. Section E has 3 source based/case based/passage based/integrated units of assessment (4 marks each)
with sub parts
SECTION A
(Multiple Choice Questions) Each question carries 1 mark
𝟏 𝟏 𝟏
Q1. If direction cosine of a line are < 𝒂 , 𝒂 , 𝒂 > then a is
(a) a< 𝟏 (b) a> 𝟏 (c) a=±𝟐 (d) a=±√𝟑
𝟕 𝟏
Q2. The inverse of matrix [ ] is
𝟒 −𝟑
𝟕 𝟏 𝟏 𝟑 𝟏 (𝒄) [𝟑 𝟏 (𝒅) [−𝟑 −𝟏

(a) [ ] (b)𝟐𝟓 [ ] ] ]
−𝟒 −𝟑 𝟒 −𝟕 𝟒 −𝟕 −𝟒 𝟕
Q3. 𝑰𝒇 𝑨 𝒊𝒔 𝒂 𝒔𝒒𝒖𝒂𝒓𝒆 𝒎𝒂𝒕𝒓𝒊𝒙 𝒐𝒇 𝒐𝒓𝒅𝒆𝒓 𝟑 𝒂𝒏𝒅 |𝑨 | = 𝟒, 𝒕𝒉𝒆𝒏 |𝒂𝒅𝒋 𝑨 | =

(a) 4 (b) 9 (c) 64 (d)16
𝟏−𝒄𝒐𝒔𝟐𝒙
,𝒙 ≠ 𝟎
Q4. The value of 'k' for which the function 𝒇(𝒙) = { 𝟐𝒙𝟐 is continuous at x=0
𝒌 ,𝒙 = 𝟎
is
(a) 0 (b)-1 (c) 1. (d) 2
⃗⃗ 𝒐𝒏 𝒕𝒉𝒆 𝒗𝒆𝒄𝒕𝒐𝒓 𝟐𝒊⃗ + 𝟔𝒋⃗ + 𝟑𝒌

Q5. The projection of the vector 𝟕𝒊⃗ + 𝒋⃗ − 𝟒𝒌 ⃗⃗ 𝒊𝒔
𝟕 𝟕 𝟖 𝟕
(a) (𝒃) (𝒄) 𝟕 (𝒅) 𝟐
√𝟏𝟒 𝟏𝟒
Q6. If m and n, respectively, are the order and the degree of the differential equation
𝒅𝒙 𝒅𝟐 𝒚
𝒚 = 𝒙(𝒅𝒚)𝟑 +𝒅𝒙𝟐 , then m + n=
(a) 5 (b) 3 (c) 2 (d) 4
Q7. The solution set of the inequality 3x+5y <4 is
(a) an open half-plane not containing the origin.
(b) an open half-plane containing the origin.
(c) the whole XY-plane not containing the line 3x+5y=4.
(d) a closed half plane containing the origin.
Q8. The direction cosine of Z-axis is
𝟏 𝟏 𝟏
(a) 1, 1, 1 (b) , , (c) 0, 0, 0 (d) 0, 0, 1
√𝟑 √𝟑 √𝟑
𝟐 𝒙𝟐
Q9. The value of ∫𝟏 𝒅𝒙 is
𝒙𝟑 +𝟏
𝟏 𝟗 𝟏 𝟐 𝟐
(a) 𝟑 𝒍𝒐𝒈 𝟐 (𝒃) 𝒍𝒐𝒈 (𝒄)𝒍𝒐𝒈 (𝒅)𝟎
𝟑 𝟗 𝟖
Q10. The corner points of the shaded unbounded feasible region of an LPP are (0, 4).
(0.6, 1.6) and (3, 0) as shown in the figure. The minimum value of the objective function
Z = 4x + 6y occurs at
(a) (0.6, 1.6) only (b) (3.0) only (c) (0.6, 1.6) and (3,0) (d) at every point of
the line-segment joining the points (0.6, 1.6) and (3,0) only
Q11. If A and B are nonsingular square matrices of the same order, then 𝑨𝑩𝑻 −
𝑩𝑨𝑻 𝒊𝒔 𝒂
(a) skew-symmetric matrix (b) symmetric matrix (c) 𝒏𝒖𝒍𝒍 𝒎𝒂𝒕𝒓𝒊𝒙 (d)
Nonsingular matrix
⃗⃗ 𝒂𝒏𝒅 𝟑𝒊⃗ − 𝟐𝒋⃗ + ⃗𝒌⃗ 𝒊𝒔

Q12. The area of triangle determined by the vectors 𝒊⃗ + 𝟐𝒋⃗ + 𝟑𝒌
(a) 4√𝟑 (b) 8√𝟑 (c) 36 (d) 18
𝟎 𝟐𝒙 − 𝟏 √𝒙
Q13. Find |𝑨| if A=[𝟏 − 𝟐𝒙 𝟎 𝟐√ 𝒙 ]
− √𝒙 −𝟐√𝒙 𝟎
(a) 4 (b) 1 (c) – 1 (d) 0
Q 14. A and B are the two independent events such that P(A)=0.3 P(B)=0.6 and
P(𝑨′ ∩ 𝑩′ ) is
(a) 0.9 (b) 0.18 (c) 0.28 (d) 0.1
Q15. The general solution of differential equation 𝒚𝒅𝒙 − 𝒙𝒅𝒚 = 𝟎 is (𝒂) 𝒙𝒚 =
𝟐
𝒄 (𝒃) 𝒙 = 𝒄𝒚 (𝒄) 𝒚 = 𝒄𝒙 (𝒅)𝒚 = 𝒄𝒙𝟐 .
Q16. 𝑻𝒉𝒆 𝒗𝒂𝒍𝒖𝒆 𝒐𝒇 |𝒂 ⃗⃗ − ⃗𝒃⃗ | 𝒊𝒇 |⃗𝒂⃗| = 𝟐, |𝒃
⃗⃗| = 𝟓 𝒂𝒏𝒅 ⃗𝒂⃗. ⃗𝒃⃗ = 𝟖
(a) 0 (b) 49 (c)√𝟏𝟎 (d) √𝟏𝟑
𝒅 𝒙
Q17. (𝒙 ) 𝒊𝒔 𝒆𝒒𝒖𝒂𝒍 𝒕𝒐
𝒅𝒙
(a) 𝒙𝒙−𝟏 (b) 𝒙𝒍𝒐𝒈𝒙 (c) 𝒙𝒙 (𝟏 + 𝒍𝒐𝒈𝒙) (d) x𝒙𝒙−𝟏
𝟏 𝟎
Q18. 𝑰𝒇 [𝒙 𝟏] [ ] = 𝑶, 𝒕𝒉𝒆𝒏 𝒙 =
−𝟐 𝟎
(a) 0 (b) 1 (c) 2 (d) -2

In the following questions, a statement of assertion (A) is followed by a statement of
Reason (R). Choose the correct answer out of the following choices.
Q19. Assertion (A): Let f : R→ 𝑹 𝒈𝒊𝒗𝒆𝒏 𝒃𝒚 𝒇(𝒙) = 𝒙, 𝒕𝒉𝒆𝒏 𝒇 𝒊𝒔 𝒐𝒏𝒆 − 𝒐𝒏𝒆 𝒇𝒖𝒏𝒄𝒕𝒊𝒐𝒏.
Reason(R): A function g : A→ 𝑩 𝒊𝒔 𝒔𝒂𝒊𝒅 𝒕𝒐 𝒃𝒆 𝒐𝒏𝒕𝒐 𝒊𝒇 𝒇𝒐𝒓 𝒆𝒂𝒄𝒉 𝒃 ∈ 𝑩, ∃ 𝒂 ∈
𝑨 𝒔𝒖𝒄𝒉 𝒕𝒉𝒂𝒕
g(a)=b.
Q20. Assertion (A): | 𝐬𝐢𝐧 𝒙 | 𝒊𝒔 𝒂 𝒄𝒐𝒏𝒕𝒊𝒏𝒖𝒐𝒖𝒔 𝒇𝒖𝒏𝒄𝒕𝒊𝒐𝒏
Reason(R): If f(x) and g(x) both are continuous functions,
then gof(x) is also a continuous function.
SECTION B
(This section comprises of very short answer type-questions (VSA) of 2 marks each)
𝟓
Q21.Find the value of x, when 𝐜𝐨𝐬 −𝟏 (𝟏𝟑) = 𝐭𝐚𝐧−𝟏 (𝒙)
OR
𝐧+𝟏
, 𝐢𝐟 𝐧 𝐢𝐬 𝐨𝐝𝐝
𝟐
𝐒𝐡𝐨𝐰 𝐭𝐡𝐚𝐭 𝐟: 𝐍 → 𝐍 𝐝𝐞𝐟𝐢𝐧𝐞𝐝 𝐛𝐲 𝐟(𝐧) = {𝐧 𝐢𝐬 𝐧𝐨𝐭
, 𝐢𝐟 𝐧 𝐢𝐬 𝐞𝐯𝐞𝐧
𝟐
𝐨𝐧𝐞
− 𝐨𝐧𝐞 , 𝐛𝐮𝐭 𝐨𝐧𝐭𝐨 𝐟𝐮𝐧𝐜𝐭𝐢𝐨𝐧.
Q22. A particle moves along the curve 𝟔𝒚 = 𝒙𝟑 + 𝟐. Find the point at which the y
coordinate is changing 8 times as fast as the 𝒙 𝒄𝒐𝒐𝒓𝒅𝒊𝒏𝒂𝒕𝒆.
Q23. Find the maximum value of slope of the curve 𝒚 = −𝒙𝟑 + 𝟑𝒙𝟐 + 𝟏𝟐𝒙 − 𝟓.
OR
Find the maximum profit that a company can make, if profit function of the company
is given by 𝑷(𝒙) = 𝟕𝟐 + 𝟒𝟐𝒙 − 𝒙𝟐 𝒊𝒇 𝒙 𝒊𝒔 𝒕𝒉𝒆 𝒏𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒖𝒏𝒊𝒕𝒔 𝒂𝒏𝒅 𝑷 𝒊𝒔 𝒕𝒉𝒆 𝒑𝒓𝒐𝒇𝒊𝒕
𝒅𝒙
Q24. Find: ∫
√𝟕−𝟔𝒙−𝒙𝟐
Q25. Check whether the function 𝒇 ∶ 𝑹 → 𝑹 𝒅𝒆𝒇𝒊𝒏𝒆𝒅 𝒃𝒚 𝒇(𝒙) = 𝒙𝟑 + 𝒙 has critical
point(s) or not. If yes, find them.
SECTION C
(This section comprises of short answer type questions (SA) of 3 marks each)
𝟐𝒙
Q26. Find: ∫ (𝒙𝟐 +𝟏)(𝒙𝟐+𝟐) 𝒅𝒙
Q27.Probabilities of solving a specific problem independently by A and B are ½ and 1/3

respectively. If both try to solve the problem independently, find the probability that
(i)the problem is solved
(ii) Exactly one of them solve the problem.
𝝅
Q28. Evaluate: ∫𝟎𝟐 𝒍𝒐𝒈𝒕𝒂𝒏𝒙𝒅𝒙
Q29. Solve the differential equation: 𝒚𝒅𝒙 + (𝒙 − 𝒚𝟑 )𝒅𝒚 = 𝟎

OR
( 𝟐 𝟐)
Solve the differential equation: 𝒙 − 𝒚 𝒅𝒙 + 𝟐𝒙𝒚 𝒅𝒚 = 𝟎
Q30. Solve the following Linear Programming Problem graphically:

Maximize Z = 400x + 300y subject to
𝒙 + 𝒚 ≤ 𝟐𝟎𝟎, 𝒙 ≤ 𝟒𝟎, 𝒙 ≥ 𝟐𝟎, 𝒚 ≥𝟎
𝒅𝟐 𝒚 𝒅𝒚
Q31 𝑰𝒇 𝒚 = 𝟑 𝐜𝐨𝐬(𝒍𝒐𝒈𝒙) + 𝟒 𝐬𝐢𝐧(𝒍𝒐𝒈𝒙) , 𝒔𝒉𝒐𝒘 𝒕𝒉𝒂𝒕: 𝒙𝟐 𝒅𝒙𝟐 + 𝒙 𝒅𝒙 + 𝒚 = 𝟎
SECTION D
(This section comprises of long answer-type questions (LA) of 5 marks each)
Q32. Make a rough sketch of the region in the first quadrant enclosed by x-axis, the line
𝒙 = √𝟑𝒚 and the circle 𝒙𝟐 + 𝒚𝟐 = 𝟒 and find its area using integration.
Q33. 𝑫𝒆𝒇𝒊𝒏𝒆𝒅 𝒂 𝒓𝒆𝒍𝒂𝒕𝒊𝒐𝒏 𝑹 𝒊𝒏 𝒔𝒆𝒕 𝑵 × 𝑵 𝒂𝒔 𝒇𝒐𝒍𝒍𝒐𝒘𝒔;

(𝒂, 𝒃), (𝒄, 𝒅) ∈ 𝑵 × 𝑵 𝒔𝒖𝒄𝒉 𝒕𝒉𝒂𝒕 (𝒂, 𝒃) 𝑹 (𝒄, 𝒅) 𝒊𝒇𝒇 𝒂𝒅 = 𝒃𝒄.
𝑷𝒓𝒐𝒗𝒆 𝒕𝒉𝒂𝒕 𝑹 𝒊𝒔 𝒂𝒏 𝒆𝒒𝒖𝒊𝒗𝒂𝒍𝒆𝒏𝒄𝒆 𝒓𝒆𝒍𝒂𝒕𝒊𝒐𝒏.

OR
𝒙−𝟏
𝑳𝒆𝒕 𝑨 = 𝑹 − {𝟐}, 𝐁 = 𝐑 − {𝟏}. 𝑰𝒇 𝒇: 𝑨 → 𝑩 𝒊𝒔 𝒂 𝒇𝒖𝒏𝒄𝒕𝒊𝒐𝒏 𝒅𝒆𝒇𝒊𝒏𝒆𝒅 𝒂𝒔 𝒇(𝒙) = .
𝒙−𝟐
𝑺𝒉𝒐𝒘 𝒕𝒉𝒂𝒕 𝒇 𝒊𝒔 𝒃𝒊𝒋𝒆𝒄𝒕𝒊𝒗𝒆.
Q34.An insect is crawling along the line ⃗𝒓⃗ = 𝝀(𝒊⃗ − 𝒋⃗ + ⃗𝒌⃗) and another insect is crawling
along the line 𝒓 ⃗⃗ = 𝒊⃗ − 𝒋 + 𝝁(−𝟐 𝒋⃗ + ⃗𝒌⃗). 𝑨𝒕 𝒘𝒉𝒂𝒕 𝒑𝒐𝒊𝒏𝒕𝒔 𝒐𝒏 𝒕𝒉𝒆 𝒍𝒊𝒏𝒆𝒔
𝒔𝒉𝒐𝒖𝒍𝒅 𝒕𝒉𝒆𝒚 𝒓𝒆𝒂𝒄𝒉 𝒔𝒐 𝒕𝒉𝒂𝒕 𝒕𝒉𝒆 𝒅𝒊𝒔𝒕𝒂𝒏𝒄𝒆 𝒃𝒆𝒕𝒘𝒆𝒆𝒏 𝒕𝒉𝒆𝒎 𝒊𝒔 𝒔𝒉𝒐𝒓𝒕𝒆𝒔𝒕 ? 𝑭𝒊𝒏𝒅 𝒕𝒉𝒆
𝒔𝒉𝒐𝒓𝒕𝒆𝒔𝒕 𝒑𝒐𝒔𝒔𝒊𝒃𝒍𝒆 𝒅𝒊𝒔𝒕𝒂𝒏𝒄𝒆 𝒃𝒆𝒕𝒘𝒆𝒆𝒏 𝒕𝒉𝒆𝒎.
OR
Find the coordinate of the image of the point (1, 6, 3) with respect to the line
̂ ) + 𝝀(𝒊̂ + 𝟐𝒋̂ + 𝟑𝒌
⃗𝒓⃗ = (𝒋̂ + 𝟐𝒌 ̂ ).
𝟏 −𝟏 𝟏
Q35. If A = [𝟐 𝟏 −𝟑], find A– 1. Using A– 1, solve the following system of equations:
𝟏 𝟏 𝟏
𝐱 + 𝟐𝐲 + 𝐳 = 𝟒, − 𝐱 + 𝐲 + 𝐳 = 𝟎, 𝐱 − 𝟑𝐲 + 𝐳 = 𝟐
SECTION E
(This section comprises of 3 case-study/passage-based questions of 4 marks each with
two sub-parts. First two case study questions have three sub-parts (i), (ii), (iii) of marks
1, 1, 2 respectively. The third case study question has two sub-parts of 2 marks each.)
Q36. Case-Study 1: Read the following passage and answer the following questions
given below:
Teams A, B, C went for playing a tug of war game. Teams A, B, C have attached
a rope tt a metal ring and is trying to pull the ring into their own area.
Team A pulls with force 𝑭𝟏 = 𝟔𝒊̂ + 𝟎𝒋̂ kN
Team B pulls with force 𝑭𝟐 = −𝟒𝒊̂ + 𝟒𝒋̂ kN
Team C pulls with force 𝑭𝟑 = −𝟑𝒊̂ − 𝟑𝒋̂ kN
(i) What is the magnitude of the force of team A?

(ii) Which team will win the game?
(iii) Find the resultant force exerted by the teams?
OR
(iii) In which direction is ring getting pulled?
Q37. Case study II:
45cm
24cm
A packaging company got the orders to make open boxes of maximum volume from
rectangular sheets of dimensions 45cm × 24cm. The execution department of company
suggested to cut squares of equal side from all corners of rectangular sheet and folding
up the flaps as shown.
If square of side x cm is cut from each corner, then answer the following:
(i) Volume of the box is

(ii) The side of the square for which volume is maximum is
(iii)Maximum volume of the box is
OR
(iii) Area of base of box is
Q38. Case-Study 3: Read the following passage and answer the questions given below.
Recent study suggest that roughly 12% of the word population is left handed.
Depending upon the parents , the chances of having a left handed childas follows:
A : When both father and mother are left handed :
Chanceges of left handed child is 24%
B : When father is right handed and mother is left handed :
Chances of left handed child is 22%
C : When father is left handed and mother is right handed :
Chanceges of left handed child is 17%.
D : When both father and mother are right handed :
Chanceges of left handed child is 9%
𝟏
Assuming that P(A)=P(B)=P(C)=P(D)=𝟒 and L denotes the event that child is left
handed
𝑳
(i) Find P(𝑪)
𝑳̅
(ii) Find 𝑷(𝑨).
𝑨
(iii) Find 𝑷 ( ).
𝑳
OR
(iii) Find the probality that a rendomly selected child is left handed given that exactly
one of the parent is left handed.
0 SQP 2023-24 Set-II
MATHEMATICS
CLASS XII
MARKING SCHEME
1.d 1
2.b 1
3.d 1
4.c 1
5.c 1
6.b 1
7.b 1
8.d 1
9.a 1
10.d 1
11.a 1
12.a 1
13.d 1
14.c 1
15.c 1
16.d 1
17.c 1
18.c 1
19.b 1
20.a 1
21.
𝟏𝟐 1
𝐭𝐚𝐧−𝟏 ( ) = 𝐭𝐚𝐧−𝟏 (𝒙)
𝟓
𝟏𝟐 1
𝐱=
𝟓 OR
𝐎𝐑
one-one 1
Not onto 1
𝑑𝑦 𝑑𝑥 1
22. 6 = 3𝑥 2
𝑑𝑡 𝑑𝑡 2
𝑑𝑦 𝑑𝑥 1
=8
𝑑𝑡 𝑑𝑡
1
−31 2
Finding (4,11) and(-4, )_
3
𝑑𝑦 1
23. Slope S= = −3𝑥 2 + 6𝑥 + 12 2
𝑑𝑥 1
𝒅𝒔
For Maximum slope =𝟎 2
𝒅𝒙 1
−𝟔𝒙 + 𝟔 = 𝟎, 𝒙 = 𝟏 2
1
∴ 𝑺𝑴𝒂𝒙 = −𝟑 + 𝟔 + 𝟏𝟐 = 𝟏𝟓
2
OR
1
𝑷, (𝒙) = 𝟒𝟐 − 𝟐𝒙 2
1
For Max P, 𝑷, (𝒙) = 𝟎
2
Finding 𝒙 = 𝟐𝟏 1
∴ 𝑴𝒂𝒙 𝑷(𝟐𝟏) =513 2
1
2
𝒅𝒙
24. finding 𝐼 = ∫
√𝟒𝟐 −(𝒙+𝟑)𝟐 1
𝒙+𝟑
𝑰= 𝐬𝐢𝐧−𝟏 ( 𝟒 ) +𝒄 1
25. 𝑓 , (𝑥 ) = 3𝑥 2 + 1 1/2
1
as 𝑥 2 ≥ 0 ⟹ 𝑓 , (𝑥 ) ≥ 0, f is monotonic
1/2
No critical point
26. substitution 𝑥 2 = 𝑡 2𝑥𝑑𝑥 = 𝑑𝑡 1

1 1
Using partial fraction and doing 𝐼 = ∫( − )𝑑𝑡 1
𝑡+1 𝑡+2
I = log(𝑥 2 + 1) − log(𝑥 2 + 2) + 𝐶 1
27. A and B are independent P(A∩ 𝐵)=P(A)P(B) 1

1
(i) P(A∪ 𝐵)=P(A)+P(B)-P(A)P(B) 2
1 1 1 2
=2+ 3−6=3
1 2 1 1 1
(ii) P(A)P(𝐵̅) + P(B)P(𝐴̅) = x + 𝑋 = 1
2 3 3 2 2
1
2
𝜋
28. 2𝐼 = ∫0 (𝑙𝑜𝑔 𝑡𝑎𝑛𝑥 + log 𝑐𝑜𝑡𝑥 )𝑑𝑥
2
2
𝜋
1
=∫0 log 1𝑑𝑥 = 0
2
𝑑𝑥 𝑥 1
29 + = 𝑦2
𝑑𝑦 𝑦
1
∫𝑦 𝑑𝑦=𝑦
IF=𝑒 1
𝑦4 1
Solving 𝑥𝑦 = +𝐶
4
OR
1
𝑑𝑦 𝑦 2 − 𝑥 2
= 1
𝑑𝑥 2𝑥𝑦
2𝑣 𝑑𝑥
∫ 𝑣 2 +1 𝑑𝑣 = − ∫ 𝑥 1
𝑦2
𝑥( + 1) = 𝐶
𝑥2
30.
𝑑𝑦 sin(𝑙𝑜𝑔𝑥) cos(𝑙𝑜𝑔𝑥) 1
31 = −3 +4
𝑑𝑥 𝑥 𝑥
𝑑𝑦 1
𝑥 = 4 cos(𝑙𝑜𝑔𝑥 ) − 3sin(𝑙𝑜𝑔𝑥)
𝑑𝑥
𝑑2 𝑦 𝑑𝑦 sin(𝑙𝑜𝑔𝑥) cos(𝑙𝑜𝑔𝑥) 𝑦 1
𝑥 + = −4 −3 =−
𝑑𝑥 2 𝑑𝑥 𝑥 𝑥 𝑥
32 fig. 1
Finding point of intersection of line and
1
circle(√3, 1)
√3 2 1
∆= + ∫√3 √4 − 𝑥 2 𝑑𝑥
2
√3 𝜋 √3 2
= + −
2 3 2
33.
1
1
1
2
1
2
2
2
1
OR
Proving one-one
Proving onto
Hence f is bijective
34. If PQ is the shortest distance between the given lines, then any point 1/2
P and Q on the lines are P(𝜆, −𝜆, 𝜆) and Q(1,-2𝜇 − 1, 𝜇).
Dr’s of PQ (𝜆 − 1, −𝜆 + 2𝜇 + 1, 𝜆 − 𝜇) 1/2
Since PQ⊥ 𝑡𝑜 𝑡ℎ𝑒 𝑔𝑖𝑣𝑒𝑛 𝑙𝑖𝑛𝑒𝑠
(𝜆 − 1).1 +(−𝜆 + 2𝜇 + 1).(-1)+ (𝜆 − 𝜇). 1 = 0 ⇒3 λ − 3μ = 2
2 1
Similarly, 3 λ − 5μ = 2 , Solving 𝜆 = 3 , 𝜇 = 0
1
2 2 2
∴ 𝑃(3 , − 3 , 3) and Q(1,-1,0) 1
1
2
Shortest Distance PQ=√3
OR
Any point on line L(𝜆, 2𝜆 + 1,3𝜆 + 2) 1
Dr of line P(1,6,3)L (𝜆 − 1,2𝜆 − 5,3𝜆 − 1) dr of given line (1,2,3) 1
PL⊥ 𝑡ℎ𝑒 𝑙𝑖𝑛𝑒 1
Finding 𝜆 = 1
Finding L(1,3,5) 1
Finding image(1,0,7) 1
35. |𝐴| = 10 ≠ 0 1
𝟒 𝟐 𝟐
𝑎𝑑𝑗𝐴 = [−𝟓 𝟎 𝟓] 𝒂𝒏𝒅 𝒇𝒊𝒏𝒅𝒊𝒏𝒈 𝑨−𝟏 2
𝟏 −𝟐 𝟑
−𝟏
𝑵𝒐𝒘 𝑺𝒚𝒔𝒕𝒆𝒎 𝒐𝒇 𝒆𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝑨𝑻 𝑿 = 𝑩, 𝑿 = (𝑨𝑻 ) 𝑩 1
𝟗 𝟐 𝟕
𝑭𝒊𝒏𝒅𝒊𝒏𝒈 𝒙 = , 𝒚 = , 𝒛 =
𝟓 𝟓 𝟓 1
36. (i)|𝐹 ⃗⃗⃗1 | =6kN 1
1
(ii)Team A 2
(iii)𝐹 = ⃗⃗⃗ ⃗⃗⃗⃗2 + 𝐹
𝐹1 + 𝐹 ⃗⃗⃗⃗3 = −𝑖̂ + 𝑗̂ ∴ |𝐹 | = √2𝑘𝑁
OR
1 𝜋 3𝜋
(iii) 𝜃 = 𝜋 − tan−1 ( )=𝜋 − =
1 4 4
37. (i) 𝑥(45 − 2𝑥)(24 − 2𝑥) (ii)5cm (iii)2450cm3 OR 1
(iii)490cm2 1,2
𝐿 17 1
38. (i) P( )=
𝐶 100
𝐿̅ 24 76
(ii) 𝑃 ( ) = 1 − = 1
𝐴 100 100
𝐿
𝐴 𝑃(𝐴).𝑃( )
𝐴
(iii) 𝑃 ( 𝐿 ) = 𝐿 𝐿 𝐿 𝐿
𝑃(𝐴).𝑃( )+𝑃(𝐴).𝑃( )+𝑃(𝐴).𝑃( )+𝑃(𝐴).𝑃( )
𝐴 𝐵 𝐶 𝐷
1 24
× 1
= 4 100
1 24 1 22 1 17 1 9 =3 2
× + × + × + ×
4 100 4 100 4 100 4 100
OR
𝐴 22 17 39
𝑃 ( 𝑜𝑟 𝐶) = + =
𝐿 100 100 100
KVS/DR/2023/VS
KENDRIYA VIDYALAYA SANGATHAN, DELHI REGION
PRE BOARD EXAMINATION- 2023 – 24
Class- XII Subject- Mathematics
M.M. – 80 Time- 3hours
6. Section E has 3 source based/case based/passage based/integrated units of assessment of 4 marks
each with sub-parts.
Section –A
(Multiple Choice Questions)
0 1  2

1. The value of x for which matrix A =  1 0 3  is a skew symmetric matrix:

 x  3 0 
a. -2 c. 3
b. 2 d. -3
2. If ⃗ = 7 ̂ ̂ ̂ and ⃗⃗ = 2 ̂ ̂ ̂ , then projection of ⃗ on ⃗⃗ is:

a. 8/7 c. 3/4
b. 7/8 d. 4/3
3. If = -2 , the principal value of is
a. c.
d.
b.
4. Integrating factor of + 2y tan x = sin x is:
a. Tan x c. Sec2x
b. Sec x d. Tan2x
5. The function f: N → N defined by f(x) = 3x , x ∈ N is:
a. One-one function c. Bijective function
b. Ono function d. None of these
1
6. Point on the curve y2 = 8x + 8 for which the abscissa and ordinate change at the same rate:
a. (1,2) c. (4,1)
b. (2,1) d. (1,4)
7. If A = [ ], then the value of | |is:
a. a27 c. a6
b. a9 d. a2
8. Function given by f(x) = | | | |+ | | : is not differentiable at
a. 1 point c. 3 points
b. points d. 4 points
9. The volume of a cube is increasing at the rate of 8 cm3/sec. The rate of increase of its surface area
when the length of an edge is 12 cm:
a. 2/3 Cm2/sec. c. 5/3 Cm2/sec.
b. 4/3 Cm2/sec. d. 8/3 Cm2/sec.
10. Vector form of the line is:
a. ⃗⃗⃗ = ̂ ̂ ̂ ̂ ̂ ̂ c. ⃗⃗⃗ = ̂ ̂ ̂ ̂ ̂ ̂
b. ⃗⃗⃗ = ̂ ̂ ̂ ̂ ̂ ̂ d. ⃗⃗⃗ = ̂ ̂ ̂ ̂ ̂ ̂
11. ∫ dx is:
a. Sin (2x + 3) +C c. Sin (2x + 3) +C

d. - Sin (2x + 3) +C
b. - Sin (2x + 3) +C
12. Let R be a relation defined on Z as follows : (x, y) R | |≤ 1, then R is:

a. Reflexive and transitive. c. Symmetric and transitive.
b. Reflexive and symmetric. d. Equivalence relation.
13. ∫ is equal to:
a. 0 c. 1/6
b. 1 d. 1/2
14. Order and degree of differential equation 2x2 ( ) – 3( ) + y = 0are:
a. 2,3 c. 1,4
b. 3,2 d. 1,2
dy
15. If x = at2 and y = 2at , then is :
dx
a. 1/t c. -2/t
b. -1/t2 d. t
2
16. If P (A) = 0.8, P (B) = 0.5 and P (B|A) = 0.4, what is the value of P (A ∩ B)?
a. 0.32 c. 0.1
b. 0.25 d. 0.5
17. Value of is:
a. c.
b. d.
18. An urn contains 10 black and 5 white balls. Two balls are drawn from the urn one after the other
without replacement. What is the probability that both drawn balls are black?
a. 2/7 c. 1/7
b. 3/7 d. 1/3
Assertion reason based questions
Read both the statement carefully and choose the correct alternative from the following:
a. Both the statement A and R are correct and R is the correct explanation of A.
b. Both the statement A and R are correct and R is not the correct explanation of A.
c. The statement A is correct but R is false.
d. The statement A is false but R is correct.
19. Assertion(A): Consider the two events E and F which are associated with the sample space of a
random experiment. P(E/F) =
Reason(R): P(E/F) =
20. Assertion(A) : If the derivative function of x is = 1, then its anti-derivatives or integral is
∫ =x+c
Reason(R): If ( ) = xn , then the corresponding integral of the function is
∫ = + C, n ≠ -1.
Section: B
(Very short questions)
Each question carries 2 marks
21. Using integration find area of shaded portion.
3
22. Find ∫
OR
Find ∫ √ dx
23. Show that the function f(x) = x3 – 3x2 + 4x, x R is increasing on R.
24. A card is drawn at random from a pack of 52 cards and it is found to be a king card. Find the
probability that the card drawn is a black card.
OR
A box of oranges is inspected by examining three randomly selected oranges drawn without
replacement. If all the selected oranges are good, then box is approved for sale otherwise it is
rejected. Find the probability that a box containing 15 oranges, out of which 12 are good and 3
are bad ones, will be approved for sale.
25. Find derivative of xx with respect to x.
Section: C
(Short questions)
 4(1  1  x )
 , if x  0
 x
26. Show that the function f(x) defined by f(x) =  2, if x  0 is continuous at x = 0
 sin x
  cos x, if x  0
 x
OR
dy log x
If xy = e x – y ,Show that 
dx log( xe) 
2
27. Evaluate ∫ dx.
OR
Evaluate ∫ dx.
28. Find x, if [ x -5 -1][ ][ ]=O
29. Using integration find area of the region bounded by the curve y2 = 4x, the X- axis and the
line x = 3.
OR
Using integration find area of the region bounded by the line y – 1 = x , the X- axis and the
ordinates x = -2 and x = 3.
4
30. Find the value of λ and μ if ( ̂ ̂ ̂) (̂ ̂ ̂) ⃗⃗
31. In answering a question on a multiple choice test, a student either knows the answer or guesses.
Let 3/4 be the probability that he knows the answer and 1/4 be the probability that he guesses.
Assuming that a student who guesses at the answer will be correct with the probability 1/4.
What is the probability that the student knows the answer given that he answered it correctly?
Section: D
(Long questions)
1  1 1 
32. If A = 2 1  3 , find A 1 and hence solve
 
1 1 1 
x + 2y + z = 4,
- x + y + z = 0,
x – 3y + z = 2.
33. Find the shortest distance between the lines: ⃗⃗⃗ = ̂ ̂ ̂ and
⃗⃗⃗ = ̂ ̂ ̂.
OR
Find the shortest distance between the lines and
34. Solve the following linear programming problem graphically:

Minimise and maximize Z = 5x + 10y subject to x + 2y ≤ 120, x + y ≥ 60, x – 2y ≥ 0, x, y ≥ 0.
Also show that maximum of Z occurs at two points.
35. Prove that the surface area of a solid cuboid, of square base and given volume, is minimum when
it is a cube.
OR
Show that the rectangle of maximum area that can be inscribed in a circle is a square.
Section: E[4MARKS]
Case study / Source based/ integrated

36. An organization conducted bike race under 2 different categories-boys and girls. Totally there
were 250 participants. Among all of them finally three from Category 1 and two from Category 2
were selected for the final race. Ravi forms two sets B and G with these participants for his
college project.
Let B = {b1, b2, b3} G={g1,g2} where B represents the set of boys selected and G the set of girls who
were selected for the final race.
Ravi decides to explore these sets for various types of relations and functions
5
Base on above information, answer the following questions.
A. Ravi wishes to form all the relations possible from B to G. How many such relations are
possible? (1)
a. 26 c. 23
b. 25 d. 0
B. Let R: B→B be defined by R = {(𝑥, 𝑦): 𝑥 and y are students of same sex}, Then this relation R
is_______ (1)
a. Equivalence relation
b. Reflexive relation only
c. Reflexive and symmetric but not transitive
d. Reflexive and transitive but not symmetric
C. Ravi wants to know among those relations, how many functions can be formed from B to G? (2)
a. 22 c. 32
b. 212 d. 23
OR
Let 𝑅: 𝐵 → 𝐺 be defined by R = { (b1,g1), (b2,g2),(b3,g1)}, then R is__________ (2)
a. One one function c. Bijective function
b. Onto function d. Neither one one nor onto function
37. A class XII student appearing for a competitive examination was asked to attempt the following
questions.
Let ⃗, ⃗⃗ and ⃗ 𝑏𝑒 𝑡ℎ𝑟𝑒𝑒 non zero vectors.

A. If ⃗, and ⃗⃗ are such that | ⃗ ⃗⃗| = | ⃗ ⃗⃗| , then (2)
a. ⃗ ⃗⃗ c. ⃗ = ⃗⃗
b. ⃗ ⃗⃗ d. None of these.
B If ⃗, and ⃗⃗ are unit vectors and θ be angle between them, then | ⃗ ⃗⃗| is: (2)
a. Sin c. 2Cos
b. 2Sin d. Cos
38. Polio drops are delivered to 50K children in a district. The rate at which polio drops are given is
directly proportional to the number of children who have not been administered the drops. By the
end of 2nd week half the children have been given the polio drops. How many will have been given
the drops by the end of 3rd week can be estimated using the solution to the differential equation
= 𝐤(𝟓𝟎 − 𝐲) where x denotes the number of weeks and y the number of children who have been
6
given the drops.
A. Order of the above given differential equation. (1)
a. 0 c. 2
b. 1 d. 3
B Degree of the above given differential equation. (1)

a. 0 c. 2
b. 1 d. 3
C, Which method of solving a differential equation can be used to solve = 𝐤(𝟓𝟎 − 𝐲)? (2)
a. Variable separable method
b. Solving Homogeneous differential equation
c. Solving Linear differential equation
d. All of the above
OR
C The solution of the differential equation = 𝐤(𝟓𝟎 − 𝐲) is given by, (2)
a. log | 50 – y| = kx + C
b. - log | 50 – y| = kx + C
c. log | 50 – y| = log| kx |+ C
d. 50 – y = kx + C
**********************************************************************************************************
7
PB23MAT03 QP
KENDRIYA VIDYALAYASANGATHAN ERNAKULAM REGION
PREBOARD EXAMINATION 2023-24
MATHEMATICS
CLASS XII
1. This Question paper contains –Five sections A,B,C,D and E. Each section is
2. Section A has 18 MCQ’s and 02 Assertion –Reason based questions of 1 mark each
3. Section B has 5 very short Answer type questions of 2 marks each
4. Section C has 6 Short Answer type questions of 3 marks each
5. Section D has 4 Long Answer type questions of 5 marks each
6. Section E has 3 source based/ case based/passage based/integrated units of
assessment(4 marks each) with sub parts
SECTION A
(Multiple choice Questions each carries 1 mark)
1. If A is a matrix of order mxn and B is a matrix such that AB1 and B1 A are both
defined, then order of matrix B is
a. mxm b.nxn c. nxm d. mxn
 1 0 0  x   1 
    
2. If  0 y 0   1   2  ,then x+y+z is
 0 0 1  z   1 
    
a.4 b. 3 c. 0 d. -2
3. Which of the following functions from Z toZ is bijective
a. f(x) = x3 b. f(x)=x+2 c. f(x)= 2x+1 d. f(x)= x2 +1
4. If A is a square matrix of order 3,|A1| = -3, then find |AA1|
a. 9 b.-9 c.-3 d.3
5. Three points P(2x,x+3), Q(0,x) and R(x+3,x+6)are collinear, then x is
a. 0 b. 2 c. 3 d. 1
1 1 2 
 
6. If Cij denotes the cofactor of elements Pij of the matrix P=  0 2  3  ,then the
3 2 4 

value of C31C23
a. 5 b. 24 c. -24 d. -5
2
d y
7. If x = t2 and y = t3 , then is equal to
dx 2
3 3 3 3
a. b. c. d. t
2 4t 2t 2
x3
8.  x  1dx is
x2 x3 x2 x3
a. x    log( x  1)  c b. x    log( x  1)  c
2 3 2 3
x2 x3 x2 x3
x   log( x  1)  c x   log( x  1)  c
c. 2 3 d. 2 3
 cos 3x  cos x
 x0
9. If f : R → R defined by f(x)=  x2 is continuous at x=0, then 𝜆 is
  x0
equal to
a. -2 b. -4 c. -6 d. -8
a
1 
10. If  dx  , then a is
0 1  4x
2
8
1 3 1 3
a. b. c. d.
2 5 2 4
11. Which of the following is not homogeneous function of x and y
x y
a.x2 –xy b. 2x-y c. cos 2    d. sinx-siny
 y x
 dy 
12. The solution of log    ax  by is
 dx 
e by e ax e by e ax
a.  c b.  c
b a b a
e by e ax e by e ax
c.  c d.  c
b a b a

13. The value of 𝜆, such that the vectors a  2iˆ  ˆj  kˆ and bˆ  iˆ  2 ˆj  3kˆ are
orthogonal is
3 5
a.0 b. 1 c. d.
2 2
   
14. If a and b are unit vectors and  be the angle between them the | a - b | is
       
. a. sin   b.2 sin   c. cos  d .2 cos 
2 2 2 2
     
15. If a =10, b =2 and a.b =12, find a  b
a.5 b. 10 c. 14 d.16

16. If the line makes an angle with each of y and z axis, then the angle whichit makes
4
with x-axis is
a.0 b. 𝜋 c. 𝜋/2 d. 𝜋/4
17. The corner points of the feasible region determined by a set of linear constraints are
P(0,5), Q(3,5),R(5,0) ,S(4,1) and the objective function Z = ax + 2by where a, b > 0,
the condition on a and b such that the maximum Z occurs at Q and S is
a. a - 5b = 0 b. a - 3b = 0 c. a - 2b = 0 d. a - 8b = 0
  3 
18. The value of tan 1 2 sin  2 cos 1  is

  2 
a.𝜋/3 b. 2𝜋/3 c. −𝜋/3 d. 𝜋/6
ASSERTION- REASON BASED QUESTIONS
In the following questions, a statement of Assertion is followed by a statement of Reason.
Choose the correct answer out of the following choices
a. Both A and R are true and R is the correct explanation of A
b. Both A and R are true and R is not the correct explanation of A
c. A is true but R is false
d. A is false but R is true
19. Assertion: For an objective function Z = 15x + 20y, corner points are (0,0),
(10,0),(0,15) and (5,5). Then optimal values are 300 and 0 respectively.
Reason: The maximum or minimum value of an objective function is known as
optimal value of LPP. These values are obtained at corner points
5
20. Assertion: If A andB are any two events such that 2P(A) = P(B) = and P(A/B) =
13
2 11
then P(AUB) =
5 26
Reason: For any two events A and B,P(AUB) = P(A) +P(B) – P(A ∩B)
SECTION B
(This section comprises of very short answer type questions (VSA) of 2 marks each
2 2
21. Find the value of cos 1 (cos )  sin 1 (cos )
3 3
OR
Show that the function f :R→R defined as f(x) = x2 is neither one one nor onto
x 2 x  1
22. If f(x) =  , then show that f is not differentiable at x = 1
 x x  1
23. The side of an equilateral triangle are increasing at the rate of 2cm/s. find the rate at
which the area increases, when the side is 10cm.
  
24. If r  xiˆ  yˆj  zkˆ , find ( r  iˆ ).( r  ˆj ) + xy
OR
 x  2 y 1 z  3 x  2 2y  8 z  5
Find the angle between   and   and check
2 7 3 1 4 4
whether the lines are parallel or perpendicular
 
25. Find a unit vector perpendicular to each of the vectors a and b where
 
a  5iˆ  6 ˆj  2kˆ and b  7iˆ  6 ˆj  2kˆ
SECTION C
3
dx x
26. Evaluate  3  2x  x 2
OR Evaluate 
1 4 x  x
dx
 cos x  sin x
27. Differentiate tan 1   with respect to x
 cos x  sin x
cos x
28. Find  dx
1  sin x 2  sin x 
dy  y
29. Solve the differential equation x  y  x tan  
dx x
OR
Find the particular solution of the differential equation (1+e 2x)dy + ex(1+y2)dx =0,
given that y(0) =1
30. Solve the following LPP graphically, minimise Z = 5x + 10y subject to constraints
x + 2y ≤ 120, x + y ≥ 60 ,x - 2y ≥ 0 , x,y ≥ 0
31. Let X denote the number of colleges where you will apply after your results and
P(X=x) denote your probability of getting admission in x number of colleges. It is
 kx x  0 and x  1
 2kx x2

given that P( x)   where k is a positive constant. Find the
k (5  x) x  3 and x  4
 0 x4
value of k. Also find the probability that you will get admission in at most 2 colleges
OR
A coin is biased so that the head is three times as likely to occur as tail. If the coin is
tossed twice, find the probability distribution of tails. Hence find the mean of the
number of tails.
SECTION D
(This section comprises of long answer type questions (LA) of 5 marks each)
2  3 5 
 
32. If A =  3 2  4  , find A-1. Use A-1 to solve the following system of equations
 1 1  2
 
2x-3y+5z=11, 3x+2y-4z = -5, x + y - 2z = -3

  

33. Show that the lines r  3iˆ  2 ˆj  4kˆ   iˆ  2 ˆj  2kˆ and r  5iˆ  2 ˆj   3iˆ  2 ˆj  6kˆ 
are intersecting. Hence find the point of intersection.
OR
Find the equation of line passing through the point (2,-1,3) and perpendicular to the

  
 
lines r  iˆ  ˆj  kˆ   2iˆ  2 ˆj  kˆ , r  2iˆ  ˆj  3kˆ   iˆ  2 ˆj  2kˆ
4
34. Sketch the graph of y=|x-1| and using integration evaluate  x  1dx
0
35. Define the relation R in the set NxN as follows

For (a,b),(c,d) NxN, (a,b)R(c,d) iff ad=bc. Prove that R is an equivalence relation in
NxN
OR
Check whether the relation R in the set Z defined as R = {(a,b): a+b is divisible by 2}
is reflexive, symmetric or transitive. Write the equivalence class containing 0
SECTION E
(This section comprises of 3 case study /passage based questions of 4 marks each with
two sub parts. First two case study questions have three sub parts of marks 1,1,2
respectively. The third case study questions has two sub parts of 2 marks each)
36. A person has to reach a company for an interview. He has four options to reach a
company i.e. by metro, by bus, by scooter or by other means of transport. The
probabilities of using these means are 3/10, 1/5,1/10 and 2/5 respectively. The
probabilities that he will be late if he comes by metro, bus or scooter are 1/4, 1/3and
1/12respectively but if he comes by other means of transport he will not be late. Using
above information answer the following questions
a.What is the conditional probability of reaching late by other means of trasports
b.What is the probability that he travelled by bus and was late
c. What is the probability of reaching late?
OR
What is the probability that he reached late to company if he used metro
as mean of transport
37. A company is launching a new product and decided to pack the product in the form of
a closed right circular cylinder of volume 432  ml and having minimum surface area
as shown. They tried different options and tried to get the solution by answering the
questions given below:
a.If r is the radius of base of cylinder and h is the height of cylinder then establish the
relation between r and h
b.Find the total surface area in terms of r only
c. Find the radius r for minimum surface area
OR
Find the minimum surface area
38. To reduce global warming, environmentalists and scientists came up with an
innovative idea of developing a special bulb that would absorb harmful gases and
thereby reduce global warming. But during the process of absorption the bulb would
get inflated and its radius would be increasing at 1cm/sec
a.Find the rate at which the volume increases when the radius is 6 cm
b.At an instant when the volume was increasing at the rate of 400πcm3 /sec, find the
rate at which its surface area is increasing.
*****************
KENDRIYA VIDYALAYA SANGATHAN, GURUGRAM REGION
(SUMMER STATION)
FIRST PRE BOARD EXAMINATION - 2023-24
CLASS: XII Maximum Marks: 80
SUBJECT: MATHEMATICS TIME ALLOWED: 3 Hours
General instruction
1 This question paper contains- five sections A, B, C, D and E. Each section is
2. Section A has 18 MCQ’s and 02 Assertion- Reason based question of 1 mark
each.
3. Section B has 5 very short answer type questions (VSA) of 2 marks each.
4. Section C has 6 short answer type questions (SA) of 3 marks each.
5. Section D has 4 long answer type questions (LA) of 5 marks each.
6. Section E has 3 source base / case based/passage based / integrated units of
aassessment 4 marks each with sub parts.
Q.No. SECTION A Marks
(MULTIPLE CHOICE QUESTIONS)
1 −1 −1 1
If x = √asin t and y = √acos t then
dy dy
(a)x +y=0 (b) x =y
dx dx
dy
(c) y =x (d) none of the above
dx
2 3𝑥 2 1
∫ 6 𝑑𝑥 𝑖𝑠 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜
𝑥 +1
(a) log (x6 +1) +c ( b) tan-1 x3 +c

(c) 3 tan -1x3 + c (d) log x2 +c
3 1 0 0 1
If 𝐴 = [0 1 0] then A2 + 2A is equal to
0 0 1
(a) 4A (b) 3A (c) 2A (d) A

4 A square matrix A = [𝑎𝑖𝑗 ] n × n is called a diagonal matrix 1
if 𝑎𝑖𝑗 = 0 for
(a) i = j (b) i < j (c) i > j (d) i ≠ j
5 The feasible region for LPP is shown shaded in the figure. 1
Let Z = 3 x – 4 y be the objective function, then maximum value
of Z is
(a) 12
(b) 8
(c) 0
(d) –18
6 The area of the feasible region for the following constraints 1
3y + x ≥ 3, x ≥ 0, y ≥ 0 will be
(a) Bounded (b) Unbounded (c) Convex (d) Concave
7 The direction cosines of the line which makes equal angles with 1
the coordinate axes are
1 1 1 1 1 1
(a) ( , , ) (b) (− ,− ,− )
√3 √3 √3 √3 √3 √3
1 1 1
(c) (± ,± ,± ) (d) none of the above
√3 √3 √3
8 √3 ⃗⃗ 1
If |𝑎⃗| = , |𝑏| = 4 and angle between 𝑎⃗⃗⃗⃗ 𝑎𝑛𝑑 𝑏⃗⃗⃗⃗ is 600 then
2
the value of 𝑎 ⃗⃗⃗⃗ is equal to
⃗⃗⃗⃗ . 𝑏
1
(a) √3 (b) (c) −√3 (d) none of the above
3 √
9 Order and degree of differential equation 1

1
2 4
𝑑2𝑦 𝑑𝑦
= [𝑦 + ( ) ]
𝑑𝑥 2 𝑑𝑥
(a) 4 and 2 (b) 1 and 2 (c) 1 and 4 (d) 2 and 4
10 The number of all possible matrices of order 3x3 with each 1

entry -1 or 1 is
(a) 512 (b) 81 (c) 27 (d) 18
𝑥−1 𝑦−3 𝑧+6 𝑥−1 𝑦−3 𝑧+6

11 If the lines = = 𝑎𝑛𝑑 = = are 1
𝑘 1 −2 1 −2 𝑘
perpendicular, then k is equal to
(a) 2 (b) 1 (c) -2 (d) 3
12 𝑑𝑦 1
Integrating factor of differential equation 𝑥 + 2𝑦 = 𝑥 2
𝑑𝑥
1 1
(a) (b) x2 (c) x (d)
𝑥2 𝑥
13 𝑑𝑦 1
The solution of differential equation 2𝑥 − 𝑦 = 3 represents:
𝑑𝑥
(a)straight lines (b) circles (c) parabola (d) ellipse
14 x 2 6 9 1
If | |= | | then x is equal
18 x 4 6
(a) 6 (b) -6 (c) ±6 (d) none of the above

15 The probability of obtaining an even prime number on each
die, when a pair of dice is rolled is 1
1 1 1
(a) 0 (b) (c) (d)
3 12 36
16 A unit vector perpendicular to both the vectors 1
𝑖̂ − 2𝑗̂ + 3𝑘̂ 𝑎𝑛𝑑 𝑖̂ + 2𝑗̂ − 𝑘̂ is
1 1
(a) ±
3
(𝑖̂ + 𝑗̂ + 𝑘̂) (b) ± (−𝑖̂ + 𝑗̂ + 𝑘̂)
√ √3
1 1
(c) ± (𝑖̂ − 𝑗̂ − 𝑘̂) (d) ± (𝑖̂ − 𝑗̂ + 𝑘̂)
√3 √3
17 1 2 1
If 𝐴 = [ ] , then find the value of |2A|
4 2
(a) -6 (b) -24 (c) 12 (d) -12
18 if α is the angle between any two vectors 1

𝑎⃗ 𝑎𝑛𝑑 𝑏⃗⃗ , 𝑡ℎ𝑒𝑛 |𝑎⃗. 𝑏⃗⃗| = |𝑎⃗ × 𝑏⃗⃗| when α is equal to
𝜋 𝜋
(a) 0 (b) (c) (d) 𝜋
4 2
ASSERTION – REASON BASED QUESTIONS
Directions: Each of these questions contains two statements,
Assertion and Reason. Each of these questions also has four
alternative choices, only one of which is the correct answer.
You have to select one of the codes (a), (b), (c) and (d) given
below.
(a) Assertion is correct, reason is correct; reason is a correct
(b) Assertion is correct, reason is correct; reason is not a
correct explanation for assertion
(c) Assertion is correct, reason is incorrect
(d) Assertion is incorrect, reason is correct.
19 Assertion: A relation R = {(a,b) : |a-b|<2} defined on the set 1
A = {1, 2, 3, 4, 5} is reflexive.
Reason : A relation R on the set A is said to be reflexive if
(a,b)𝜖 𝑅 𝑎𝑛𝑑 (𝑏, 𝑐)𝜖𝑅 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑎, 𝑏 𝜖 𝐴.
20 Assertion: The intervals in which f(x) = log sin x , 0 ≤ 𝑥 ≤ 𝜋 is 1

𝜋
Increasing is (0, ).
2
Reason: A function is increasing in (a,b) if f’ (x) > 0 for each
x𝜖(𝑎, 𝑏).
SECTION B
21 Find the value of 2
1 1
cos −1 + 2 sin−1
2 2
OR
7π
tan−1 (tan )
6
22 𝑥 2 +𝑥+1 2
Find ∫ (𝑥+2)(𝑥 2 +1) 𝑑𝑥 .
23 𝑥 + 𝑘, 𝑖𝑓 𝑥 < 3 2
If function 𝑓(𝑥) = { 4 , 𝑥 = 3 is continuous function at
3𝑥 − 5, 𝑥 > 3
x=3, then find the value of k.
24 The volume of the cube is increasing at the rate of 9 cubic 2
centimeters per second. How fast is the surface area increasing
when the length of an edge is 10 centimeters?
OR
Find the maximum profit that a company can make, if the point
function is given by
p(x) = 41-72x-18x2
25 Find the intervals in which the function f given by 2

f(x) = 4x3 -6x2 -72x +30
(a) Strictly increasing (b) strictly decreasing
SECTION C
26 𝑑𝑦 3
Find , 𝑖𝑓 𝑦 𝑥 + 𝑥 𝑦 + 𝑥 𝑥 = 𝑎𝑏
𝑑𝑥
OR
𝑑2 𝑦
If x= a (cost +t sint) and y = a (sint-t cost), find .
𝑑𝑥 2
27 Let a pair of dice be thrown and the random variable X be the 3

sum of the numbers that appear on the two dice. Find the
expectation of X.
OR
A die is thrown twice and the sum of the numbers appearing is
observed to be 6. What is the conditional probability that the
number 4 has appeared at least once?
28 Find a particular solution of the differential equation 3
𝑑𝑦
+ 𝑦𝑐𝑜𝑡𝑥 = 4𝑥 𝑐𝑜𝑠𝑒𝑐𝑥 , 𝑥 ≠ 0, 𝑔𝑖𝑣𝑒𝑛 𝑡ℎ𝑎𝑡
𝑑𝑥 𝜋
𝑦 = 0 𝑤ℎ𝑒𝑛 𝑥 =
2
OR
Find a general solution of the differential equation
ex tan y dx + (1-ex) sec2 y dy =0
29 Evaluate the definite integrals 3

4
∫ [|𝑥 − 1| + |𝑥 − 2| + |𝑥 − 3|] 𝑑𝑥
1
30 Evaluate the integrals 3
𝑥+3
∫ 𝑑𝑥
√5 − 4𝑥 + 𝑥 2
31 Solve the following Linear Programming Problems graphically 3

Maximise Z = 5x + 3y
Subject to 3x + 5y ≤ 15, 5x + 2y ≤ 10, x ≥ 0, y ≥ 0.
SECTION D
32 Find the area of the region bounded by the curve y2 = x and the 5
lines x=1, x=4 and x-axis in the first quadrant.
33 Let A be the set of all the triangle in a plane and R be the

relation defined on R as R = {(T1 , T2 ): T1 is similar to T2}
1. Show that the relation R is an equivalence relation. 4

2. Consider three right angle triangle T1 with sides 3, 4, 5, +1
T2 with sides 5, 12, 13 and T3 with sides 6,8,10. Which
triangle among T1, T2, and T3 are related?
OR
Show that f :R→ {𝑥𝜖𝑅 ∶ −1 < 𝑥 < 1} 3

𝑥
𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑏𝑦 𝑓(𝑥) = , x𝜖𝑅 is one – one and onto +2
|𝑥|1+
function.
34 Two factories decided to award their employee for three

values of (a) adaptable to new situation, (b) careful and alert in
difficult situations and (c) keeping calm in tense situations, at
the rate of ₹ x, ₹ y and ₹ z per person respectively. The first
factory decided to honour respectively 2, 4 and 3 employees
with total prize money of ₹ 29000. The second factory decided
to honour respectively 5, 2 and 3 employees with a total prize
money of ₹ 30500. If three prizes per person together cost
₹ 9500 then
(i) Represents the above situation by a matrix equation 1
and form linear equations using matrix multiplication,
(ii) Solve these equation using matrices. 4
35 By computing the shortest distance determine whether the 5

lines intersect or not. If not then find the shortest distance
between the lines.
𝑥−1 𝑦−2 𝑧−3 𝑥−2 𝑦−4 𝑧−5
= = 𝑎𝑛𝑑 = =
2 3 4 3 4 5
OR
Find the vector equation of the line passing through the point
(1, 2, – 4) and perpendicular to the two lines:
𝑥 − 8 𝑦 + 19 𝑧 − 10 𝑥 − 15 𝑦 − 29 𝑧 − 5
= = 𝑎𝑛𝑑 = =
3 −16 7 3 8 −5
SECTION E
This section comprises of 3 case study questions of 4 marks
having sub parts
36 A doctor is to visit a patient. From the past experience, it is
known that the probabilities that he will come by cab, metro,
bike or by other means of transport are respectively 0.3, 0.2,
0.1 and 0.4. The probabilities that he will be late are 0.25, 0.3,
0.35 and 0.1 if he comes by cab, metro, bike and other means
of transport respectively.
(i)What is the probability that the doctor is late by other 1
means?
(ii)When the doctor arrives late, what is the probability that he 1
comes by metro?
(iii) When the doctor arrives late, what is the probability that 2
he comes by bike or other means?
OR
2
When the doctor arrives late, what is the probability that he
comes by cab or metro?
37
Gitika house is situated at Shalimar Bag at O, going to Aloke’s

house she first travel 8 km in the east, here at point A a
hospital is situated. From the hospital she takes auto and goes
6 km in the north. Here at point B a school is situated. From 1
school she travels by bus to reach Aloke’s house which is 300 of 1
east and 6 km from point B. 2
(i) What is vector distance from Gitika’s house to school?
(ii) What is vector distance from school to Aloke’s house?
(iii) What is vector distance from Gitika’s house to Aloke’s 2
house?
OR
What is the total distance travel by Gitika from her house to
Aloke’s house?
38 A telephone company in a town has 500 subscribers on its list

and collect fixed charges of ₹ 300 per subscriber. The company
proposes to increase the annual subscription and it is believed
that every increase of ₹ 1, one subscriber will discontinue the
service.
2
(i) Based on above information find out
how much amount can be increased
for maximum revenue.
2
(ii) Find out maximum revenue
received by the telephone company.
================================================================
KENDRIYA VIDYALAYA SANGATHAN
GURUGRAM REGION
MARKING SCHEME
(SUMMER STATION)
FIRST PRE BOARD EXAMINATIONS 2023-24
CLASS XII
SUBJECT MATHEMATICS
Q.no. SECTION A Marks
1 dy 1
(a) x + y = 0
dx
2 (b) tan-1 x3 +c 1
3 (b) 3A 1
4 (d) i ≠ j 1
5 (c) 0 1
6 (b) Unbounded 1
7 1 1 1 1
(c) (± , ± , ± )
√3 √3 √3
8 (a) √3 1
9 (d) 2 and 4 1
10 (a) 512 1
11 (c) -2 1
12 (b) x2 1
13 (c) parabola 1
14 (c) ±6 1
1
15 (d) 1
36
1
16 (b) ± (−𝑖̂ + 𝑗̂ + 𝑘̂) 1
√3
17 (b) -24 1
𝜋
18 (b) 1
4
19 (c) Assertion is correct, reason is incorrect 1
20 (a) Assertion is correct, reason is correct; reason is a correct 1
SECTION B
21 π π
cos −1 (cos ) + 2sin−1 (sin )
3 6 1
π π 2π
+2× = 1
3 6 3
OR
π
tan−1 (tan π + ) 1
6
π π π
∈ (− , )
6 2 2 1
22 𝑥 2 +𝑥+1 𝐴 𝐵𝑥+𝑐 ½
= +
(𝑥+2)(𝑥 2 +1) 𝑥+2 𝑥 2 +1
A=3/5 , B= 2/5 and C = 1/5
2 1 1/2
𝑥2 + 𝑥 + 1 3 𝑥+
2
= + 52 5
(𝑥 + 2)(𝑥 + 1) 5(𝑥 + 2) 𝑥 + 1
2
𝑥 +𝑥+1
∫ 𝑑𝑥
(𝑥 + 2)(𝑥 2 + 1)
3 1 1
= log|𝑥 + 2| + 𝑙𝑜𝑔|𝑥 2 + 1| + tan−1 𝑥 + 𝐶
5 5 5 1
23 lim f(x) = lim− f(x) = f(3) ½

x→3+ x→3
f(3) = 4 ½
lim f(x) = lim+ 3x − 5 = 4

x→3+ x→3 ½
lim f(x) = lim− x + k = 3 + k
x→0− x→3
3+k = 4 , k = 1 1/2
24 the volume of a cube with radius “x” is given by V = x3 ans
surface area = 6x2 ½
Hence, the rate of change of volume “V’ with respect to the
𝑑𝑉
time “t” is given by: = 9 cm3/s
𝑑𝑡
𝑑𝑉 𝑑 𝑑𝑥 𝑑𝑥 3
9= = 𝑥 = 3𝑥 2 . , By using the chain rule =
3
1/2
𝑑𝑡 𝑑𝑡 𝑑𝑡 𝑑𝑡 𝑥2
𝑑𝑠 𝑑(6𝑥 2 ) 𝑑𝑥 3 36
= = 12 = 12 × 2 =
𝑑𝑡 𝑑𝑡 𝑑𝑡 𝑥 𝑥
𝑑𝑠 2
At x= 10 , = 3.6 cm /s 1/2
𝑑𝑡
OR ½
p(x) = 41-72x-18x2
P’(x) = -72 -36
P’’ = -36
½
For maximum or minina or critical points p’(x) =0
½
-72 -36x =0 , x=-2
x=-2, p’’ (-2) = -36 <0 ½
hence x= -2 is a point of local maxima,
maximum profit = 113
½
3 2
25 f(x) = 4x -6x -72x +30
𝑓 ′ (𝑥) = 12𝑥 2 − 12𝑥 − 72 ½
For critical points 𝑓 ′ (𝑥) = 0, 1/2
12𝑥 2 − 12𝑥 − 72 = 0
12(x-3)(x+2) = 0
X= 2, x= -3 ½
(−∞, −2) and (3, ∞) the function is increasing 1/2
(-2,3) the function is decreasing
SECTION C
26 𝑦 + 𝑥 𝑦 + 𝑥 𝑥 = 𝑎𝑏
𝑥
𝑑 𝑑 𝑏
(𝑦 𝑥 + 𝑥 𝑦 + 𝑥 𝑥 ) = 𝑎
𝑑𝑥 𝑑𝑥 1/2
𝑑
(𝑦 𝑥 + 𝑥 𝑦 + 𝑥 𝑥 ) = 0
𝑑𝑥
𝑑𝑢 𝑑𝑣 𝑑𝑤 ½
+ + =0
𝑑𝑥 𝑑𝑥 𝑑𝑥
Taking log one by one for u = yx ,v= xy, w= xx
𝑑𝑢 𝑥 𝑑𝑦
= 𝑦𝑥 ( + 𝑙𝑜𝑔𝑦), ½
𝑑𝑥 𝑦 𝑑𝑥
𝑑𝑣 𝑦 𝑑𝑦
= 𝑥 𝑦 ( + 𝑙𝑜𝑔𝑥 ),
𝑑𝑥 𝑥 𝑑𝑥 ½
𝑑𝑤
= 𝑥 𝑥 (1 + 𝑙𝑜𝑔𝑥)
𝑑𝑥 1/2
𝑑𝑢 𝑑𝑣 𝑑𝑤
+ +
𝑑𝑥 𝑑𝑥 𝑑𝑥
𝑥 𝑑𝑦 𝑦 𝑑𝑦 ½
= 𝑦𝑥 ( + 𝑙𝑜𝑔𝑦) + 𝑥 𝑦 ( + 𝑙𝑜𝑔𝑥 ) + 𝑥 𝑥 (1 + 𝑙𝑜𝑔𝑥)
𝑦 𝑑𝑥 𝑥 𝑑𝑥
=0
OR
𝑑2𝑦
If x= a(cost +t sint) and y = a(sint-t cost), find 2
𝑑𝑥
𝑑𝑥 𝑑𝑦 1+1
= 𝑎𝑡𝑐𝑜𝑠𝑡 𝑎𝑛𝑑 = 𝑎𝑡 𝑐𝑜𝑠𝑡
𝑑𝑡 𝑑𝑡
𝑑𝑦 1/2
= 𝑡𝑎𝑛𝑡
𝑑𝑥
𝑑2𝑦 2
𝑑𝑡
= 𝑠𝑒𝑐 𝑡
𝑑𝑥 2 𝑑𝑥 1/2
𝑑2𝑦 𝑠𝑒𝑐 2 𝑡
=
𝑑𝑥 2 𝑎𝑡
27
xi 2 3 4 5 6 7 8 9 10 11 12
4 4 2
Pi(x) 1 2 3 5 6 5 3 2 1
36 36 36 36 36 36 36 36 36 36 36
1
E(X) = µ = ∑𝑛𝑖=1 𝑥𝑖 𝑃𝑖 = 7
OR
Let E be the event that ‘number 4 appears at least once’ and F ½
be the event that ‘the sum of the numbers appearing is 6’.
Then,
E = {(4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (1,4), (2,4), (3,4), (5,4), ½
(6,4)}
F = {(1,5), (2,4), (3,3), (4,2), (5,1)} 1/2
P(E) = 11/36 and P(F) = 5/36 , ½

E∩F = {(2,4), (4,2)}, P(E∩F) = 2/36 1/2
𝑃(E∩F ) 2 1/2
P(E/F) = =
𝑃(𝐹) 5
28 𝑑𝑦
+ 𝑦𝑐𝑜𝑡𝑥 = 4𝑥 𝑐𝑜𝑠𝑒𝑐𝑥
𝑑𝑥
P= cot x , Q = 4x cosec x ½
I.F = 𝑒 ∫ cot 𝑥𝑑𝑥 = 𝑠𝑖𝑛𝑥 ½

y I.F. = ∫ 𝑄. 𝐼𝐹 𝑑𝑥 1/2
y sinx = ∫ 4x cosec x sinxdx =∫ 4x dx ½

𝜋2 ½
y sinx = 2x2 +C, C = −
2
2 𝜋2
y sinx = 2x − ½
2
OR
e tan y dx + (1-ex) sec2 y dy =0
x
(1-ex) sec2 y dy = - ex tan y dx ½

sec2 y −ex ½
dy = dx
tany 1−ex
log |tany|= log |1-ex| + log|c| 1
log |tany|- log |1-ex| = log|c| ½
Taking exponential both side 1/2

tany = c (1-ex)
4
29
∫ [|𝑥 − 1| + |𝑥 − 2| + |𝑥 − 3|] 𝑑𝑥
1
4 4 4
∫ |𝑥 − 1|𝑑𝑥 + ∫ |𝑥 − 2|𝑑𝑥 + ∫ |𝑥 − 3|𝑑𝑥 ½
1 1 1
4 2 4 3
= ∫1 (𝑥 − 1)𝑑𝑥 + ∫1 −(𝑥 − 2)𝑑𝑥 + ∫2 (𝑥 − 2) + + ∫1 −(𝑥 −
4
3) + ∫3 (𝑥 + 3) 1
For calculation 1
4
19
∫ [|𝑥 − 1| + |𝑥 − 2| + |𝑥 − 3|] 𝑑𝑥 = 1/2
1 2
30 𝑥+3
∫ 𝑑𝑥
√5 − 4𝑥 + 𝑥 2
𝑑 ½
X+3 = A (5 − 4𝑥 + 𝑥 2 ) + 𝐵
𝑑𝑥
A=-1/2 and B = 1
𝑥+3 ½
∫ 𝑑𝑥
√5 − 4𝑥 + 𝑥 2
1 (−4 − 2𝑥) 𝑑𝑥
= − ∫ 𝑑𝑥 + ∫
2 √5 − 4𝑥 + 𝑥 2 √5 − 4𝑥 + 𝑥 2 ½
1
For calculation
𝑥+3 𝑥+2
∫ 𝑑𝑥 = −√5 − 4𝑥 + 𝑥 2 + sin−1 +𝐶 1/2
√5 − 4𝑥 + 𝑥 2 3
31
½
Corner points (0,0), (0,3) , (2,0), (20/19 , 45/19)
Max. of Z = 235/19, at (20/19 , 45/19) ½

SECTION D
32 For labelled and correct diagram 2
4
Required area = ∫1 √𝑥𝑑𝑥
1/2
For calculation
14 2
= 1/2
3
33 To prove reflexive 1
To prove symmetric 1
To prove transitive 2
3 4 5 1
Proving T1 related with T3 = =
6 8 10
0R
Showing one one 3
Taking cases, x1,x2 both are even , x1,x2 both are odd and x1,x2
one is odd another is even 2
Showing onto by any method
34 (i) 2x +4y +3z = 29000 1

5x +2x +3x = 30500
x +y + z = 9500
(ii) matrix method AX=B

X=A-1B ½
IAI = -1 ½
−1 −1 6
adj A=[−2 −1 9 ] 2
3 2 −16 ½
calculation
X= 2500, y= 3000 and z =4000 ½
35 𝑥−1 𝑦−2 𝑧−3 𝑥−2 𝑦−4 𝑧−5

= = 𝑎𝑛𝑑 = =
2 3 4 3 4 5
⃗⃗⃗ ̂
a1 = 𝑖̂ + 2𝑗̂ + 3𝑘 , ⃗⃗⃗⃗
a2 = 2𝑖̂ + 4𝑗̂ + 5𝑘 ̂ 1
⃗⃗⃗⃗
b1 = 2𝑖̂ + 3𝑗̂ + 4𝑘̂ , ⃗⃗⃗⃗
b2 = 3𝑖̂ + 4𝑗̂ + 5𝑘̂
a1 = 𝑖̂ + 2𝑗̂ + 2𝑘̂
a⃗⃗⃗⃗2 − ⃗⃗⃗ ½
⃗⃗⃗⃗
b1 × ⃗⃗⃗⃗⃗ b2 = −𝑖̂ + 2𝑗̂ − 𝑘̂ 1
⃗⃗⃗⃗1 × ⃗⃗⃗⃗⃗
|b b2 | = √6
½
⃗⃗⃗⃗⃗⃗ b2 ) . ⃗⃗⃗⃗⃗⃗
(b1 × ⃗⃗⃗⃗⃗ (a2 − ⃗⃗⃗
a1 ) = 1
⃗⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗ ) .(a

⃗⃗⃗⃗⃗⃗ −a
½
(b ×b ⃗⃗⃗⃗ )
S.D. = d = | 1 ⃗⃗⃗⃗2 ⃗⃗⃗⃗⃗⃗2 1 |
|b1 ×b2 |
1 ½
Shortest distance d = ½
√6
The lines do not intersect 1/2
OR
Eq. of line 𝑟 = 𝑎 + ⅄𝑏⃗
Line passes through (1,2,-4) and let (a,b,c ) be the D,Ratio of ½
line then
Eq of line is
𝑟 = 𝑖̂ + 2𝑗̂ − 4𝑘̂ + ⅄(𝑎𝑖̂ + 𝑏𝑗̂ + 𝑐𝑘̂ )
Line is perpendicular to the lines 1/2
𝑥 − 8 𝑦 + 19 𝑧 − 10 𝑥 − 15 𝑦 − 29 𝑧 − 5
= = 𝑎𝑛𝑑 = =
3 −16 7 3 8 −5
𝑎 × 𝑏⃗ 𝑖𝑠 𝑝𝑒𝑟𝑝𝑒𝑛𝑑𝑖𝑐𝑢𝑙𝑎𝑟 𝑡𝑜 𝑎 𝑎𝑛𝑑 𝑏⃗ 𝑏𝑜𝑡ℎ
𝑖̂ 𝑗̂ 𝑘̂
𝑎 × 𝑏⃗ = |3 −16 7 |
3 8 −5 1
̂ 2
= 24𝑖̂ + 36𝑗̂ + 72𝑘 2
Hence D’ Ratio of line is (24,36,72)
Eq. of line
𝑥−1
=
𝑦−2
=
𝑧+4 ½
24 36 72
½
𝑟 = 𝑖̂ + 2𝑗̂ − 4𝑘̂ + ⅄(24𝑖̂ + 36𝑗̂ + 72𝑘̂ )
½
𝑟 = 𝑖̂ + 2𝑗̂ − 4𝑘̂ + ⅄(2𝑖̂ + 3𝑗̂ + 6𝑘̂ )
SECTION E
36 Let A be the event that the doctor visit the patient late and let
E1, E2, E3, E4 be the events that the doctor comes by cab,
metro, bike and other means of transport respectively.
P(E1)=0.3, P(E2)=0.2, P(E3)=0.1, P(E4)=0.4
P(A I E1) = Probability that the doctor arriving late when he

comes by cab = 0.25
Similarly, P ( A I E2) = 0.3, P (A I E3) = 0.35 and
P ( A I E4) = 0.1
A A A
P(A) = P (𝐸1 )P ( ) + P(𝐸2 )P ( ) + P(𝐸3 )P ( ) +
𝐸1 𝐸2 𝐸3
A
P(𝐸4 )P ( )
𝐸 4
P(A) = 0.25x0.3 + 0.3x0.2 + 0.1x0.35 + 0.4x0.1 = 0.21
(i)P(𝐸4 /𝐴) =
P (E4 )P(A/E4 )
=
4 1
A A A A
P (E1 )P( )+ P(E2 )P( )+P(E3 )P( )+P(E4 )P( ) 21
E1 E2 E3 E4
P (𝐸2 )P(A/𝐸2 ) 2
(i)P(𝐸2 /𝐴) = A A A A =
P (E1 )P( )+ P(E2 )P( )+P(E3 )P( )+P(E4 )P( ) 7
E1 E2 E3 E4 1
(iii) P(𝐸3 /𝐴) + P(𝐸4 /𝐴) =

A
P (𝐸3 )P( )+ P (𝐸4 )P(A/𝐸4 ) 5
𝐸3
A A A A = 2
P (E1 )P( )+ P(E2 )P( )+P(E3 )P( )+P(E4 )P( ) 14
E1 E2 E3 E4
OR
P(𝐸1 /𝐴) + P(𝐸2 /𝐴) =
A
P (𝐸1 )P(𝐸 )+ P (𝐸2 )P(A/𝐸2 ) 9
1
= 2
A A A A 14
P (E1 )P( )+ P(E2 )P( )+P(E3 )P( )+P(E4 )P( )
E1 E2 E3 E4
37 (i)we have ⃗⃗⃗⃗⃗
𝑂𝐴 = 8𝑖̂ 𝑘𝑚 ⃗⃗⃗⃗⃗
𝐴𝐵 = 6𝑗̂ 1
vector distance from Gitika’s house to school =8𝑖̂ + 6𝑗̂
(ii)vector distance from school to Aloke’s house
= 6cos300𝑖̂ + 6𝑠𝑖𝑛300 𝑗̂ 1
= 3√3𝑖̂ + 3𝑗̂
(iii)vector distance from Gitika’s house to Aloke’s house=
8𝑖̂ + 6𝑗̂ + 3√3𝑖̂ + 3𝑗̂ 2
=(8 + 3√3)+9𝑗̂
OR
The total distance travel by Gitika from her house to Aloke’s
2
house= 8 + 6 + 6 = 20 km
38 Let the increase of ₹ x in annual subscription of ₹ 300
maximize the profit of the company. Due to this increase of ₹
x, x subscriber will discontinue. Therefore
Number of subscriber = 500-x
Annual subscription = ₹ (300+x)
R be the total revenue =(500-x)(300-x)
𝑑𝑅 𝑑2𝑅 2
= 200 − 2𝑥 𝑎𝑛𝑑 2 = −2
𝑑𝑥 𝑑𝑥
𝑑𝑅
For critical point = 200 − 2𝑥 = 0 , x= 100
𝑑𝑥
𝑑2𝑅
< 0 𝑎𝑡 x=100 2
𝑑𝑥 2
So R is maximum at x=100 and maximum R = 400x400=160000
================================================================
KENDRIYA VIDYALAYA SANGATHAN: HYDERABAD REGION
FIRST PRE-BOARD EXAMINATION 2023-24
CLASS XII
MATHEMATICS (041)
1. This Question paper contains - five sections A, B, C, D and E. Each section is
2. Section A has 18 MCQs and 02 Assertion-Reason based questions of 1 mark each.
assessment (4 marks each) with sub parts.
SECTION A
(This section comprises of Multiple-choice questions (MCQ) of 1 mark each.)
1. The function 𝒇: 𝑹 → 𝑹 𝑖𝑠 𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑏𝑦 𝒇(𝒙) = 𝒙𝟐 is 1
(a) One-One but not Onto (b) Onto but not One-one
(c) Both One-One and Onto (d) Neither One-One nor Onto
2. 1 2𝑥 5 1
If 𝐴 = 38 3 12< is a symmetric matrix then the value of 𝒙 + 𝒚 is
5 4𝑦 7
(a) 4 (b) 3 (c) 7 (d) 1
3. 2𝑥 3 4 3 1
If ? ?=? ? then find the value of x
4 𝑥 4 18
(a) 2 (b) 18 (c) 6 (d) ±6
4. 1 2 3 1
The determinant of the matrix 34 5 6< is
7 8 9
(a) 1 (b) 0 (c) 5 (d) 9
5. The derivative of the function 𝑓(𝑥) = 𝑥 " 𝑤𝑖𝑡ℎ respect to 𝑥 is 1
(a) 𝑥 " (b) 𝑥 " . 𝑙𝑜𝑔𝑥
(c) 𝑥 " (1 + 𝑙𝑜𝑔𝑥) (d) 𝑥 " (1 − 𝑙𝑜𝑔𝑥)
$
6. # 1
K sin# 𝑥 𝑑𝑥 𝑖𝑠
%
$ $
(a) 0 (b) 𝜋 (c) #
(d) &
7. "! (! 1
The area of the shaded region in the given figure is P ' + )* = 1Q
(a) 4𝜋 (b) 3𝜋
(c) 12𝜋 (d) 6𝜋
8. The function 𝑓(𝑥) = |𝑥| 𝑖𝑠 1
Page 1 of 6
(a) Continuous at 𝑥 = 0 but not differentiable at 𝑥 = 0.
(b) Continuous at 𝑥 = 0 and differentiable at 𝑥 = 0.
(c) Neither continuous nor differentiable at x=0.
(d) Continuous at 𝑥 = 1 and differentiable at 𝑥 = 1.
9. A point 𝑥 = 𝑎 is said to be critical point for the function 𝑓(𝑥) 𝑖𝑓 1
(a) 𝑓(𝑥) is differentiable at x=a. i.e. 𝑓 + (𝑎)𝑒𝑥𝑖𝑠𝑡.
(b) 𝑓 + (𝑎) = 0.
(c) 𝑓 + (𝑎) = 0 or 𝑓(𝑥) is not differentiable at 𝑥 = 𝑎.
(d) 𝑓 + ′(𝑎) = 0.
10. If 𝑓(𝑥) is differentiable at c and 𝑓(𝑥) attains maximum value at c if identify 1
the correct statement
(a) The sign of 𝑓 ! (𝑥) changes from negative to positive as 𝑥 increasing through c.
(b) The sign of 𝑓 ! (𝑥) changes from positive to negative as 𝑥 increasing through c.
(c) The sign of 𝑓 + (𝑥) changes does not changes.
(d) The sign of 𝑓 ! ′(𝑥) changes from negative to positive as 𝑥 increasing through c.
11. 2 0 1
If A = W X = P + Q, where P is symmetric and Q is a skew symmetric
5 4
matrix, then 𝑃 is equal to
, , , ,
0 −# 2 #
0 #
2 −#
(a) \ , ] (b) \ , ] (c) \ , ] (d) \ , ]
#
0 #
4 −# 0 #
4
12. 0 1 1
If 𝐴 = W X, then 𝐴#%#- is equal to:
0 0
0 2023 0 1 0 0 2023 0
(a) W X (b) W X (c) W X (d) W X
0 0 0 0 0 0 0 2023
13. .! / 1( - 1
The sum of order and degree of the differential equation .0!
+3P1" Q =e0 is
(a) 2 (b) 3 (c) 5 (d) 4

14. 𝒅𝒚 1
Integrating factor of the deferential equation 𝒅𝒙
= 𝒙 + 𝒚 is
(a) 𝑒 ( (b) 𝑒 5( (c) 𝑒 5" (d) 𝑒 "

15. The projection of the vector 𝚤̂ − 2𝚥̂ + 𝑘c on 3𝚤̂ + 5𝚥̂ + 7𝑘c is 1
(a) 0 (b) √6 (c) √83 (d) √38

16. Two events A and B will be independent, then 1
(a) 𝐴 𝑎𝑛𝑑 𝐵+ 𝑎𝑟𝑒 𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 (b) 𝐴+ 𝑎𝑛𝑑 𝐵 𝑎𝑟𝑒 𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡
(c) 𝐴+ 𝑎𝑛𝑑 𝐵+ 𝑎𝑟𝑒 𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 (d) All of them correct
Page 2 of 6
17. For the following LPP Maximize 𝑧 = 4𝑥 + 𝑦 subject to the 1
conditions 𝑥 + 𝑦 ≤ 50 ,3𝑥 + 𝑦 ≤ 90, 𝑥 ≥ 0, 𝑦 ≥ 0.
The feasible solution for the given LPP as follows. Then
the maximum value of z is
(a) 120 (b) 110
(c) 50 (d) 130
18. Let 𝑎l 𝑎𝑛𝑑 𝑏l are two vectors. In the following does not assure m𝑎 ∥ 𝑏l. 1
(a) 𝑎l = 𝛼 𝑏l (b) 𝑎l = − 𝑏l
(c) 𝑎l 𝑎𝑛𝑑 𝑏l are collinear (d) |𝑎l| = p𝑏lp
Instructions for Q19 & Q20 are as follows
Two statements are given below - one labelled Assertion (A) and the other labelled Reason
(R). Read the statements carefully and choose the option that correctly describes
statements (A) and (R).
(a) Both (A) and (R) are true and (R) is the correct explanation for (A).
(b) Both (A) and (R) are true but (R) is not the correct explanation for (A).
(c) (A) is false but (R) is true.
(d) Both (A) and (R) are false.
19. Let 𝑙, 𝑚, 𝑛 are direction cosines of the line which makes angles with 𝑥, 𝑦 𝑎𝑛𝑑 𝑧 1
axis respectively 𝛼, 𝛽 𝑎𝑛𝑑 𝛾.
Assertion (A): sin# 𝛼 + sin# 𝛽 + sin# 𝛾 = 2
Reason (R): 𝑙 # + 𝑚# + 𝑛# = 1
20. Consider, the graph of constraints stated as linear 1
inequalities as below: 5𝑥 + 𝑦 ≤ 100, 𝑥 + 𝑦 ≤ 60, 𝑥, 𝑦 ≥ 0.
Assertion (A): The points (10, 50), (0, 60), (10, 10) and
(20, 0) are feasible solutions.
Reason (R): Points within and on the boundary of the
feasible region represent feasible solutions of the constraints.
SECTION B
(This section comprises of very short answer type-questions (VSA) of 2 marks
each.)
21. Find the value of tan √3 – cot (−√3).
–) –) 2
(or)
) √-
Find the value of tan5) (1) + sin5) P− #Q + cos 5) P # Q
Page 3 of 6
22. The radius r of a right circular cone is decreasing at the rate of 3 cm/minute 2
and the height h is increasing at the rate of 2 cm/minute. When r = 9 cm
and h = 6 cm, find the rate of change of its volume.
(or)
The total revenue received from the sale of x units of a product is given by
𝑅(𝑥) = 3𝑥 # + 36𝑥 + 5 in rupees. Find the marginal revenue when x = 5,
where by marginal revenue we mean the rate of change of total revenue with
respect to the number of items sold at an instant.
23. Find the interval in which the function 𝑓(𝑥) = 3𝑥 # − 12𝑥 + 9 is increasing. 2
24. 𝒆𝒙 (𝒙5𝟑) 2
Evaluate ∫ (𝒙5𝟏)𝟑
𝒅𝒙
25. If 𝑎l = 𝚤̂ + 2𝚥̂ + 𝑘c , 𝑏l = 2𝚤̂ + 𝚥̂ and 𝑐̅ = 3𝚤̂ − 4𝚥̂ − 5𝑘c, then find a vector 2
perpendicular to both of the vectors ~𝑎l − 𝑏l• 𝑎𝑛𝑑 ~𝑏l − 𝑐̅•.
(or)
If 𝑎l, 𝑏l, 𝑐̅ are unit vectors such that 𝑎l + 𝑏l + 𝑐̅ = 0l. Then find the value of
𝑎l. 𝑏l + 𝑏l. 𝑐̅ + 𝑐̅. 𝑎l
SECTION C
26. Solve the following linear programming problem graphically : 3
Minimize : z = 3x + 9y When: 𝑥 + 3𝑦 ≤ 60, 𝑥 + 𝑦 ≥ 10, 𝑥 ≤ 𝑦, 𝑥 ≥ 0, 𝑦 ≥ 0
27. 𝑑𝑦 3
𝐼𝑓 𝑥 = 𝑎 (𝑐𝑜𝑠 𝑡 + 𝑡 𝑠𝑖𝑛 𝑡)𝑎𝑛𝑑 𝑦 = 𝑎 (𝑠𝑖𝑛 𝑡 − 𝑡 𝑐𝑜𝑠 𝑡), 𝑡ℎ𝑒𝑛 𝑓𝑖𝑛𝑑 .
𝑑𝑥
(or)
If 𝑦 = (tan5) 𝑥)# , 𝑡ℎ𝑒𝑛 𝑝𝑟𝑜𝑣𝑒 𝑡ℎ𝑎𝑡 (𝑥 # + 1)# 𝑦 ++ + 2𝑥(𝑥 # + 1)𝑦 + − 2 = 0
28. Evaluate ∫ ("5)) dx 3
("5#)("5-)
(or)
Evaluate ∫ √𝑥 # − 2𝑥𝑑𝑥
$
29. 3
Evaluate ∫%% log(1 + 𝑡𝑎𝑛𝑥) 𝑑𝑥
30. 1( # 3
Solve the differential equation 𝑥. 𝑙𝑜𝑔𝑥. 1" + 𝑦 = " . 𝑙𝑜𝑔𝑥 .
(or)
& &
Solve differential equation 2𝑦𝑒 𝑑𝑥 + „𝑦 − 2𝑥𝑒 ' … 𝑑𝑦 = 0.
'
31. A coin is tossed three times. 3

=
Determine 𝑃 P> Q 𝑓𝑜𝑟 𝑡ℎ𝑒 𝑓𝑜𝑙𝑙𝑜𝑤𝑖𝑛𝑔.
(i) E : head on third toss , F : heads on first two tosses
(ii) E : at least two heads , F : at most two heads
(iii) E : at most two tails , F : at least one tail
Page 4 of 6
SECTION D
(This section comprises of long answer-type questions (LA) of 5 marks each)
32. Given below are two lines 𝑙) 𝑎𝑛𝑑 𝑙# .
"?) (?) A?)
𝑙) : @
= =
5*
𝑙# : 𝑟̅ = ~3𝚤̂ + 5𝚥̂ + 7𝑘c• + 𝜆(𝚤̂ − 2𝚥̂ + 𝑘c)
)
2
(i) Find the angle between the lines.
3
(ii) Find the shortest distance between the lines.
Show your work.
(or)
5
Find the co-ordinates of the foot of perpendicular drawn from the point A
(1, 8, 4) to the line joining the points B (0, –1, 3) and C (2, –3, –1).
Show your work.
33. Electrons in an atom move around the nucleus in a 5
"! (!
path -*
+ #, = 1. Find the area covered by one of this path
using integration. Show your work.
34. In a ground students are playing with a ball 5
with their positions on a two dimensional
plane. They pass the ball or related to each
other from one player to another player if and
only if the sum of the x-coordinate of the first
student and the y-coordinate of the second
student is equal to the sum of the y-coordinate
of the first student and the x-coordinate of the
second student. Verify whether the relation is
(i) Reflexive or not?
(ii) Symmetric or not?
(iii) Transitive or not?
(iv) Equivalence Relation or not?
Show your work.
(or)
Let 𝑓: 𝑁 → 𝑁( 𝑆𝑒𝑡 𝑜𝑓 𝑁𝑎𝑡𝑢𝑟𝑎𝑙 𝑁𝑢𝑚𝑏𝑒𝑟𝑠) be defined by
𝑛 + 1 𝑖𝑓 𝑛 𝑖𝑠 𝑎𝑛 𝑜𝑑𝑑 𝑛𝑢𝑚𝑏𝑒𝑟
𝑓(𝑛) = Š .
𝑛 − 1 𝑖𝑓 𝑛 𝑖𝑠 𝑎𝑛 𝑒𝑣𝑒𝑛 𝑛𝑢𝑚𝑏𝑒𝑟
(i) Check if 𝑓 is one-one.
(ii) Check if f is onto.
Show your work.
35. The theatre charges in a village ₹ 20 for children, 5
₹ 30 for students and ₹ 50 for adults. One day the
movie “Life of Pi” is playing in that theatre. There
are half as many as many adults as there are
students and theatre was filled with full capacity of
400 members. If the total ticket sales was ₹ 13000,
How many children, students, and adults are
attended?
Page 5 of 6
SECTION E
(This section comprises of 3 case-study/passage-based questions of 4 marks each
with sub-questions. First two case study questions have three sub questions of
marks 1, 1, 2 respectively. The third case study question has two sub questions of
2 marks each.)
36. A student plots four points on a 3-D plane such that the four points form a
parallelogram. He plots A (3,-1,2), B(1,2,-4) and
C(-1,1,2).
(i) Find the coordinate of the fourth 1
vertex.
(ii) Find the two adjacent sides of 1
parallelogram in vector form.
(iii) Find area of the parallelogram using 2
vector form of the two adjacent sides.
37. Of the students in a college, it is known that 70% reside in hostel and 30%
reside outside the hostel. Previous year results report that 40% of hostellers
attain A grade and 20% of those who reside outside attain A grade in the
annual exam. At the end of the year, a student is chosen at random from
the college.
(i) Express the given information in terms of probability. 1
(ii) What is the probability that he has an A grade. 1
(iii) If the selected has an A grade, what is the probability that the
student is a hosteler. 2
(OR)
(iii) If the selected student has an A grade, what is the probability that
the student is residing outside? 2
38. An architect designs a building for a multi-

national company. The floor consists of a
rectangular region with semi-circular ends
having a perimeter of 200m as shown
below:
Based on the above information answer the
following:
(i) If x and y represent the length and breadth of the rectangular region, 2
then find the relation between the variables x and y. Also find the area
of the rectangular region A expressed as a function of x?
(ii) Find the maximum value of area A? 2
***** ALL THE BEST *****
Page 6 of 6
KENDRIYA VIDYALAYA SANGATHAN: HYDERABAD REGION
PRE-BOARD EXAMINATION 2023-24
CLASS XII
SUBJECT: MATHEMATICS (041)
Marking Scheme
Instruction:
All alternative methods are accepted and marks will be given according to the procedure.
1.d 2.c 3.d 4.b 5.c 6.d 7.b 8.a 9.c 10.b
11.b 12.c 13.b 14.c 15.a 16.d 17.a 18.d 19.a 20.a
𝜋
21. tan–1 √3 = ---- ½ M
3
5𝜋
cot –1(−√3) = ---- ½ M
6
𝜋 5𝜋 𝜋
tan–1 √3 – cot (−√3) = 3 − –1
= −2 ---- 1M
6
(or)
𝜋
tan −1 (1)
= ---- ½ M
4
1 𝜋
sin−1 (− 2) = − 6 ---- ½ M
√3 𝜋
cos −1 ( 2 ) = ---- ½ M
6
𝜋
Answer : ---- ½ M
4
22. 1 𝑑𝑣 1
𝑣 = 3 𝜋𝑟 2 ℎ 𝑎𝑛𝑑 𝑑𝑡 = 3 𝜋 (𝑟 2 𝑑𝑡 + ℎ. 2𝑟. 𝑑𝑡 )
𝑑ℎ 𝑑𝑟
---- ½ M
Substitution of values ---- ½ M
Simplification and Answer −54𝜋 𝑐𝑚2 / min ---- 1M
(or)
𝑑𝑅
Marginal Revenue = 6𝑥 + 36 ---- ½ M
𝑑𝑥
Substitution of x value ---- ½ M
Marginal revenue=6(5)+36=66 ---- 1M
23. 𝑓 ′ (𝑥) = 6𝑥 − 12 ---- ½ M
𝑥=2 ---- ½ M
Given function is increasing in the interval (2, ∞) ---- 1M
24. ∫
𝒆𝒙 (𝒙−𝟑)
𝒅𝒙
𝟏
= ∫ 𝒆𝒙 [(𝒙−𝟏)𝟐 − (𝒙−𝟏)𝟑 ] 𝒅𝒙
𝟐
---- ½ M
(𝒙−𝟏)𝟑
Formula ---- ½ M
𝑒𝑥
Required answer (𝑥−1)2
+𝑐 ---- 1 M
25. (𝑎̅ − 𝑏̅) = −𝑖̂ + 𝑗̂ + 𝑘̂ ---- ½ M
(𝑏̅ − 𝑐̅) = −𝑖̂ + 5𝑗̂ + 5𝑘̂ ---- ½ M
Required vector 4𝑗̂ − 4𝑘̂ ---- 1 M
(or)
𝑎̅, 𝑏̅, 𝑐̅ are unit vectors ⇒ |𝑎 ̅ | = |𝑏̅| = |𝑐̅| = 1 ---- ½ M
(𝑎̅ + 𝑏̅ + 𝑐̅). (𝑎̅ + 𝑏̅ + 𝑐̅) = 0 ---- ½ M
3
𝑎̅. 𝑏̅ + 𝑏̅. 𝑐̅ + 𝑐̅. 𝑎̅ = − 2 ---- 1 M
26. For finding feasible region graphically
Graph ---- 1M
Corner Points ---- 1M
Minimum value at (5,5) is 60 ---- 1M
27. 𝑑𝑥
= 𝑎𝑡𝑐𝑜𝑠𝑡 ---- 1M
𝑑𝑡
𝑑𝑦
= 𝑎𝑡𝑠𝑖𝑛𝑡 ---- 1M
𝑑𝑡
𝑑𝑦
= 𝑡𝑎𝑛𝑡 ---- 1M
𝑑𝑥
(or)
𝑦= (tan−1 𝑥) 2
𝑑𝑦 1
= 2 tan−1 𝑥. 1+𝑥 2 ----1M
𝑑𝑥
(𝑥 2 + 1)𝑦 ′ = 2 tan−1 𝑥
2
(𝑥 2 + 1)𝑦 ′′ + 2𝑥𝑦 ′ = ----1M
1+𝑥 2
(𝑥 2 + 1)2 𝑦 ′′ + 2𝑥(𝑥 2 + 1)𝑦 ′ − 2 = 0 ----1M
28. (𝑥−1) 𝐴 𝐵
= 𝑥−2 + 𝑥−3 & finding 𝐴 = −1 & 𝐵 = 2 ---- 1 ½ M
(𝑥−2)(𝑥−3)
Required solution -㏑( x - 2 ) + 2㏑( x - 3 ) + c ---- 1 ½ M
(or)
∫ √𝑥 2 − 2𝑥𝑑𝑥 = ∫ √(𝑥 − 1)2 − 1𝑑𝑥 ---- 1M
Formula ---- 1M
𝑥−1 1
Required answer √𝑥 2 − 2𝑥 − 2 ln| (𝑥 − 1) + √𝑥 2 − 2𝑥| + 𝑐 ---- 1M
2
𝜋
29. 4
𝐼 = ∫ log(1 + 𝑡𝑎𝑛𝑥) 𝑑𝑥
0
Applying formula ---- ½ M
𝜋
2
=∫0 log (1+𝑡𝑎𝑛𝑥) 𝑑𝑥
4 ---- ½ M
𝜋 𝜋
=∫0 log 2 𝑑𝑥 − ∫0 log(1 + 𝑡𝑎𝑛𝑥) 𝑑𝑥
4 4 ---- 1M
𝜋
=4 𝑙𝑜𝑔2 − 𝐼
𝜋
I= 8 𝑙𝑜𝑔2 ---- 1M
30. 𝑑𝑦 1
+ 𝑥.𝑙𝑜𝑔𝑥 . 𝑦 = 𝑥 2
2
---- ½ M
𝑑𝑥
Identification of linear DE ---- ½ M
𝐼𝐹 = 𝑙𝑜𝑔𝑥 ---- 1M
2
General solution 𝑦. 𝑙𝑜𝑔𝑥 = − 𝑥 (1 + 𝑙𝑜𝑔𝑥) + 𝑐 ---- 1M
(or)
𝑥 𝑥
2𝑦𝑒 𝑦 𝑑𝑥 + (𝑦 − 2𝑥𝑒 𝑦 ) 𝑑𝑦 =0
𝑥
𝑑𝑥 2𝑥𝑒 𝑦 −𝑦
Writing = 𝑥 ---- ½ M
𝑑𝑦
2𝑦𝑒 𝑦
Identification of Homogeneous DE ---- ½ M
1
2𝑒 𝑣 𝑑𝑣 = − 𝑦 𝑑𝑦 ---- 1 M
𝑥
2𝑒 𝑦 = −𝑙𝑜𝑔𝑦 + 𝑐 ---- 1 M
31. (i)
𝐸
𝑃 (𝐹 ) = 2
1
---- 1 M
𝐸 3
(ii) 𝑃 (𝐹 ) = 7 ---- 1 M
𝐸 6
(iii) 𝑃 (𝐹 ) = 7 ---- 1 M
32. Writing 𝑝𝑜𝑖𝑛𝑡𝑠 𝑎𝑛𝑑 𝑑𝑟𝑠 𝑜𝑓 𝑏𝑜𝑡ℎ 𝑡ℎ𝑒 𝑙𝑖𝑛𝑒𝑠 ---- ½ M
Writing formula ---- ½ M
(𝑎 𝑎1 = 4𝑖̂ + 6𝑗̂ + 8𝑘̂
̅̅̅2 − ̅̅̅) ---- ½ M
(𝑏̅1 × ̅̅̅
𝑏2 ) = −4𝑖̂ − 6𝑗̂ − 8𝑘̂ ---- ½ M
Substitution and simplification distance =√116 𝑢𝑛𝑖𝑡𝑠 = 2√29 𝑢𝑛𝑖𝑡𝑠 ---- 1 M
Angle between two vectors formula ---- ½ M
10
Angle between the lines 𝜃 = cos −1 ( ) ---- 1 ½ M
√129
(or)
𝑥 𝑦+1 𝑧−3
Finding line equation = = ---- 1 M
2 −2 −4
Find general point P on the line (2𝜆, −2𝜆 − 1, −4𝜆 + 3) A ---- ½ M
Direction ratios of AP (𝟐𝝀 + 𝟏, −𝟐𝝀 − 𝟗, −𝟒𝝀 − 𝟏) ---- ½ M
AP⊥BC ⇒ 𝟐(𝟐𝝀 + 𝟏) − 𝟐(𝟐𝝀 − 𝟗) − 𝟒(−𝟒𝝀 − 𝟏) = 𝟎 B P C ---- 1 M
Finding λ=−1 ---- 1 M
Coordinates of foot of perpendicular is (−2,1,7) ---- 1 M
33. Figure ---- 1 M
5
𝑦 = 6 √36 − 𝑥 2 ---- 1 M
65
Area = 4 ∫0 √36 − 𝑥 2 𝑑𝑥 ---- 1 M
6
10 𝑥 36 𝑥 6
Area= (2 √36 − 𝑥 2 + sin−1 (6)) ---- 1 M
3 2 0
Substitution and simplification 30𝜋 𝑠𝑞𝑢𝑎𝑟𝑒 𝑢𝑛𝑖𝑡𝑠 ---- 1 M
34. Expressing given statement in mathematical form
(𝑎, 𝑏) 𝑅 (𝑐, 𝑑) 𝑖𝑓𝑓 𝑎 + 𝑑 = 𝑏 + 𝑐 ---- 1 M
For reflexive ---- 1 M
For symmetric ---- 1 M
For transitive ---- 1 M
For conclusion of equivalence relation ---- 1 M
(or)
For proving one-one (in all three cases) ---- 2 ½ M
For proving onto ---- 2 ½ M
35. Writing Equations ---- 1M
Writing Matrix form ---- 1M
Finding 𝐴−1 ---- 2M
Finding number of children=100, Students=200, Adults=100 ---- 1M
36. The coordinates of the fourth vertex are (1,−2,8) ---- 1 M
̅̅̅̅ = −2𝑖̂ + 3𝑗̂ − 6𝑘̂ & 𝐵𝐶
The side 𝐴𝐵 ̅̅̅̅ = −2𝑖̂ − 𝑗̂ + 6𝑘̂ ---- 1 M
Area of the parallelogram= 28 Square units ---- 2 M
37. 7 3 𝐴
𝑃(𝐻) = 10,𝑃(𝑂) = 10,𝑃 (𝐻) = 10 , 𝑃 (𝑂) = 10
4 𝐴 2
---- 1 M
34
𝑃(𝐴) = 100 ---- 1 M
𝐻 28 14
𝑃 (𝐴 ) = 34 = 17 ---- 2 M
(or)
1
(i) ---- 1 M
3
1
(ii) ---- 1 M
6
2
(iii) ---- 1 M
3
38. (i) 2𝑥 + 𝜋𝑦 = 200 ---- 1 M
2
(ii) 𝜋 (100𝑥 − 𝑥 2 ) ---- 1 M
5000
(iii) 𝑚2 ---- 2 M
𝜋
Kendriya Vidyalaya Sangathan, Jaipur Region
First Pre-Board Exam 2023-24
CLASS- XII
SUBJECT- MATHEMATICS (041)
Time: 3 Hours SET-A Max.marks: 80
1. This Question paper contains - five sections A, B, C, D and E. Each section is
assessment (4marks each) with sub parts.
SECTION A
1. For what value of 𝑥, are the determinants |
2𝑥 −3
| and |
10 −3
| equal?
5 𝑥 1 2
(a) 5 (b) 2 (c) ± 5 (d) ± 2

2 2 𝜆 −4
If A = |0 2 5 |, then A-1 exists, if
1 1 3
(a) 𝜆 = 2 (b) 𝜆 ≠ 2 (c) 𝜆 = −2 (d) 𝜆 ≠ −2

3 𝑇
If A is a skew- symmetric matrix then 𝐵 𝐴𝐵 will be
(a ) Unit matrix ( b) Symmetric matrix
(c ) Skew symmetric matrix ( d) zero matrix
4 1
𝑒 −2𝑥 , 𝑥 < ln 2
A function f: R -> R is defined by: 𝑓(𝑥) = { 4 , 1
ln 2 ≤ 𝑥 ≤ 0
𝑒 −2𝑥 , 𝑥>0
1
Which of the following statements is true about the function at the point 𝑥 = ln 2
(a) f(x) is not continuous but differentiable.
(b) f(x) is continuous but not differentiable.
(c) f(x) is neither continuous nor differentiable.
(d) f(x) is both continuous as well as differentiable
5
In which of these interval is the function f(x) = x2 - 4x strictly decreasing?
(a) ) ( −∞ , 0 ) (b) ) ( 0 , 4 ) (c) ) ( 2 , ∞ ) (d) ( −∞ , ∞ )
6 L1 and L2 are two skew lines. How many lines joining L1 and L2 can be drawn such that
the line is perpendicular to both L1 and L2 ?
(a) exactly one (b) exactly two (c) zero (d) infinitely many
1
7 If the value of a third order determinant is 12, then the value of the determinant formed by
replacing each element by its cofactors will be
(a) 12 (b) 144 (c) -12 (d) 13
8 If 𝑥 𝑦 = 𝑒 𝑥−𝑦 𝑡ℎ𝑒𝑛
𝑑𝑦
𝑖𝑠 ….
𝑑𝑥
1+𝑥 1−log 𝑥 log 𝑥
(a) (b) (c) (d) not defined
1+𝑙𝑜𝑔𝑥 1+𝑙𝑜𝑔𝑥 (1+log 𝑥)2
1
2+𝑥
9 The value of ∫ log ( ) 𝑖𝑠
−1 2−𝑥
(a) 0 (b) 1 (c) 2 (d) e
10
The points D, E and F are the mid-points
of AB, BC and CA respectively.
Where A (0,0) B ( 2,2 ) and C (4, 6 )
What is the area of the shaded region?
(a) 0.5 sq units (b) 1.0 sq unit (c) 1.5 sq unit ( d) 2.0 sq unit
11 Probability that A speaks truth is 4/5. A coin is tossed. A report that a head appears. The
probability that actually there was head is
1 1 2 4
(a) (b) 5 (c) (d) 5
2 5
12 A linear programming problem (LPP)

along with the graph of its constraints is
shown here. The corresponding objective
function is
Minimize: Z = 3x + 2y.
The minimum value of the objective
function is obtained at the corner point
(2, 0).
The optimal solution of the above linear
programming problem .
(a) does not exist as the feasible region is unbounded.
(b) does not exist as the inequality 3x + 2y < 6 does not have any point in common with
the feasible region.
(c) exists as the inequality 3x + 2y > 6 has infinitely many points in common with the
feasible region.
(d) exists as the inequality 3x + 2y < 6 does not have any point in common withthe feasible
region.
2
13 The feasible region of a linear programming problem is bounded. The corresponding
objective function is Z = 6x - 7y.
The objective function attains ________________in the feasible region.
(a) only minimum
(b) only maximum
(c) both maximum and minimum
(d) either maximum or minimum but not both
14 Unit vector which makes equal angle with 𝑖̂, 𝑗̂ 𝑎𝑛𝑑 𝑘̂ is

(a) 𝑖̂ + 𝑗̂ + 𝑘̂ ( b ) 𝑖̂
( c) 𝑘̂ ( d ) none of these
15 The vector which is perpendicular to 𝑎⃗ = 𝑖 − 2𝑗 + 3𝑘 and 𝑏⃗⃗ = 2𝑖 + 3𝑗 − 5𝑘 is
(a) 𝑖̂ − 11𝑗̂ + 7𝑘̂ ( b ) 𝑖̂ + 11𝑗̂ + 7𝑘̂
( c) 𝑖̂ − 11𝑗̂ − 7𝑘̂ ( d ) 11 𝑖̂ + 𝑗̂ + 7𝑘̂
16 The value of ( 𝑘̂ × 𝑗̂ ). 𝑖̂ + 𝑗̂. 𝑘̂ is

(a) 0 (b) 1 (c) -1 (d) 2 𝑖̂
17 The projection of 2𝑖̂ − 𝑗̂ − 4 𝑘̂ on vector 7𝑘̂ is

(a) 4 28 28
(b) (c) − (d) 0
√21 √21
18 The difference of the order and degree of the differential equation

2
𝑑2𝑦 𝑑𝑦 3
(𝑑𝑥 2 ) + (𝑑𝑥 ) + 𝑥 4 = 0 𝑖𝑠
(a) 1 (b) 2 (c) -1 (d) 0

In the following questions 19 & 20, a statement of Assertion (A) is followed by a statement
of Reason (R).
Choose the correct answer out of the following choices :
(A) Both (A) and (R) are true and (R) is the correct explanation of (A).
(B) Both (A) and (R) are true, but (R) is not the correct explanation of (A).
(C) (A) is true, but (R) is false.
(D) (A) is false, but (R) is true.
19 Assertion (A): If n (A) = p and n (B) = q then the number of relations fromA to B is 2pq
Reason (R) : A relation from A to B is a subset of A x B
20 Assertion (A): Degree of differential equation: 𝑥 − 𝑐𝑜𝑠 (𝑑𝑦/𝑑𝑥) = 0 is 1.
Reason (R): Differential equation 𝑥 − 𝑐𝑜𝑠 (𝑑𝑦/𝑑𝑥) = 0 can be converted in the
polynomial equation of derivative.
SECTION B
(This section comprises of very short answer type-questions (VSA) of 2 markseach.)
21 Find the domain of the function cos −1|𝑥 − 1| .

OR
Draw the graph of the following function: 𝑦 = 2 sin−1 𝑥 , − 𝜋 ≤ 𝑦 ≤ 𝜋
3
22 𝑑2 𝑦 𝜋
𝐼𝑓 𝑥 = sin 𝜃 , 𝑦 = cos 𝜃 𝑓𝑖𝑛𝑑 2
𝑤ℎ𝑒𝑛 𝜃 =
𝑑𝑥 4
23 Find the rate of change of the volume of a sphere with respect to its surface area when the
radius is 2 cm.
24 Iqbal, a data analyst in a social media platform is tracking the number of active users
on their site between 5 pm and 6 pm on a particular day.
The user growth function is modelled by 𝑁(𝑡 ) = 1000 𝑒 0.1 𝑡
where N(t) represents the number of active users at time t minutes during that period.
Find how fast the number of active users are increasing or decreasing at 10 minutes past
5 pm.
OR
The population of rabbits in a forest is modelled by the function below:
24 𝟐𝟎𝟎𝟎
𝑷(𝒕) = , where P represents the population of rabbits in t years.
𝟏+ 𝒆−𝟎.𝟓 𝒕
Determine whether the rabbit population is increasing or not, and justify your answer.
25 Evaluate : ∫(𝑠𝑖𝑛−1 𝑥)2 𝑑𝑥
SECTION C
26 If 𝑦 = 𝑒 𝑎𝑐𝑜𝑠
−1 𝑥 𝑑2 𝑦
, −1 ≤ 𝑥 ≤ 1 then show that: (1 − 𝑥 2 ) 𝑑𝑥 2 − 𝑥
𝑑𝑦
− 𝑎2 𝑦 = 0
𝑑𝑥
𝜋
27
Evaluate: 𝐼 = ∫02 𝑙𝑜𝑔 𝑠𝑖𝑛 𝑥 𝑑𝑥
28 2 + 𝑠𝑖𝑛 2𝑥
Evaluate ∫ 𝑒 𝑥 ( ) 𝑑𝑥
1 + 𝑐𝑜𝑠2𝑥
OR
𝑒𝑥
Evaluate: ∫ 𝑑𝑥
√5 − 4 𝑒 𝑥 − 𝑒 2𝑥
29 Find the particular solution of differential equation :

𝑑𝑦 𝜋
+ 𝑦 𝑐𝑜𝑡 𝑥 = 2𝑥 + 𝑥 2 cot 𝑥 , given that 𝑦 = 0 when 𝑥 = 2
𝑑𝑥
OR
Find the particular solution of the differential equation:
𝑦 𝑦
(x dy − y dx) 𝑦 𝑠𝑖𝑛 = ( 𝑦 𝑑𝑥 + 𝑥 𝑑𝑦) 𝑥 𝑐𝑜𝑠 , given that y = 𝜋 when x = 3
𝑥 𝑥
In adjacent figure the feasible region of

30 a maximization problem whose
objective function is given by
Z = 5x + 3y.
i) List all the constraints the
problem is subjected to.
ii) Find the optimal solution of the
problem.
4
A company follows a model of bifurcating the tasks into the categories shown below
31
At the beginning of a financial year, it was noticed that:
URGENT NOT URGENT
urgent and not urgent but

IMPORTANT important important
NOT IMPORTANT urgent but not urgent and
not important not important
➢ 40% of the total tasks were urgent and the rest were not
➢ half of the urgent tasks were important, and
➢ 30% of the tasks that were not urgent, were not important
What is the probability that a randomly selected task that is not important is urgent?
OR
31 Out of a group of 50 people, 20 always speak the truth. Two persons are selected at
random from the group, without replacement. Find the probability distribution of selected
persons who always speak the truth.
Section – D
32 The Earth has 24 time zones, defined by dividing the Earth into 24 equal longitudinal
segments. These are the regions on Earth that have the same standard time. For
example, USA and India fall in different time zones, but Sri Lanka and India are in
the same time zone.
A relation R is defined on the set U = {All people on the Earth} such that
R = {(x, y) : the time difference between the time zones x and y reside in is 6 hours}.
i) Check whether the relation R is reflexive, symmetric and transitive.
ii) Is relation R an equivalence relation?
OR
𝐿𝑒𝑡 𝑓 ∶ [ 1, ∞ ) → [ 1 , ∞ ) 𝑖𝑠 𝑔𝑖𝑣𝑒𝑛 𝑏𝑦 𝑓(𝑥) = (𝑥 2 + 1)2 − 1 .
Check whether the function is bijective.
1 0 3 −4 18 12
33 Find the product of [−1 2 −2] [ 0 4 2]
2 −3 4 2 −6 −4
Hence solve the system of equations:
x – y + 2z = 1, 2y – 3z = 1, 3x – 2y + 4z = 2
34 Find the area of the region in the first quadrant enclosed by the
𝑥 − 𝑎𝑥𝑖𝑠 , 𝑡ℎ𝑒 𝑙𝑖𝑛𝑒 𝑦 = 𝑥 𝑎𝑛𝑑 𝑡ℎ𝑒 𝑐𝑢𝑟𝑣𝑒 𝑥 2 + 𝑦 2 = 32.
35 Find the vector and Cartesian equations of the straight line passing through the
point (-5, 7, -4) and in the direction of (3, -2, 1).
Also find the point where this straight li ne crosses the XY - plane.
OR
Given below are two lines 𝐿1 𝑎𝑛𝑑 𝐿2
𝐿1 ∶ 2𝑥 = 3𝑦 = −𝑧 and 𝐿2 ∶ 6𝑥 = −𝑦 = −4𝑧
i. Find the angle between the two lines.
ii. Find the shortest distance between the two lines.
5
SECTION E
with two sub-questions. First two case study questions have three sub questions
of marks 1, 1, 2 respectively. The third case study question has two sub questions
of 2 marks each.)
36
The flight path of two airplanes in a
flight simulator game are shown
here. The coordinates of the
airports P ( -2, 1, 3) and Q (3, 4, -1)
are given.
Airplane 1 flies directly from

P to Q
Airplane 2 has a layover at R and
then flies to Q.
The path of Airplane -2 from P to R can be represented by the vector 5𝑖̂ + 𝑗̂ − 2𝑘̂
(Note: Assume that the flight path is straight and fuel is consumed uniformly
throughout the flight.)
i) Find the vector that represents the flight path of Airplane 1.
ii) Find the vector representing the path of Airplane 2 from R to Q.
iii) Find the angle between the flight paths of Airplane 1 and Airplane 2 just after
take off?
OR
iii) Consider that Airplane- 1 started the flight with a full fuel tank.
Find the position vector of the point where one third of the fuel runs out if the
entire fuel is required for the flight.
37 Rubiya, Taksh, Shanti, and Lilly

entered a spinning zone for a fun
game, but there is a twist: they
don't know which spinner will
appear on their screens until it is
their turn to play. They may
encounter one of the following
spinners, or perhaps even both:
Different combinations of
numbers will lead to exciting
prizes. Below are some of the
rewards they can win:
6
➢ Get the number '5', from Spinner A and '8' from Spinner B, and you'll win a music
player!
➢ You win a photo frame if Spinner A lands on a value greater than that of
Spinner B!
i) Taksh spuns both the spinners, A and B in one of his turns.

What is the probability that Taksh wins a music player in his turn?
ii) Lilly spuns spinner B in one of her turns.
What is the probability that the number she got is even given that it is a multiple of 3?
iii) Rubiya spuns both the spinners.

What is the probability that she wins a photo frame? .
OR
iii) As Shanti steps up to the screen, the game administrator reveals that for her turn, the
probability of seeing Spinner A on the screen is 65%, while that of Spinner B is 35%.
What is the probability that Shanti gets the number '2'?
38 A cylindrical tank of fixed volume of 144 𝜋 m3 is

to be constructed with an open top to throw all the
garbage in an orphanage. The manager of the
orphanage called a contractor for the construction
ensure that a tank to dispose off biodegradable
waste can be constructed at a minimum cost.
i) Find the cost of the least expensive tank that can be constructed if it costs
Rs. 80 per sq. m for base and Rs. 120 per sq. m for walls.
ii) Find the radius and height as well.
7
CLASS- XII
SET - B
Time: 3 Hours Max.marks: 80
6. Section E has 3 source based/case based/passage based/integrated units of assessment (4
marks each) with sub parts.
SECTION A
1 Direction Ratios of a line perpendicular to XZ - plane are
(a) 1 , 0, 1 (b) 0 , 5 , 0 (c) 1 , 1 , 1 1 1 1
(d) , ,
√3 √ 3 √ 3
2 A is a matrix of order 3 such that |adj A| = 7. Then find |A|
(a) 7 (b) 49 (c) √7 (d)

1
7
3
If y=log tan
[ ( )] π x
+
4 2
dy
, then is :
dx
(a) sec x (b) cosec x (c) tan x (d) sec x tan x
4 1
The value of ∫ log
−1
( 2−x
2+ x
) is
(a) 0 (b) 1 (c) 2 (d) e
5 For what value of 𝑥, are the determinants |25x −3x | and |101 −32 | equal?
(a) 5 (b) 2 (c) ± 5 (d) ± 2
| |
6 2 λ −4
If A = 0 2 5 , then A-1 exists, if
1 1 3
(a) λ=2 (b) λ ≠ 2 (c) λ=−2 (d) λ ≠−2
1
7 If A is a skew- symmetric matrix then BT AB will be
(a ) Unit matrix ( b) Symmetric matrix
(c ) Skew symmetric matrix ( d) zero matrix
{
8 −2 x 1
e , x <ln
2
A function f: R -> R is defined by: f ( x )= 1
4 , ln ≤ x ≤0
2
−2 x
e , x >0
1
Which of the following statements is true about the function at the point x=ln
2
9
3 1
In which of these intervals is the function f ( x )=x + 3
, x >0 is decreasing?
x
(a) ) [−1 ,1] (b) ) (−1 , 1) (c) [ −1 ,1 ]−{0 } (d) {−1,1 }
10 The points D, E and F are the mid-points of AB, BC

and CA respectively.
Where A (0,0) B ( 2,2 ) and C (4, 6 )
11 Probability that A speaks truth is 4/5. A coin is tossed. A reports that a head appears. The probability that
actually there was head is
1 1 2 4
(a) (b) (c) (d)
2 5 5 5
12 The value of ( k^ × ^j ). i^ + ^j . k^ is
(a) 0 (b) 1 (c) - 1 (d) 2 i^

^ ^ ^
13 The projection of 2 i− j−4 k on vector 7 k is ^
(a) 4 28 −28 (d) 0
(b) (c)
√21 √ 21
14 dy 2
Integrating factor of the differential equation x −4 x =2 y is
dx
(a) x 2 (b) −x 2 −1 1
(c) (d)
x2 x2
2
15 A linear programming problem (LPP) along with the
graph of its constraints is shown below. The
corresponding objective function is
The minimum value of the objective function is
obtained at the corner point (2, 0).
programming problem

(b) does not exist as the inequality 3x + 2y < 6 does not have any point in common with the
feasible region.
(c) exists as the inequality 3x + 2y > 6 has infinitely many points in common with the feasible
region.
(d) exists as the inequality 3x + 2y < 6 does not have any point in common with the feasible region.
16 The feasible region of a linear programming problem is bounded. The corresponding objective
function is Z = 6x - 7y.
(a) only minimum
(b) only maximum
If a⃗ , b⃗ and ( a⃗ + b⃗ ) are all unit vectors and θ is the angle between a⃗ and b⃗ , then the value of θ
17 is
2π π π 5π
(a) (b) ( c) (d)
3 3 6 6
⃗
18 The vector which is perpendicular to a⃗ =i−2 j+3 k and b=2 i+3 j−5 k is
^
(a) i−11 ^j+7 k^ ^
( b ) i+11 ^j+7 k^
^
( c) i−11 ^j−7 k^ ( d ) 11 i+^ ^j+7 k^

In the following questions 19 & 20, a statement of Assertion (A) is followed by a statement of Reason
(R).
19 Assertion (A): If n (A) = p and n (B) = q then the number of relations from A to B is 2pq
Assertion (A): Degree of differential equation: x−cos (dy /dx ) = 0 is 1.
20
Reason (R): Differential equation x−cos (dy /dx ) = 0 can be converted in the polynomial equation of
derivative.
3
SECTION B
(This section comprises of very short answer type-questions (VSA) of 2 marks each.)
21 Find the domain of the function cos−1| x−1| .

OR
Draw the graph of the following function: y=2sin −1 x ,−π ≤ y ≤ π
22
For what value of ‘k’ is the function continuous at x = 0?
23 Find the rate of change of the volume of a sphere with respect to its surface area when the
radius is 2 cm.
24 Iqbal, a data analyst in a social media platform is tracking the number of active users on their
site between 5 pm and 6 pm on a particular day.
The user growth function is modelled by N ( t )=1000 e0.1 t
5 pm.
OR
2000
P (t)= −0.5 t , where P represents the population of rabbits in t years.
1+ e
2
25 Evaluate : ∫|1−x2|dx
−2
SECTION C
√
26 dy 1− y
2
If √ 1−x 2+ √ 1− y 2=a ( x− y ) , show that =
dx 1−x
2
π
2
27 Evaluate: ∫ ( 2 logsin x−logsin 2 x ) dx
0
28
∫ e x( 1+cos 2x )
2+ sin 2 x
dx
|
OR
x
e
Evaluate: ∫ dx
√ 5−4 e x −e2 x
4
29 Find the particular solution of differential equation :
dy π
+ y cot x=2 x+ x 2 cot x , given that y=0when x=
dx 2
OR
Find the particular solution of the differential equation:
y y
( x dy− y dx ) y sin = ( y dx + x dy ) x cos , given that y = π when x = 3
x x
30 In adjacent figure the feasible region of

a maximization problem whose
Z = 5x + 3y.
problem.
A company follows a model of bifurcating the tasks into the categories shown below
31
URGENT NOT URGENT

urgent but not urgent and
NOT IMPORTANT not important not important
 40% of the total tasks were urgent and the rest were not
 half of the urgent tasks were important, and
 30% of the tasks that were not urgent, were not important
OR
Out of a group of 50 people, 20 always speak the truth. Two persons are selected at random
from the group, without replacement. Find the probability distribution of selected persons
who always speak the truth.
Section – D
[ ][ ]
1 0 3 −4 18 12
32 Find the product of −1 2 −2 0 4 2
2 −3 4 2 −6 −4
Hence solve the system of equations:
5
x – y +2 z =1, 2 y – 3 z=1 ,3 x – 2 y+ 4 z=2
33 Using integration, find the area of the region bounded by the triangle whose vertices are
(−1,2 ) , ( 1,5 )∧(3,4)
segments. These are the regions on Earth that have the same standard time. For example,
USA and India fall in different time zones, but Sri Lanka and India are in the same
time zone.
R = {(x, y)| the time difference between the time zones x and y reside in is 6 hours}.
OR
2
Let f : [ 1 , ∞ ) → [ 1, ∞ ) is given by f ( x ) =( x 2+1 ) −1.
Check whether the function is bijective.
35 Find the vector and cartesian equations of the straight line passing through the
Also find the point where this straight line crosses the XY-plane.
OR
Given below are two lines L1∧L2
L1 :2 x=3 y=−z and L2 :6 x=− y=−4 z
ii. Find the shortest distance between the two lines..
SECTION E
with two sub-questions. First two case study questions have three sub questions of
marks 1, 1, 2 respectively. The third case study question has two sub questions of
2 marks each.)
36
The flight path of two airplanes in
a flight simulator game are shown
here. The coordinates of the
airports P ( -2, 1, 3) and Q (3, 4, -1)
are given.
Airplane 1 flies directly from

P to Q
Airplane 2 has a layover at R and
then flies to Q.
^ ^j−2 k^
The path of Airplane-2 from P to R can be represented by the vector 5 i+
(Note: Assume that the flight path is straight and fuel is consumed uniformly throughout
the flight.)
6
iii) Find the angle between the flight paths of Airplane 1 and Airplane 2 just
after take off?
OR
Priya , Sahil, Annu, and Prachi
37 entered a spinning zone for a fun
Different combinations of numbers
will lead to exciting prizes. Below are
some of the rewards they can win:
 Get the number '5', from Spinner A and '8' from Spinner B, and you'll win a music
player!
 You win a photo frame if Spinner A lands on a value greater than that of Spinner
B!
i) Sahil spun both the spinners, A and B in one of his turns.
Find the probability that Sahil wins a music player in his turn?
ii) Prachi spun spinner B in one of her turns.
Find the probability that the number she got is even given that it is a multiple of 3? .
iii) Priya spun both the spinners.
What is the probability that she wins a photo frame?
OR
iii) As Annu steps up to the screen, the game administrator reveals that for her turn, the
Find the probability that Annu gets the number '2'?.
38 A cylindrical tank of fixed volume of 144 π m3 is to

be constructed with an open top to throw all the
garbage in an orphanage. The manager of the
orphanage called a contractor for the construction
ensure that a tank to dispose off biodegradable
waste can be constructed at a minimum cost.
7
i) Find the cost of the least expensive tank that can be constructed if it costs
Rs. 80 per sq. m for base and Rs. 120 per sq. m for walls.
ii) Find the radius and height as well.
8
CLASS- XII
SET - C
Time: 3 Hours Max.marks: 80
1. This Question paper contains - five sections A, B, C, D and E. Each section is compulsory.
However, there are internal choices in some questions.
assessment (4 marks each) with sub parts.
SECTION A
1 The points D, E and F are the mid-points of AB,

BC and CA respectively.
Where A (0,0) B ( 2,2 ) and C (4, 6 )
2 Probability that A speaks truth is 4/5. A coin is tossed. A reports that a tail appears. The
probability that actually there was head is
1 1 2 3
(a) (b) (c) (d)
2 5 5 5
3
[ ( )]
If y=log tan
π x
+
4 2
dy
, then is :
dx
(a) sec x (b) cosec x (c) tan x (d) sec x tan x
1
4 The value of ∫ log

−1
( )
2+ x
2−x
is
(a) 0 (b) 1 (c) 2 (d) e
1
5 A linear programming problem (LPP) along with
the graph of its constraints is shown below. The
corresponding objective function is
The minimum value of the objective function is
obtained at the corner point (2, 0).
programming problem

(b) does not exist as the inequality 3x + 2y < 6 does not have any point in common with
the feasible region.
(c) exists as the inequality 3x + 2y > 6 has infinitely many points in common with the feasible
region.
(d) exists as the inequality 3x + 2y < 6 does not have any point in common with the feasible
region.
6 The feasible region of a linear programming problem is bounded. The corresponding

objective function is Z = 6x - 7y.
(a) only minimum
(b) only maximum
7 a⃗ ∧b⃗ are vectors suc h t h at |⃗a|=2 ,|b⃗|=3∧⃗a . b=4
⃗ ,t h en t h e value
of |⃗a−b⃗|is :
(a) √ 5 (b) √ 21 (c) -1 (d) 1

8 ^ ^ ^
The projection of 2 i− j−4 k on vector 7 k is ^
(a) 4 28 −28 (d) 0
(b) (c)
√21 √ 21
9 dy 2
Integrating factor of the differential equation x −4 x =2 y is
dx
(a) x 2 (b) −x 2 −1 1
(c) (d)
x2 x2
10 Direction Ratios of a line perpendicular to xz plane are
(a) 1 , 0, 1 (b) 0 , 5 , 0 (c) 1 , 1 , 1 1 1 1

(d) , ,
√3 √ 3 √ 3
11 A is a matrix of order 3 such that |adj A| = 7. Then find |A|
2
(a) 7 (b) 49 (c) √7 (d)
1
7
12 If A is a square matrix, such that A2=I , then A−1 is equal to :
(a) A (b) 2 A (c) A+ I (d) I
| |
2 λ −4
13 If A = 0 2 5 , then A-1 exists, if
1 1 3
(a) λ=2 (b) λ ≠ 2 (c) λ=−2 (d) λ ≠−2
| |
14 1 1 1
Minimum value of 1 1+ sinθ 1 is
1 1 1+cos θ
(a) 0 (b) −1 −1 1
(c) (d)
2 2
15 A function f: R -> R is defined by:
1
Which of the following statements is true about the function at the point x=ln
2
16
1 3
In which of these intervals is the function f ( x )=x +
3
, x >0 is decreasing?
x
(a) ) [−1 ,1] (b) ) (−1 , 1) (c) [ −1 ,1 ]−{0 } (d) {−1,1 }
17 If a⃗ , b and ( a⃗ + b ) are all unit vectors and θ is the angle between a⃗ and b⃗ , then the value
⃗ ⃗
of θ is
2π π π 5π
(a) (b) ( c) (d)
3 3 6 6
18 The vector which is perpendicular to a⃗ =i−2 j+3 k and b=2 ⃗ i+3 j−5 k is
(a) ^
i−11 ^j+7 k^ ^
( b ) i+11 ^j+7 k^
( c) ^
i−11 ^j−7 k^ ^ ^j+7 k^
( d ) 11 i+
In the following questions 19 & 20, a statement of Assertion (A) is followed by a statement of
Reason (R).
3
19 Assertion (A): If n (A) = p and n (B) = q then the number of relations from A to B is 2pq
Assertion (A): Degree of differential equation: x−cos (dy /dx ) = 0 is 1.
20
Reason (R): Differential equation x−cos (dy /dx ) = 0 can be converted in the polynomial
equation of derivative.
SECTION B
(This section comprises of very short answer type-questions (VSA) of 2 marks each.)
21 2
Evaluate : ∫|1−x2|dx
−2
22 Find the domain of the function cos−1| x−1| .
OR
Draw the graph of the following function: y=2sin −1 x ,−π ≤ y ≤ π
23 If the circumference of circle is increasing at the constant rate, prove that rate of change
of area of circle is directly proportional to its radius.
{
24 1 – cos kx
,if x ≠ 0
x sin x
Find the value (s) of k so that the following function f ( x )= , is
1
, if x=0
2
continuous at x = 0.
25 Iqbal, a data analyst in a social media platform is tracking the number of active users
on their site between 5 pm and 6 pm on a particular day.
The user growth function is modelled by N ( t )=1000 e0.1 t
5 pm.
OR
2000
P (t)= , where P represents the population of rabbits in t years.
1+ e−0.5 t
SECTION C
4
26 In adjacent figure the feasible region of
a maximization problem whose
Z = 5x + 3y.
problem.
27 Find the general solution of the differential equation

e tany dx + ( 1−e ) sec y dy=0
x x 2
OR
2 dy
Find the general solution of the differential equation :- x =x 2−2 y 2+ xy
dx
[ ( )]
2 3
dy 2
28 1+
dx
If ( x −a)2+( y−b)2=c 2 , for c >0 , prove that is independent of a and b .
d2 y
2
dx
30 ∫ e x( 1+cos 2x )
2+ sin2 x
|dx
OR
ex
Evaluate: ∫ dx
√ 5−4 e x −e2 x
31 A company follows a model of bifurcating the tasks into the categories shown below
URGENT NOT URGENT

urgent but not urgent and

NOT IMPORTANT not important not important
 40% of the total tasks were urgent and the rest were not
 half of the urgent tasks were important, and
 30% of the tasks that were not urgent, were not important
OR
Out of a group of 50 people, 20 always speak the truth. Two persons are selected at
5
random from the group, without replacement. Find the probability distribution of selected
persons who always speak the truth.
Section – D
[ ]
1 1 1
32 Find A ,
-1
If A = 1 2 −3 .
2 −1 3
Use the result to solve the following system of linear equation:
x + y + 2z = 0 : x + 2y – z = 9 : x – 3y + 3z = -14
33 Using integration, find the area of the region bounded by the triangle whose vertices are
(−1,2 ) , ( 1,5 )∧(3,4)
segments. These are the regions on Earth that have the same standard time. For
example, USA and India fall in different time zones, but Sri Lanka and India are in
the same time zone.
R = {(x, y)| the time difference between the time zones x and y reside in is 6 hours}.
OR
Let f : 𝑅+ → [-9,∞) be a function defined as : f(x) = 5𝑥2+ 6x – 9
Show that f(x) is bijective.
35 Find the vector and Cartesian equations of the straight line passing through the
Also find the point where this straight line crosses the XY-plane.
OR
Given below are two lines L1∧L2
L1 :2 x=3 y=−z and L2 :6 x=− y=−4 z
ii. Find the shortest distance between the two lines.
SECTION E
with two sub-questions. First two case study questions have three sub questions
of marks 1, 1, 2 respectively. The third case study question has two sub questions
of 2 marks each.)
6
36
The flight path of two airplanes in a flight
simulator game are shown here. The
coordinates of the airports P ( -2, 1, 3) and
Q (3, 4, -1) are given.
Airplane 1 flies directly from P to Q

Airplane 2 has a layover at R and then
flies to Q.
The path of Airplane-2 from P to R can be represented by the vector 5 i+ ^ ^j−2 k^

(Note: Assume that the flight path is straight and fuel is consumed uniformly throughout
the flight.)
iii) Find the angle between the flight paths of Airplane 1 and Airplane 2
just
after take off?
OR
Mamta, Rahul, Shreya, and Preeti

37 entered a spinning zone for a fun
Different combinations of numbers
will lead to exciting prizes. Below
are some of the rewards they can
win:
 Get the number '5', from Spinner A and '8' from Spinner B, and you'll win a music
player!
 You win a photo frame if Spinner A lands on a value greater than that of
Spinner B!
i) Rahul spun both the spinners, A and B in one of his turns.
Find the probability that Rahul wins a music player in his turn?
ii) Preeti spun spinner B in one of her turns.
Find the probability that the number she got is even given that it is a multiple of
7
3.
iii) Mamta spun both the spinners.
Find the probability that she wins a photo frame?
OR
iii) As Shreya steps up to the screen, the game administrator reveals that for her turn, the
What is the probability that Shreya gets the number '2'?
38 Engine displacement is the measure of the cylinder volume swept by all the pistons of a
piston engine. The piston moves inside the cylinder bore. The cylinder bore in the form of
circular cylinder open at the top is to be made from a metal sheet of area 75 π cm2
Based on the above information, answer the following questions :

dV
( i ) Find
dr
(ii ) Find the radius of cylinder when its volume is maximum.
8
KENDRIYA VIDYALAYA SANGATHAN
REGION: LUCKNOW
CHAPTERWISE QUESTIONS OF PRE BOARD EXAM
CHAP 1 : RELATIONS AND FUNCTIONS
SECTION :A

Assertion (A) : f: N → N given by f(x) = 5x is injective but not surjective
Reason (R) : If co-domain ≠ range , then the function is not surjective.
Ans .
CHAP 1 : INVERSE TRIGONOMETRIC FUNCTIONS
SECTION :B
7𝜋
1. Find the value of cos −1 ⁡ (cos⁡ ( )).
6
SECTION D
1.Let 𝐴 = {𝑥⁡𝜖⁡𝑍; 0 ≤ 𝑥 ≤ 12⁡}. Check whether the relation 𝑅 = {(𝑎, 𝑏); 𝑎, 𝑏 ∈ 𝐴⁡|𝑎 − 𝑏| is
divisible by 4} in the set A is reflexive , symmetric or transitive.
OR
Let L be the set of lines in XY plane and R be the relation in L defined as
R={(𝑙1, 𝑙2 ):⁡𝑙1, 𝑖𝑠⁡𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙⁡𝑡𝑜⁡𝑙2 }. Prove that the relation R is an equivalence relation. Find the set
of all the lines which are related to the line 𝑦 = 2𝑥 + 4.
CHAP 3 : MATRICES
SECTION : A
1.If A is any square matrix of order 3 × 3 such that |𝐴| = 3 , Then the value of |𝑎𝑑𝑗𝐴| is
1
(a) 3 (b) (c) 9 (d) 27
3
2. If A = [ aij] is a symmetric matrix of order n, then
(a) aij = 1/aij for all i,j (b) aij ≠ 0 for all i,j
(c) aij = aji for all i,j (d) aij = 0 for all i, j
CHAP 4: DETERMINANT
SECTION : A
i− j
1.Write the element a23 of a 3 x 3 matrix A = (aij) whose elements aij are given by aij=
2 .
1 −1
(a) 2 × 3 (b) (c) (d) None of these
2 2
6 0
2. If for any square matrix A, A(adjA) = [ ] then value of |𝐴|
0 6
(a) 3 (b) 6 (c) 8 (d) 1
3. If a matrix A is both symmetric and skew symmetric then matrix A is
(a) scaler matrix (b) a diagonal matrix (c) a zero matrix (d) rectangular matrix
SECTION D
− 4 4 4 1 −1 1
1. Find the product of A= − 7 1 3  and B = 1 −2 − 2 and hence solve the
 5 − 3 − 1 2 1 3 
system of linear equations :
x - y+ z = 4
x - 2y – 2z = 9
2x + y + 3z = 1
CHAP 5 : CONTINUITY AND DIFFERENTIABILITY

SECTION :A
𝑑𝑦
1. If x = t 2 and y = t 3 , then 𝑑𝑥 is equal to
2𝑡 3𝑡
(a) (b) 2t (c)3t (d)
3 2
2. If the function f(x) is continuous at x= 0, then the value of k is
−3 3 2 4
(a) (b) (c)  (d)
2 2 3 9
SECTION :B
1.Find dy/dx if 2x + 3y = sin y.
SECTION :C
dy
1.If y =xcosx + (cos x)x, find .
dx
OR
If y = (tan −1 x ) , show that (x 2 + 1) y 2 + 2 x( x 2 + 1) y1 = 2 .
2 2
CHAP 6 : APPLCATION OF DERIVATIVES

SECTION :B
1.Find the rate of change of the area of a circle with respect to its radius 'r' when r = 6 cm
OR
Show that the function f given by f(x) = x3 – 3x2 + 4x, x ∈ R is increasing on R
SECTION E
(This section comprises of 3 case-study/passage-based questions of 4 marks each)
1.Case study 1: Read the following passage and answer the questions given below:
You want to make two gardens in the shape of square and circle in front of your house. If you
purchase a wire of length 28m to fence these gardens and you have used x meters of wire to fence
circular garden.
(i) What is the Radius of the circular garden and side of squared garden?
(ii) If you want to minimize the combined area of both gardens without wasting the wire of
length 28m. Then How much length of the wire will be needed to fence the circular
garden. And how much length of the wire will be needed to fence the squared garden
2. Case study 2: Read the following passage and answer the questions given below.
The Relation between the height of the plant (y in cm) with respect to exposure to sunlight is
1
governed by the following equation 𝑦 = 4𝑥 − ⁡ 2 𝑥 2 where x is the number of days exposed to
sunlight.
(i) What is the number of days it will take for the plant to grow to the maximum height?
(ii) What is the maximum height of the plant?
(iii) What will be the height of the plant after 2 days?
OR
7
If the height of the plant is 2 cm, then what is the number of days it has been exposed
to the sunlight ?
CHAP 7 : INTEGRALS
SECTION :A
𝑎
1.The value of ∫−𝑎 𝑠𝑖𝑛3 𝑥 ⁡𝑑𝑥 is
(a) a (b) a/3 (c) 1 (d) 0
2
2. Evaluate∫ 1+cos 2𝑥 𝑑𝑥
(a)tan 𝑥+c (b) 𝑙𝑜𝑔 tan 𝑥 +c (c) tan(x/2) +c (d) log(1+cos2x)+c
SECTION :C
𝑥 2 +x+1
1. Find : ∫ 𝑑𝑥
(⁡𝑥+2)(𝑥 2 +1)
𝜋
√sin 𝑥⁡
2. Evaluate : ∫02 √sin 𝑥+√cos 𝑥 ⁡𝑑𝑥
OR

Evaluate sin x + cos x
 0
4
9 + 16 sin 2 x
dx
CHAP 8 : APPLCATION OF INTEGRALS

SECTION :D
1.Using integration, find the area bounded by the curves y = x 2 , and⁡y = |x|.
OR
Find the area of the region bounded by the line y = 3x + 2, the x-axis and the ordinates x = –1
and x = 1.
CHAP 9: DIFFERENTIAL EQUATIONS

SECTION :A
𝑑𝑦
1. The integrating factor of differential equation cos 𝑥 𝑑𝑥 + y sin x = 1 is
(a) cos x (b) tan x (c) sec x (d) sin x
2. If m is the order and n is the degree of given differential equation,
3
𝑑2 𝑦 𝑑𝑦 5
(𝑑𝑥 2 ) + ⁡ (𝑑𝑥 ) + ⁡ 𝑥 4 = 0 then what is the value of m + n
(a) 5 (b) 9 (c) 4 (d) 7
SECTION :C
𝑑𝑦
1.Solve the differential equation:2𝑥 2 𝑑𝑥 − 2𝑥𝑦 + 𝑦 2 = 0.
OR
𝑑𝑦 𝜋
Solve differential equation 𝑐𝑜𝑠 2 𝑥 𝑑𝑥 + 𝑦 = 𝑡𝑎𝑛𝑥⁡(0 ≤ 𝑥 < 2 )⁡
CHAP 10: VECTORS

SECTION :A
1.If |𝑎 ⃗ | = 2 and 𝑎
⃗ | = 10 ,|𝑏 ⃗ = 12 , then the value of
⃗ .𝑏 |𝑎 ⃗ | is
⃗ ×𝑏
(a) 5 (b) 10 (c) 14 (d) 16
2. The value of λ for which the vectors 3iˆ − 6 ˆj + kˆ and 2iˆ − 4 ˆj + kˆ are parallel is
( a) 2/3 (b). 3/2 (c) 5/2 (d) 2/5
3. The scalar projection of the vector 2 𝑖̂ + 3𝑗̂ - 5𝑘̂ on the vector 5 𝑖̂ + 5𝑗̂ + 5𝑘̂ is
2
(a) (b) 0 (c) 25 (d) None of these
5√3
4. The value of i. (𝑗 × 𝑘) + 𝑗. (𝑘 × 𝑖) + 𝑘. (𝑗 × 𝑖) is
(a) 1 (b) 3 (c) 0 (d) −1
SECTION :B
1.For what value of 'a' the vectors: 2 𝑖̂ - 3𝑗̂ + 4𝑘̂ and a𝑖̂ + 6𝑗̂ - 8𝑘̂ are collinear?
OR
Find the value of λ if 𝑎 + ⃗⃗⃗
𝑏⁡ and 𝑎 - ⃗⃗⃗
𝑏⁡ are orthogonal given that 𝑎 = 𝑖̂ − 𝑗̂ + ⁡7𝑘̂ and ⃗⃗⃗
𝑏⁡=5𝑖̂ −
𝑗̂ + ⁡𝜆𝑘̂.
CHAP 11: THREE DIMENSIONAL GEOMETRY

1. Assertion (A) : The direction-cosines of the line joining the points (1,0,0) and (0,1,1) is
−1 1 1
( , , )
√3 √3 √3
Reason (R) : direction ratios = direction cosines.
SECTION :B
1.Find the Vector equation of the line which passes through the point (-2,4,-5) and is parallel to
𝑥+3 2−𝑦 2𝑧+3
the line 2 = 5 = 6
SECTION :D
1.Find the coordinates of the image of the point (1, 6, 3) with respect to the line
r = (j^ + 2k^ ) + 𝜆 ( i^ + 2 j^ + 3k^ ) ; where '⁡𝜆 ' is a scalar. Also, find the distance of the image from
the y axis.
CHAP 12: LINEAR PROGRAMMING

SECTION :A
1.The corner points of the feasible region determined by the following system of linear
inequalities: 2x + y ≤ 10, x + 3y ≤ 15, x, y ≥ 0 are (0,0), (5,0), (3,4), (0,5). Let Z= px + qy,
where
p,q > 0. Condition on p and q so that the maximum of Z occurs at both (3,4) and (0,5) is
(a) p = q ( b) p = 2q (c) p = 3q (d) q = 3p
2.. The Solution set of system 3x + 6y ≥ 80, 4x + 3y ≥ 100, x, y ≥ 0 is
(a) Lies in I quadrant (b) Lies in II quadrant
(c) Lies in III quadrant (d) Lies in IV quadrant
SECTION :C
1.Solve the following Linear Programming Problem graphically:
Maximize Z = 17.5x + 7y
subject to the constraints,
x + 3y ≤ 12
3x + y ≤ 12
x, y ≥ 0
CHAP 12: PROBABILITY
SECTION :A
1.Three balls are drawn from a bag containing 2 red and 5 black balls, if the random variable X
represents the number of red balls drawn, then X can take values
(a) 0, 1, 2 (b) 0, 1, 2, 3 (c) 0 (d) 1, 2
SECTION :C
1.A die is thrown twice and the sum of numbers appearing is observed to be 7. What is the
conditional probability that the number 2 has appeared at least once.
SECTION E
(This section comprises of 3 case-study/passage-based questions of 4 marks each)
1.. Case study : Read the text carefully and answer the questions:
A doctor is to visit a patient. From the past experience, it is known that the probabilities
that he will come by cab, metro, bike or by other means of transport are respectively 0.3, 0.2, 0.1,
and 0.4. The probabilities that he will be late are 0.25, 0.3, 0.35, and 0.1 if he comes by cab,
metro, bike and other means of transport respectively.
i. When the doctor arrives late, what is the probability that he comes by metro?
ii. When the doctor arrives late, what is the probability that he comes by other
means of transport?
KENDRIYA VIDYALAYA SANGATHAN, RAIPUR REGION
FIRST PRE-BOARD EXAM (2023-24)
CLASS- XII SUBJECT–MATHEMATICS(041)
Time Allowed: 3 Hours Max.Marks:80
1. This Question paper contains - five sections A, B, C, D and E. Each section is compulsory. However, there are
internal choices in some questions.
6. Section E has 3 source based/case based/passage based/integrated units of assessment of 4 marks each with
sub-parts.
Section A
1. If 𝐴 = [𝑎 − 9 4 9 𝑏 3 − 4 − 3 𝑐 ] is skew symmetric matrix , then the value of 𝑎 + 𝑏 + 𝑐 is 1
(a) 2 (b). −2 (c) 0 (d) can’t be determined
2. If A = [3 − 5 2 0 ] and B = [1 17 0 − 10 ] , then |𝐴𝐵|= 1
(a) 100 (b). −100 (c) 1000 (d) −1000

1−𝑐𝑜𝑠 4𝑥
3. The value of ‘k’ for which the function 𝑓(𝑥) ={ 𝑖𝑓 𝑥 ≠ 0 𝑘 𝑖𝑓 𝑥 = 0 is 1
8𝑥 2
continuous at 𝑥 = 0 is
(a) 0 (b). −1 (c) 1 (d) 2
4. If m and n , respectively are the order and the degree of deferential 1
𝑑 𝑑𝑦 4
equation 𝑑𝑥 [(𝑑𝑥 )] = 0, then m+n =
(a). 1 (b). 2 (c) 3 (d) 4
5. If the two lines 1

𝑦 𝑧
𝐿1 : 𝑥 = 5, =
3 − 𝛼 −2
𝑦 𝑧
𝐿2 : 𝑥 = 2, =
−1 2 − 𝛼
are perpendicular , the the value of 𝛼 =
2 7
(a) .3 (b). 3 (c) 4 (d) 3
6. 1+𝑥 𝑑𝑦 1
If (1−𝑦) = 𝑎 , then 𝑑𝑥 is equal to
𝑥−1 𝑥−1 𝑦+1 𝑦−1

(a) 𝑦−1
(b). 𝑦+1 (c) 𝑥−1 (d) 𝑥+1
7. The point which lies in the half plane 2𝑥 + 𝑦 − 4 ≤ 0 1
(a) (0,1) (b) (3,1) (c) (5,5) (d) (2,2)
8. If 𝑎⃗ , 𝑏⃗⃗ and ( 𝑎⃗ +𝑏⃗⃗ ) are all unit vectors and 𝜃 is the angle between 𝑎⃗ and 𝑏⃗⃗ then the value of 𝜃 is : 1
2𝜋 5𝜋 𝜋 𝜋
(a) 3
(b) 6 (c) 3 (d) 6
9. 1 1
∫ 𝑠𝑖𝑛2 𝑥.𝑐𝑜𝑠 2 𝑥
𝑑𝑥 =
(a) . 𝑡𝑎𝑛 𝑥 + 𝑐𝑜𝑡 𝑥 + 𝑐 (b). 𝑡𝑎𝑛 𝑥 − 𝑐𝑜𝑡 𝑥 + 𝑐
(c) 𝑐𝑜𝑡 𝑥 = 𝑡𝑎𝑛 𝑥 + 𝑐 (d) −𝑡𝑎𝑛 𝑥 . 𝑐𝑜𝑡 𝑥 + 𝑐
10. The general solution of the differntial equation 𝑦𝑑𝑥 − 𝑥𝑑𝑦 = 0 is 1
(𝑎). 𝑥𝑦 = 𝑐 (b). 𝑥 = 𝑐𝑦 2 (c) 𝑦 = 𝑐𝑥 (d) 𝑦 = 𝑐𝑥 2
11. If the area of the triangle with vertices (−3,0), (3,0) 𝑎𝑛𝑑 (0, 𝑘) is 9 sq. Units , then the value/s of k 1
will be
(a) ±3 (b) ±6 (c) 0 (d). 12
12. ABCD is parallalogram such that ⃗⃗⃗⃗⃗⃗ 𝐵𝐶 = 𝑏⃗⃗, then ⃗⃗⃗⃗⃗⃗⃗

𝐴𝐵 = 𝑎⃗ and ⃗⃗⃗⃗⃗⃗ 𝐵𝐷 = 1
(a) 𝑎⃗ +𝑏⃗⃗ (b) −𝑎⃗ +𝑏⃗⃗ (c) 𝑎⃗ −𝑏⃗⃗ ⃗⃗

(d). 0
13. The corner points of the bounded feasible region of an LPP are O(0, 0), A(250, 0), B(200, 50) and 1
C(0, 175). If the maximum value of the objective function Z = 2ax + by occurs at the points A(250, 0)
and B(200, 50), then the relation between a and b is :
(a) 2a = b (b) 2a = 3b (c) a = b (d) a = 2b

14. ̂ on (𝑖̂ + 2𝑗̂ + 2𝑘̂ )
The magnitude of projection of (2𝑖̂ − 𝑗̂ + 2𝑘) 1
4 12 8 8
(a) (b) (c) (d)
81 9 9 81
𝜋
15. For what value of 𝑥 ∈ [0, 2 ], is A + AT = √3 I , where A =[𝑐𝑜𝑠 𝑥 𝑠𝑖𝑛 𝑥 − 𝑠𝑖𝑛𝑥 𝑐𝑜𝑠 𝑥 ] 1
𝜋 𝜋 𝜋
(𝑎) (b) (c) 0 (d)
3 6 2
𝑥
16. The vector equation of a line which passes through the point (2, 4, 5) and is parallel to the line 3 = 1
3−𝑦 2𝑧+2
=
2 4
(a) 2𝑖̂ + 4𝑗̂ + 5𝑘̂ +𝜇 (3𝑖̂ − 2𝑗̂ + 2𝑘̂)
(b) 2𝑖̂ + 4𝑗̂ + 5𝑘̂ +𝜇 (3𝑖̂ + 2𝑗̂ + 2𝑘̂)
(c) 2𝑖̂ + 4𝑗̂ + 5𝑘̂ +𝜇 (3𝑖̂ + 2𝑗̂ + 4𝑘̂)
(d) 3𝑖̂ + 42 + 4𝑘̂ +𝜇 (3𝑖̂ + 4𝑗̂ + 5𝑘̂)
17. An urn contains 6 balls of which two are red and four are black. Two balls are drawn at random. 1
Probability that they are of the different colours is
6 4 8 1
(a). 15 (b). 15 (c) 15 (d) 15
18. A function f defined from N to N such that 𝑓(𝑥) = 𝑥 2 , where N is set of natural numbers. Which 1
of the following is true
(a) . f is one-one function but not onto (b). f is onto function but not one one
(c). f is one one and onto function (d). f is neither one one nor onto
In the following question, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the
correct answer out of the following choices.
(a) Both (A) and (R) are true and (R) is the correct explanation of (A).
(b) Both (A) and (R) are true but (R) is not the correct explanation of(A).
(c) (A) is true but (R) is false.
(d) (A) is false but (R) is true.
19. (Assertion ): If det(A) = 2, where A is a 3 × 3 matrix, then det{𝑎𝑑𝑗(𝑎𝑑𝑗𝐴)} =16 1
(Reason): If A is a square matrix of order n, then |𝑎𝑑𝑗 𝐴| = |𝐴|𝑛−1
20. (Assertion ): The function 𝑓(𝑥) = 𝑥 2 − 6𝑥 + 8 is strictly increasing in the interval (4,∞) 1
(Reason): A function has maximum value at point 𝑥 = 𝑎, if 𝑓 ′(𝑎) = 0 and f ’’(a)>0
Section B
𝑐𝑜𝑠 𝑥
21. (A).Simplify: (1−𝑠𝑖𝑛 𝑥 ) 2
OR
1
(B).Solve for 𝑥 ∶ 𝑠𝑖𝑛 𝑠𝑖𝑛 ( + 𝑥) = 1
5
22. (A).Prove that the function 𝑓(𝑥) = 𝑥 3 − 3𝑥 2 + 3𝑥 + 4 is increasing function 2
OR
(B). Find the interval/s in which the function 𝑓 ∶ 𝑅 → 𝑅 defined by 𝑓(𝑥) = 𝑥𝑒 𝑥 is increasing
23. 𝑑2 𝑦 𝑑𝑦 2
If 𝑦 = 𝑎𝑒 2𝑥 + 𝑏𝑒 −𝑥 , prove that 𝑑𝑥 2 − 𝑑𝑥 − 2𝑦 = 0
24. A particle moves along the curve 𝑥 2 = 2𝑦 . At what point, ordinate increases at the same rate as 2
abscissa increases?
25. (𝑥 2+𝑠𝑖𝑛2𝑥)𝑠𝑒𝑐 2𝑥 2
Evaluate : ∫ 𝑑𝑥
1+𝑥 2
Section C
𝜋
26. Evaluate ∫04 𝑙𝑜𝑔(1 + 𝑡𝑎𝑛𝑥)𝑑𝑥 3
27. 𝑥 2+1 3
Differentiate 𝑥 𝑐𝑜𝑠𝑥 + (𝑥 2−1) with respect to 𝑥
28. 𝑑𝑦 𝑦 𝑦 3
Find particular solution of differential equation 𝑑𝑥 − 𝑥 + 𝑐𝑜𝑠𝑒𝑐 (𝑥 ) = 0 with the conditions x=1 , y
=0.
OR
Find the particular solution of the differential equation
𝑑𝑦 1
(1+𝑥 2 ) + 2𝑥𝑦 = 2 given that y = 0 when x = 1.
𝑑𝑥 1+𝑥
29. (A).Solve the following Linear Programming Problem graphically : 3

Maximise 𝑍 = 3𝑥 + 9𝑦
subject to the constraints: 𝑥 + 3𝑦 ≤ 60, 𝑥 + 𝑦 ≥ 10, 𝑥 ≤ 𝑦 , 𝑥 ≥ 0, 𝑦 ≥ 0
OR
(B). Solve the following Linear Programming Problem graphically:
Minimize:𝑍 = 𝑥 + 2 𝑦 ,
subject to the constraints: 𝑥 + 2𝑦 ≥ 100 , 2𝑥 − 𝑦 ≤ 0, 2𝑥 + 𝑦 ≤ 200, 𝑥, 𝑦 ≥ 0
30. 𝑐𝑜𝑠 𝑥 3
(A). Evaluate: ∫ √6−6𝑠𝑖𝑛 𝑥+𝑐𝑜𝑠 2 𝑥
𝑑𝑥
OR
1 𝑥+|𝑥|+1
(B). Evaluate:∫−1 𝑑𝑥
𝑥 2+2|𝑥|+1
31. Out of a group of 50 people, 20 always speak the truth. Two persons are selected at random from 3
the group (without replacement). Find the probability distribution of number of selected persons
who always speak the truth.
Section D
32. If A = [3 1 2 3 2 − 3 2 0 − 1 ] find A-1 .Hence, solve the system of equations: 5
3𝑥 + 3𝑦 + 2𝑧 = 1 , 𝑥 + 2𝑦 = 4, 2𝑥 – 3𝑦 – 𝑧 = 5
33. (A) Let N be the set of all natural numbers and R be a relation on N×N defined by 5
𝑎, 𝑏) 𝑅(𝑐, 𝑑) ⇔ 𝑎𝑑 = 𝑏𝑐 for all 𝑎, 𝑏), (𝑐, 𝑑 𝑁 × 𝑁. Show that R is an
equivalence relation on N×N.
OR
(B) .Prove that a function 𝑓 ∶ [0, ∞ ) → [ −5, ∞) defined as𝑓 (𝑥) = 4𝑥 2 + 4𝑥 − 5 𝑖𝑠 both
one-one and onto
34. (A).Find the co-ordinates of the foot of perpendicular drawn from the point A (1, 8, 4) to the line 5
joining the points B (0, –1, 3) and C (2, –3, –1).
OR
(B).Find the value of b so that the lines
𝑥−1 𝑦−𝑏 𝑧−3

𝐿1 : = =
2 3 4
𝑥−4 𝑦−1
𝐿2 : = =𝑧
5 2
are intersecting lines. Also, find the point of intersection of these given lines
35. The area of the region bounded by line 𝑦 = 𝑚𝑥 (𝑚 > 0), the curve 5
𝜋
𝑥 2 + 𝑦 2 = 4 and the x-axis in the first quadrant is 2 units. Using integration , find the value of m.
Section E
36. 2+2
A shopkeeper sells three types of flower seeds 𝐴1 , 𝐴2 𝑎𝑛𝑑 𝐴3 . They are sold as a mixture where the
proportions are 4 : 4 : 2 respectively. The germination rates of the three types of seeds are 45%,
60% and 35%.

(i) What is the probability of a randomly chosen seed to germinate ?
(ii) What is the probability that the randomly selected seed is of type A 1, given that it germinates ?
37. Engine displacement is the measure of the cylinder volume swept by all the pistons of a piston 1+1+
engine. The piston moves inside the cylinder bore 2
The cylinder bore in the form of circular cylinder open at the top is to be made from a metal sheet
of area 75𝜋 cm2.
(i) If the radius of cylinder is r cm and height is h cm, then write the volume V of cylinder in terms of
radius r.
𝑑𝑉
(ii) Find .
𝑑𝑟
(iii) (A) Find the radius of cylinder when its volume is maximum.
OR
(iii) (B) For maximum volume, h > r. State true or false and justify.
38. There are two insects in a spider web. The positions of these two insects are , B(–1, 0, 0) and C(0, 1, 1+1+
2).The position of spider is A(1,2,3) 2
Based on above information answer the following questions
(i) Which insect is nearer to the spider.

(ii) What are the direction cosines of ⃗⃗⃗⃗⃗⃗
𝐵𝐶
(iii) (a). Find the value of angle A
OR
(iii)(b).Find the area of triangle ABC

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PM SHRI KENDRIYA VIDYALAYA, MALKAPURAM

2. Section A has 18 MCQ’s and 02 Assertion-Reason based questions of 1 mark

3. Section B has 8 Response based MCQ’s of 2 marks each.

4. Section C has 4 Short Answer (SA) type questions of 3 marks each.

5. Section D has 4 Long Answer (LA) type questions of 5 marks each.

6. Section C has 3 case based/source based/integrated/other type questions

(Multiple Choice Question)

Each question carries 1 mark

(a) f is one-one and onto

(b) f is many- one and onto

(c) f is one-one but not onto

(d) f is neither one-one nor onto

(a) 0 (b) -1 (c) 1 (d) 2

Q8. Let the function f : R →R be defined by f(x)=2x + cos x. Then f(x)

(a) 2(sin x+ x cos θ) +c (b) 2(sin x- x cos θ) + c

( a ) 1 and 1 ( b ) 1 and not defined

(a)12 (b) 14 (c) 16 (d) none of these

(a) 12 (b) 5 (c) 13 (d) 60

and (3, 4, 6) is:

(a). i + 2k + λ(4i + 4j + 4k) (b). i – 2k + λ(4i + 4j + 4k)

( c). -i+2k+ λ(4i + 4j + 4k) (d). -i+2k+ λ(4i – 4j – 4k)

Q16. The linear programming problem minimize Z= 3x+2y, subject to constraints

x + y ≤ 8, 3x+5y ≤ 15, x, y ≥0, has

(a) One solution (b) No feasible solution

(c) Two solutions (d) Infinitely many solutions

(a) Infeasible solution (b) Feasible solution

Q18. .The value of P(E|F), where E: no tail appears, F: no head appears,

Q19. Assertion: The function f(x) = x2 is differentiable function ∀ x ∈ R

sin2 x−cos2 x 3−5 sinx

Q30. Solve the following linear programming problem graphically:

subject to the constraints: x + y ≤ 50 , 3x + y ≤ 90 , x ≥ 0, y ≥ 0

A problem in mathematics is given to 3 students whose chances of solving it are 1/2,

(This section comprises of long answer type-questions (LA)of 5 marks each)

is divisible by 3 is an equivalence relation. OR

x 1 y  3 z  5 x2 y4 z6

(ii) Find the critical point of the function.

Based on the above information, answer the following questions.

1ST PRE – BOARD EXAMINATION - 2024

SUBJECT: MATHEMATICS MAX. MARKS: 80 MARKS

CLASS: XII TIME: 3 HOURS

Q. 8: The integrating factor of the differential equation

3 𝑖̂ + 𝜆 𝑗̂ + 𝑘̂ are perpendicular is:

Q. 15: Which of the following is true for the function 𝑥 + 1 ?

(a) local maximum value = 2

Reason (R): The Differential coefficient of the function with

5 units from the point (1 , 3 , 3) .

But he reaches in time if he uses any other mode of transport.

Q. 29: Solve the following Linear Programming Problem graphically:

𝑟⃗ = (3 𝑖̂ + 2 𝑗̂ − 4 𝑘̂) + λ (𝑖̂ + 2 𝑗̂ + 2 𝑘̂)

(i) Find the probability of winning of Dinesh?

₹ 45 𝑝𝑒𝑟 𝑚2 for sides, what is the cost of the least

केंद्रीय विद्यालय संगठन, अहमदाबाद संभाग 2023-24

KENDRIYA VIDYALAYA SANGATHAN, AHMEDABAD REGION 2023-24

कक्षा बारहिीं के ललए संचयी परीक्षा

CUMULATIVE TEST FOR CLASS-XII

Q.NO MARKING POINTS MARKS

21. Any point on the line 𝑥 + 2

such that 𝑃𝑄 = 5 where 𝑄 (1 , 3 , 3) .

22. 𝑐𝑜𝑠 −1 𝛼 + 𝑐𝑜𝑠 −1 𝛽 + 𝑐𝑜𝑠 −1 𝛾 = 3 𝜋 1

⇒ 𝑐𝑜𝑠 −1 𝛼 = 𝜋, 𝑐𝑜𝑠 −1 𝛽 = 𝜋 , 𝑐𝑜𝑠 −1 𝛾 = 𝜋