Grade-12,PB-1,Edited

SETH ANANDRAM JAIPURIA SCHOOL, LUCKNOW
PRE-BOARD-1 (2024-25) SUBJECT: MATHEMATICS (041) MAX. MARKS : 80

CLASS : XII DURATION: 3 HRS
General 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.
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
1. The Greatest integer function f: Z → Z, given by f(x) = [x], is
(a) both one-one and onto. (b) neither one-one nor onto.
(c) one-one but not onto. (d) onto but not one-one
2. The maximum number of equivalence relations on the set A = {1, 2, 3} are
(a) 1 (b) 2 (c) 3 (d) 5
3. Simplest form of cot

−1
{√ a
x −a 2
2 } is
x −1 −1 x −1 x −1 x
(a) cosec (b) sec (c) cosec (d) cos
a a a a
−1 −1
4. tan ¿ ¿ - sec ¿ is equal to
π
(a) π (b)−π (c) (d) None of these
3
5. Suppose P, Q and R are different matrices of order 3 × 5, a × b and c x d respectively, then value
of ac + bd is, if matrix 2P + 3Q – 4R is defined
(a) 9 (b) 30 (c) 34 (d) None of these
6. ∫ ¿ ¿ ¿ = …….
x x x x
(a) 10 −10 +c (c)
10 +10 +c
−1
( b ) ( 10 x + x 10) + c (d) log ( x 10 +10x ) + c
x
e (1+ x)
7.∫ dx
cos 2 (x e x )
(a)−cot ( e x x ) + c (c) tan ( x e x ) +c
( b ) tan ( e x ) +c (d) None of these
8. The function f(x)= ax + b is strictly decreasing for all x∈R iff:
(a) a=0 (b) a<0 (c) a>0 (d) None of these
9. A cylindrical vessel of radius 0.5 m is filled with oil at the rate of 0.25 π cubic metre/min. The rate
at which oil is rising is
(a)1m/min (b)2m/min (c)5m/min (d)None of these
( )
2 2
dy d y
10. The order and degree of the differential equation + 4 2 + 5=0 is
dx d x
(a) order 1 and degree 2 (b) order 2 and degree 2
(c) order 2 and degree 1 (d) order 1 and degree 1
dy y 2
11. The Integrating Factor of the differential equation − =2 x is
dx x
−1 1
(a) x 2 (b) x (c) (d)
x x
2
12. The area bounded by the parabola y =36 x and the line x = 1 above the x-axis is ______ sq units.
(a) 2 (b) 4 (c) 6 (d) None of these
13. For an L.P.P. the objective function is Z = 4x + 3y, and the feasible region determined by a set of
constraints (linear inequations) is shown in the graph.
Which one of the following statements is true?

(a) Maximum value of Z is at R. (b) Maximum value of Z is at Q.
(c) Value of Z at R is less than the value at P. (d) Value of Z at Q is less than the value at R.
14. If A is a square matrix of order 3 such that |A| = - 5 , then value of |− A A' ∨¿ is
(a) 125 (b) – 125 (c) 25 (d) – 25
15. If the corner points of the feasible region are 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) and 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
dy
16. If y = I sinx -cosx I then at x= π /6 is
dx
(a)0 (b)-1 (c)1 (d) None of these
17. If A is a non-singular matrix of order 3 and |A| = – 4, find |adj A|.
1
(a) 4 (b) 16 (c) 64 (d)
4
π /2
18. ∫ log ( cotx ) dx=…
0
(a) π /4 log tanx (b) π /8 log 2 (c) 0 ( d ) π /8 log 8
In the following questions 19 and 20, a statement of assertion (A) is followed by a statement of
reason (R). Mark the correct choice as:
(a) Both Assertion (A) and Reason (R) are true, and Reason(R) is the correct explanation of Assertion
(A).
(b) Both Assertion (A) and Reason (R) are true, but Reason(R) is not the correct explanation of
Assertion (A).
(c) Assertion (A) is true, but Reason (R) is false.
(d) Assertion (A) is false, but Reason (R) is true.
19. Assertion (A): The value of is 2.

Reason (R): Since,
−12 −1 1 π
20. Assertion (A) : The value of expression sec + tan−1 1 + sin is
√3 2 4
−π π π
Reason (R) : Principal value branch of sin−1 x is [ , ] and that of sec−1 x is [0, ] −¿ { }.
2 2 2
SECTION B
1−x 7 y −14 z−3 7−3 x y−5 6−z
21. Find the value of p so that the lines = = and = = are
3 2p 2 3p 1 5
at right angles.
22. Find the interval on which the function f(x)= 2x3+9x2+12x-1 is decreasing
OR
Find the point(s) on the curve y=x2, at which y coordinate is changing six times as fast as x
coordinate.
dy
23. Find the solution of the differential equation log( ) = 2x + y.
dx
24. If and , then find the value of .
(x−3) x
25. Find ∫ e dx
( x −1)3
SECTION C
π
2
26. Evaluate:
∫ ( 2 log sin x−log sin 2 x ) dx
0
OR ∫ √ x2 +1[ log ( x 2 +1 )−2 logx]

dx
x4
27. Minimize Z = x + 2y subject to the following constraints:
2x+y≥ 3, x+2y ≥ 6, x ≥ 0, y ≥ 0 .Show graphically that minimum of Z occurs at more than two
points.
28. If x = 3 sin t - sin 3t and y = 3cost - cos3t ; find dy/dx at x = π /6.
29. A trust invested some money in two types of bonds. The first bond pays 10% interest and second
bond pays 12% interest. The trust received Rs 2400 as interest. However, if trust had interchanged
money in bonds they would have got Rs 100 less. Let the amount invested in first type and second
type of bond be Rs x and Rs y. Based on the above information, answer the following questions;
(i) Write the equations in terms of x and y representing the given information.
(ii) Write the matrix equation representing the given information. Find the amount invested by trust
in first and second bond respectively.
dy
30. Solve the differential equation +y cot x =2x+x2 cot x
dx
31. If vertices A, B and C of a triangle have the co-ordinates (1,2,3) ,(-1,0,0) and (0,1,2) respectively.
Find the value of angle enclosed between sides BA and BC (consider interior angle).
SECTION D
32. Using the method of integration find the area bound by the curve IxI + IyI = 1.
OR
The area between x = y2 and x = 4 is divided into two equal parts by the line x = a, find the value of a.
[ ]
1 −1 1
33. Determine the inverse 2 1 −3 and hence use it to solve the equations
1 1 1
𝑥 + 2𝑦 + 𝑧 = 4; −¿𝑥 + 𝑦 + 𝑧 = 0 ; 𝑥 −¿ 3𝑦 + 𝑧 = 2.
34. Show that the relation R in the set N of Natural numbers given by R = {(a, b ): |a - b| is a multiple
of 4 } is an equivalence relation.
OR
d ∈ N}. Prove that R is an equivalence relation and obtain the equivalence class [(2, 5)].
Let R be a relation defined in N × N defined by: R = {(a, b) R (c, d) if and only if a +d = b + c and a, b, c,
x 1− y z−2
35. Find the co-ordinates of image of the point (1,6,3) in the line = = .
1 2 3
SECTION E
36. Case-Study 1: Read the following passage and answer the questions given below.
Rohan, a student of class XII, visited his uncle’s flat with his father.
His observations:
Window of the house is in the form of a rectangle surmounted by a
semicircular opening. The perimeter of this window is 10 m (refer fig.)
(i)If x and y represent the length and breadth of the rectangular region of the
window, then find the relation between x and y.
(ii) Express area of the window in terms of x (alone).
(iii) Find the value of x for maximizing the Area (A) of this window.
(iv) Find the maximum area of the window.
OR
(iv) For maximum value of A, find the breadth of the rectangular part of the
window.
37. Observe the diagram shown alongside in which two lines l 1∧l 2 are shown in space and AB is the
only line which is perpendicular to both the lines l 1∧l 2 .
(i)Based on above information, answer (any four) the following questions:

Which of the following is true?
( a ) l 1∧l 2 are parallel lines.
( b ) l 1∧l 2 are intersecting lines.
(c )l 1∧l 2 are skew lines.
(d)¿
x−1 y −2 z−3 x−2 y −4 z−5
(ii)If equations of l 1∧l 2 are respectively = = and = = , then a
2 3 4 3 4 5
vector perpendicular to both the lines is given by
^ ^j−k^
(a) i+2 ^ 2 ^j−k^
( b )−i+ ^ ^j+ k^
(c) i+2 (d) ¿
(iii) Two points on the lines l 1∧l 2 are
(a) ( 1 , 2, 3 ) ∧( 2 , 4 , 5 ) (b) ( 2 , 3 , 4 ) ∧( 3 , 4 , 5 ) ( c ) (−1 ,−2 ,−3 )∧(−2 ,−4 ,−5 )
(d) ¿
(iv) What are direction ratios of vector ⃗ AB ?
(a)<2 , 3 , 4> ¿
(b)<3 , 4 ,5> ¿
(c )←1, 2 ,−1>¿
(d ) ¿
(v) What is the magnitude of ⃗
AB ?
(a)√ 6 units
1
(b) units
√6
(c)2 √ 6units
1
(d) units .
2 √6
38. Location of three houses of a society is represented by the points A(-1, 0), B(1, 3) and
C(3, 2) as shown in figure. Based on the above information, answer the following question
(i) Find the equation of line(through) AB, BC and AC.

(ii) Find the area(∆𝐴BC) using integrations.

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SETH ANANDRAM JAIPURIA SCHOOL, LUCKNOW

PRE-BOARD-1 (2024-25) SUBJECT: MATHEMATICS (041) MAX. MARKS : 80

3. Simplest form of cot

Which one of the following statements is true?

19. Assertion (A): The value of is 2.

OR ∫ √ x2 +1[ log ( x 2 +1 )−2 logx]

(i)Based on above information, answer (any four) the following questions:

(i) Find the equation of line(through) AB, BC and AC.

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