Sample Question Paper

Class XII
Session 2022-23
Mathematics (Code-041)

Time Allowed: 3 Hours Maximum Marks: 80

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
(Multiple Choice Questions)
Each question carries 1 mark

Q1. If A =[aij] is a skew-symmetric matrix of order n, then

(a) 𝑎 = ∀ 𝑖, 𝑗 (b) 𝑎 ≠ 0 ∀ 𝑖, 𝑗 (c)𝑎 = 0, 𝑤ℎ𝑒𝑟𝑒 𝑖 = 𝑗 (d) 𝑎 ≠ 0 𝑤ℎ𝑒𝑟𝑒 𝑖 = 𝑗
Q2. If A is a square matrix of order 3, |𝐴′| = −3, then |𝐴𝐴′| =
(a) 9 (b) -9 (c) 3 (d) -3
Q3. The area of a triangle with vertices A, B, C is given by
(a) 𝐴𝐵⃗ × 𝐴𝐶⃗ (b) 𝐴𝐵⃗ × 𝐴𝐶⃗
(b) 𝐴𝐶⃗ × 𝐴𝐵⃗ (d) 𝐴𝐶⃗ × 𝐴𝐵⃗
, 𝑖𝑓 𝑥 ≠ 0
Q4. The value of ‘k’ for which the function f(x) = is continuous at x = 0 is
𝑘, 𝑖𝑓 𝑥 = 0
(a) 0 (b) -1 (c) 1. (d) 2
Q5. If 𝑓 (𝑥) = 𝑥 + , then 𝑓(𝑥) is
(a) 𝑥 + log |𝑥| + 𝐶 (b) + log |𝑥| + 𝐶 (c) + log |𝑥| + 𝐶 (d) − log |𝑥| + 𝐶
Q6. If m and n, respectively, are the order and the degree of the differential equation
= 0, then m + n =

(a) 1 (b) 2 (c) 3 (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.

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Q8. The scalar projection of the vector 3𝚤̂ − 𝚥̂ − 2𝑘 𝑜𝑛 𝑡ℎ𝑒 𝑣𝑒𝑐𝑡𝑜𝑟 𝚤̂ + 2𝚥̂ − 3𝑘 is
(a) (b) (c) (d)
√

Q9. The value of ∫ dx is

(a) log4 (b) 𝑙𝑜𝑔 (c) 𝑙𝑜𝑔2 (d) 𝑙𝑜𝑔

Q10. If A, B are non-singular square matrices of the same order, then (𝐴𝐵 ) =
(a)𝐴 𝐵 (b)𝐴 𝐵 (c)𝐵𝐴 (d) 𝐴𝐵

Q11. 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) only
(d) at every point of the line-segment joining the points (0.6, 1.6) and (3, 0)

2 4 2𝑥 4
Q12. If = , 𝑡ℎ𝑒𝑛 𝑡ℎ𝑒 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 value(s) of ‘x’ is/are
5 1 6 𝑥
(a) 3 (b) √3 (c) -√3 (d) √3, −√3

Q13. If A is a square matrix of order 3 and |A| = 5, then |𝑎𝑑𝑗𝐴| =

(a) 5 (b) 25 (c) 125 (d)

Q14. Given two independent events A and B 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 the differential equation 𝑦𝑑𝑥 − 𝑥𝑑𝑦 = 0 𝑖𝑠

(a) 𝑥𝑦 = 𝐶 (b) 𝑥 = 𝐶𝑦 (c) 𝑦 = 𝐶𝑥 (d) 𝑦 = 𝐶𝑥

Q16. If 𝑦 = 𝑠𝑖𝑛 𝑥, then (1 − 𝑥 )𝑦 𝑖𝑠 equal to

(a) 𝑥𝑦 (b) 𝑥𝑦 (c) 𝑥𝑦 (d) 𝑥

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Q17. If two vectors 𝑎⃗ 𝑎𝑛𝑑 𝑏⃗ are such that |𝑎⃗| = 2 , 𝑏⃗ = 3 𝑎𝑛𝑑 𝑎⃗. 𝑏⃗ = 4, 𝑡ℎ𝑒𝑛 𝑎⃗ − 2𝑏⃗ is
equal to
(a) √2 (b) 2√6 (c) 24 (d) 2√2

Q18. P is a point on the line joining the points 𝐴(0,5, −2) and 𝐵(3, −1,2). If the x-coordinate
of P is 6, then its z-coordinate is

(a) 10 (b) 6 (c) -6 (d) -10

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 is false.
(d) A is false but R is true.

Q19. Assertion (A): The domain of the function 𝑠𝑒𝑐 2𝑥 is −∞, − ∪ [ , ∞)

Reason (R): 𝑠𝑒𝑐 (−2) = −
Q20. Assertion (A): The acute angle between the line 𝑟̅ = 𝚤̂ + 𝚥̂ + 2𝑘 + 𝜆(𝚤̂ − 𝚥̂) and the x-axis
is
Reason(R): The acute angle 𝜃 between the lines
𝑟̅ = 𝑥 𝚤̂ + 𝑦 𝚥̂ + 𝑧 𝑘 + 𝜆 𝑎 𝚤̂ + 𝑏 𝚥̂ + 𝑐 𝑘 and
| |
𝑟̅ = 𝑥 𝚤̂ + 𝑦 𝚥̂ + 𝑧 𝑘 + 𝜇 𝑎 𝚤̂ + 𝑏 𝚥̂ + 𝑐 𝑘 is given by 𝑐𝑜𝑠𝜃 =

SECTION B
This section comprises of very short answer type-questions (VSA) of 2 marks each

Q21. Find the value of 𝑠𝑖𝑛 [𝑠𝑖𝑛 ]

OR
Prove that the function f is surjective, where 𝑓: 𝑁 → 𝑁 such that
𝑛+1
, 𝑖𝑓 𝑛 𝑖𝑠 𝑜𝑑𝑑
𝑓(𝑛) = 2
𝑛
, 𝑖𝑓 𝑛 𝑖𝑠 𝑒𝑣𝑒𝑛
2
Is the function injective? Justify your answer.

Q22. 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. At what rate is the tip of his shadow moving? At what rate is his shadow
lengthening?

Q23. If 𝑎⃗ = 𝚤̂ − 𝚥̂ + 7𝑘 𝑎𝑛𝑑 𝑏⃗ = 5𝚤̂ − 𝚥̂ + 𝜆𝑘, then find the value of 𝜆 so that the vectors
𝑎⃗ + 𝑏⃗ 𝑎𝑛𝑑 𝑎⃗ − 𝑏⃗ are orthogonal.
𝑶𝑹

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 Find the direction ratio and direction cosines of a line parallel to the line whose equations
are
6𝑥 − 12 = 3𝑦 + 9 = 2𝑧 − 2
Q24. If 𝑦√1 − 𝑥 + 𝑥 1 − 𝑦 = 1 , 𝑡ℎ𝑒𝑛 𝑝𝑟𝑜𝑣𝑒 𝑡ℎ𝑎𝑡 = −

Q25. Find |𝑥⃗| if (𝑥⃗ − 𝑎⃗). (𝑥⃗ + 𝑎⃗) = 12, where 𝑎⃗ is a unit vector.

SECTION C
(This section comprises of short answer type questions (SA) of 3 marks each)

Q26. Find: ∫
√

Q27. Three friends go for coffee. They decide who will pay the bill, by each tossing a coin and
then letting the “odd person” pay. There is no odd person if all three tosses produce the
same result. If there is no odd person in the first round, they make a second round of
tosses and they continue to do so until there is an odd person. What is the probability
that exactly three rounds of tosses are made?
OR
Find the mean number of defective items in a sample of two items drawn one-by-one
without replacement from an urn containing 6 items, which include 2 defective items.
Assume that the items are identical in shape and size.
Q28. Evaluate: ∫
√

OR

Evaluate: ∫ |𝑥 − 1| 𝑑𝑥

Q29. Solve the differential equation: 𝑦𝑑𝑥 + (𝑥 − 𝑦 )𝑑𝑦 = 0

OR
Solve the differential equation: 𝑥𝑑𝑦 − 𝑦𝑑𝑥 = 𝑥 + 𝑦 𝑑𝑥

Q30. Solve the following Linear Programming Problem graphically:

Maximize Z = 400x + 300y subject to 𝑥 + 𝑦 ≤ 200, 𝑥 ≤ 40, 𝑥 ≥ 20, 𝑦 ≥ 0

Q31. Find ∫ ( )
𝑑𝑥
SECTION D
(This section comprises of long answer-type questions (LA) of 5 marks each)

Q32. Make a rough sketch of the region {(𝑥, 𝑦): 0 ≤ 𝑦 ≤ 𝑥 , 0 ≤ 𝑦 ≤ 𝑥, 0 ≤ 𝑥 ≤ 2} and find
the area of the region using integration.
Q33. Define the relation R in the set 𝑁 × 𝑁 as follows:
For (a, b), (c, d) ∈ 𝑁 × 𝑁, (a, b) R (c, d) iff ad = bc. Prove that R is an equivalence
relation in 𝑁 × 𝑁.
OR

Page 4
 Given a non-empty
empty set X, define the relation R in P(X) as follows:
For A, B ∈ 𝑃(𝑋), (𝐴, 𝐵) ∈ 𝑅 iff 𝐴 ⊂ 𝐵. Prove that R is reflexive, transitive and not
symmetric.

Q34. An insect is crawling along the line 𝑟̅ = 6𝚤̂ + 2𝚥̂ + 2𝑘 + 𝜆 𝚤̂ − 2𝚥̂ + 2𝑘 and another
insect is crawling along the line 𝑟̅ = −4𝚤̂ − 𝑘 + 𝜇 3𝚤̂ − 2𝚥̂ − 2𝑘 . At what points on the
lines should they reach so that the distance between them is the shortest? Find the shortest
possible distance between them.

OR
The equations of motion of a rocket are:
ar
𝑥 = 2𝑡, 𝑦 = −4𝑡, 𝑧 = 4𝑡, where the time t is given in seconds, and the coordinates of a
moving point in km. What is the path of the rocket? At what distances will the rocket be
from the starting point O(0,
(0, 0, 0) and from the following line in 10 seconds?
𝑟⃗ = 20𝚤̂ − 10𝚥̂ + 40𝑘 + 𝜇((10𝚤̂ − 20𝚥̂ + 10𝑘 )

2 −3 5
Q35. If A = 3 2 −4 , find 𝐴 . Use 𝐴 to solve the following system of equations
1 1 −2
2𝑥 − 3𝑦 + 5𝑧 = 11, 3𝑥 + 2
2𝑦 − 4𝑧 = −5, 𝑥 + 𝑦 − 2𝑧 = −3

SECTION E
(This
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
sub parts (i), (ii), (iii)
of marks 1, 1, 2 respectively. The third case study question has two sub
sub-parts of 2
marks each.)

Q36. Case-Study 1: Read the following passage and answer the questions given below.

The temperature of a person during an intestinal illness is given by

𝑓(𝑥) = −0.1𝑥 + 𝑚𝑥 + 98 98.6,0 ≤ 𝑥 ≤ 12, m being a constant, where f(x) is the
temperature in °F at x days.
(i) Is the function differentiable in the interval (0, 12)? Justify your answer.
(ii) If 6 is the critical point of thee function, then find the value of the constant m.
 (iii) Find the intervals in which the function is strictly increasing/strictly decreasing.
OR
(iii) Find the points of local maximum/local minimum, if any, in the interval (0, 12) as
well as the points of absolute maximum/absolute minimum in the interval [0, 12].
Also, find the corresponding local maximum/local minimum and the absolute
maximum/absolute minimum values of the function.

Q37. Case-Study 2: Read the following passage and answer the questions given below.

In an elliptical sport field the authority wants to design a rectangular soccer field
with the maximum possible area. The sport field is given by the graph of
+ = 1.
(i) If the length and the breadth of the rectangular field be 2x and 2y respectively,
then 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.
OR
(iii) Use Second 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
Q38. Case-Study 3: Read the following passage and answer the questions given below.

There are two antiaircraft guns, named as A and B. The probabilities that the shell fired
from them hits an airplane are 0.3 and 0.2 respectively. Both of them fired one shell at an
airplane at the same time.
(i) What is the probability that the shell fired from exactly one of them hit the plane?
(ii) If it is known that the shell fired from exactly one of them hit the plane, then what is
the probability that it was fired from B?

Page 7
 CBSE
Additional Practice Questions
Subject: Mathematics (041)
Class: XII 2023-24
Time Allowed: 3 Hours Maximum Marks: 80
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 MCQs 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
(This section comprises of Multiple-choice questions (MCQ) of 1 mark each.)

Serial
No. Question Marks
1 For any 2 × 2 matrix P, which of the following matrices can be Q such that PQ 1
= QP?

(a)

(b)

(c)

(d)
2 V is a matrix of order 3 such that |adj V| = 7. 1

Which of these could be |V|?

(a)
(b)
(c)
(d)

3 The points D, E and F are the mid-points of AB, BC and CA respectively. 1

(Note: The figure is not to scale.)

What is the area of the shaded region?

(a) 2 sq units
3
(b) 2 sq units
1
(c) 2 sq units
(d) (2√26 - 1) sq units

4 If f(x) = cos-1 √x, 0 < x < 1, which of the following is equal to f'(x)? 1

(a)
 (b)

(c)

(d)

5 A function f: R -> R is defined by: 1

Which of the following statements is true about the function at the point
1
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.

6 In which of these intervals is the function f(x) = 3x 2 - 4x strictly decreasing? 1

(a) (-∞, 0)
(b) (0, 2)
2
(c) (3 , ∞)
(d) (-∞, ∞)

7 Which of these is equal to ∫ 𝑒 (𝑥 log 5) 𝑒 𝑥 𝑑𝑥, where C is the constant of 1

integration?

(a)
(b)
(c)
(d)
8 Shown below is the curve defined by the equation y = log (x + 1) for x ≥ 0. 1

Which of these is the area of the shaded region?

(a) 6log(2) - 2
(b) 6log(2) - 6
(c) 6log(2)
(d) 5log(2)

9 In which of the following differential equations is the degree equal to its order? 1

(a)

(b)

(c)

(d)
10 Kapila is trying to find the general solution of the following differential 1
equations.

Which of the above become variable separable by substituting y = b.x, where

b is a variable?

(a) only (i)

(b) only (i) and (ii)
(c) all - (i), (ii) and (iii)
(d) None of the above

11 1

(a) only (i)

(b) only (ii)
(c) only (i) and (ii)
(d) only (ii) and (iii)

12 1

3
(a) λ = , σ = 0
5
5
(b) λ = 3 , σ = 5
(c) λ = 3, σ = 0
(d) (cannot be found as there are two unknowns and only one equation)
13 1

(a) (2, √2, 2)

(b) (√2, 2, √2)
1 1 1
(c) (2 , , )
√2 2
(d) (2√2, 2√2, 2√2)

14 A line m passes through the point (-4, 2, -3) and is parallel to line n, given by: 1

The vector equation of line m is given by:

Which of the following could be the possible values for p, q and r?

(a) p = 4, q = (-2), r = 3
(b) p = (-4), q = (-2), r = 3
(c) p = (-2), q = 3, r = (-6)
(d) p = 8, q = 4, r = (-3)

15 L1 and L2 are two skew lines. 1

 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) infinitely many
(d) (there cannot be a line joining two skew lines such that it is perpendicular
to both)

16 A linear programming problem (LPP) along with the graph of its constraints is 1
shown below. 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 with
the feasible region.
17 The feasible region of a linear programming problem is bounded. The 1
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

18 M and N are two events such that P(M ∩ N) = 0. 1

Which of the following is equal to P(M|(M ∪ N))?

(a)

(b)

(c)

(d)

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)}.

Based on the above information, 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).

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.

(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 true but (R) is false.
(d) (A) is false but (R) is true.

20 Two statements are given below - one labelled Assertion (A) and the other 1
labelled Reason (R). Read the statements carefully and choose the option that
correctly describes statements (A) and (R).
 Assertion (A): The maximum value of the function f(x) = x 5 , x ∈ [-1, 1], is
attained at its critical point, x = 0.

Reason (R): The maximum of a function can only occur at points where
derivative is zero.

(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.

SECTION B
(This section comprises of very short answer type-questions (VSA) of 2 marks
each.)

Serial
No. Question Marks
21 Find the domain of the function y = cos-1 (|x - 1|). Show your steps. 2

OR

Draw the graph of the following function: 2

y = 2sin-1 (x), -π ≤ y ≤ π

22 2

23 If x = cot t and y = cosec2 t, find: 2

 Show your steps.

24 Iqbal, a data analyst in a social media platform is tracking the number of active 2
users on their site between 5 pm and 6 pm on a particular day.

The user growth function is modelled by N(t) = 1000e0.1t, 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. Show your steps.

OR

The population of rabbits in a forest is modelled by the function below: 2

Determine whether the rabbit population is increasing or not, and justify your
answer.

25 Solve the integral: 2

Show your steps.

SECTION C
(This section comprises of short answer type questions (SA) of 3 marks each)

Serial
No. Question Marks
3
26
 Solve the integral:

Show your work.

27 Evaluate the integral: 3

Show your steps.

OR

Using the properties of definite integrals, prove the following: 3

State the property used.

28 When an object is thrown vertically upward, it is under the effect of gravity 3

and air resistance. For small objects, the force due to air resistance is
numerically equal to some constant k times v, where v is the velocity of the
object (in m/s) at time t (s).

This situation can be modelled as the differential equation shown below.

A tennis ball of mass 0.050 kg is hit upwards with a velocity of 10 m/s. An air
resistance numerically equal to 0.4v acts on the ball.

(i) Model the above situation using a differential equation.

(ii) Write an expression for the velocity of the ball in terms of the time.

Show your work.

29 Shown below is a curve. 3

L1 is the tangent to any point (x, y) on the curve.

L2 is the line that connects the point (x, y) to the origin.

The slope of L1 is one third of the slope of L2 .

Find the equation of the curve. Show your work.

OR

(i) Solve the differential equation and show that the solution represents a
family of circles.

(ii) Find the radius of a circle belonging to the above family that passes
through the origin.

Show your work.

30 Each unit of Product A that a company produces, is sold for Rs 100 with a 3
production cost of Rs 60 and each unit of Product B is sold for Rs 150 with
production cost of Rs 90. On a given day, the company has a budget of Rs
8000 to spend on production. The production process makes it such that they
can only produce a maximum of 100 units each day. Also, the number of
product B produced cannot be more than twice as many of Product A.

Frame a linear programming problem to determine how many units of

Product A and B should the company produce in a day in order to maximize
their profit?
 (Note: No need to find the feasible region and optimal solution.)

OR

Shown below is the feasible region of a maximisation problem whose 3

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.

Show your work.

31 A company follows a model of bifurcating the tasks into the categories shown 3
below.
 At the beginning of a financial year, it was noticed that:

♦ 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? Use Bayes' theorem and show your steps.

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

Serial
No. Question Marks
32 The Earth has 24 time zones, defined by dividing the Earth into 24 equal 5
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?

Show your work.

 OR

A function f : R - {-1, 1} -> R is defined by: 5

i) Check if f is one-one.
ii) Check if f is onto.

Show your work.

33 Abdul threw a basketball in the direction of the basketball hoop which 5

traversed a parabolic path in a vertical plane as shown below.

(Note: The image is for representation purpose only.)

The equation of the path traversed by the ball is y = ax 2 + bx + c with respect

to a xy-coordinate system in the vertical plane. The ball traversed through the
points (10, 16), (20, 22) and (30, 25). The basketball hoop is at a horizontal
distance of 70 feet from Abdul. The height of the basketball hoop is 10 feet
from the floor to the top edge of the rim.

Did the ball successfully go through the hoop? Justify your answer.

(Hint: Consider the point where Abdul is standing as the origin of the
xy-coordinate system.)

34 Shown below are concrete elliptical water pipes, each 10 feet in length. 5
 The graph given above represents the inner circumference of the elliptical
pipe, where x and y are in feet. Assume that the water flows uniformly and
fully covers the inner cross-sectional area of the pipe.

Find the volume of water in the pipe at a given instant of time, in terms of π.
Use the integration method and show your steps.

(Note: Volume = Area of the base × Height)

35 i) Find the vector and cartesian equations of the straight line passing through 5
the point (-5, 7, -4) and in the direction of (3, -2, 1).

ii) Find the point where this straight line crosses the xy-plane.

Show your work.

OR

Given below are two lines L1 and L2 : 5

L1 : 2x = 3y = -z
L2 : 6x = -y = -4z

i) Find the angle between the two lines.

ii) Find the shortest distance between the two lines.

Show your work.

 SECTION E
(This section comprises of 3 case-study/passage-based questions of 4 marks
each 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.)

Serial
No. Question Marks

36 Answer the questions based on the given information.

The flight path of two airplanes in a flight simulator game are shown below.
The coordinates of the airports P and Q are given.

Airplane 1 flies directly from P to Q.

Airplane 2 has a layover at R and then flies to Q.

(Note: Assume that the flight path is straight and fuel is consumed uniformly
throughout the flight.)
1
i) Find the vector that represents the flight path of Airplane 1. Show your
steps.

ii) Write the vector representing the path of Airplane 2 from R to Q. Show 1
your steps.
 iii) What is the angle between the flight paths of Airplane 1 and Airplane 2 2
just after takeoff? Show your work.

OR

iii) Consider that Airplane 1 started the flight with a full fuel tank. 2

Find the position vector of the point where a third of the fuel runs out if the
entire fuel is required for the flight. Show your work.

37 Answer the questions based on the given information.

Rubiya, Thaksh, Shanteri, 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:

♦ 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) Thaksh spun both the spinners, A and B in one of his turns. 1

What is the probability that Thaksh wins a music player in that turn? Show
your steps.

ii) Lilly spun spinner B in one of her turns. 1

What is the probability that the number she got is even given that it is a
multiple of 3? Show your steps.

iii) Rubiya spun both the spinners. 2

What is the probability that she wins a photo frame? Show your work.

OR

iii) As Shanteri steps up to the screen, the game administrator reveals that for 2
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 Shanteri gets the number '2'? Show your steps.

38 Answer the questions based on the given information.

Two metal rods, R1 and R2 , of lengths 16 m and 12 m respectively, are

insulated at both the ends. Rod R1 is being heated from a specific point while
rod R2 is being cooled from a specific point.

The temperature (T) in Celsius within both rods fluctuates based on the
distance (x) measured from either end. The temperature at a particular point
along the rod is determined by the equations T = (16 - x)x and T = (x -
12)x for rods R1 and R2 respectively, where the distance x is measured in
meters from one of the ends.

i) Find the rate of change of temperature at the mid point of the rod that is
being heated. Show your steps. 2

ii) Find the minimum temperature attained by the rod that is being cooled. 2
Show your work.
 BAL NIKETAN PUBLIC SCHOOL
END TERM EXAMINATION (2023-24)
MATHEMATICS
CLASS- XI
DATE: 04.03.2024 DURATION: 3 Hours
NO. OF PAGES: 07 M. MARKS: 80

General Instructions:
1. This Question paper contains five sections- A, B, C, D and E. Each section is compulsory.
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
of 4 marks each with sub-parts.
7. All questions are compulsory. Although internal choices are given in some questions.

SECTION A (20 marks)

MULTIPLE CHOICE QUESTIONS
Q. No. Questions Marks

1. For any set, (𝐴′)′ is equal to: 1 mark

[A] A′ [B] A [C] ∅ [D] U-A

2. The solution set of inequation |x+2|≤5 is: 1 mark

[A] (-7,5) [B] [-7,3] [C] [-5,5] [D] (-7,3)
3. The symmetric difference of A and B is: 1 mark
[A] (A-B) ∩(B-A) [B] (A-B)U(B-A)
[C] (AUB)-(A∩B) [D] A-B
4. In an ellipse, if c=a and b = 0, then we will obtain a: 1 mark
[A] circle [B] ellipse
[C] line segment [D] can’t say anything
5. If R is a relation on a finite set having n elements, then the number of relations 1 mark
on A is:
[A] 2n [B] 2𝑛 2 [C] n2 [D] n4

Q.P.CODE-M111 Page 1
 𝑥
6. The range of the function 𝑓(𝑥) = |𝑥| is: 1 mark

[A] R-{0} [B] R-{-1,1} [C] {-1,1} [D] {1}

7. The value of (24)3 is: 1 mark

[A] 13825 [B] 13826 [C] 13827 [D] 13824

8. The value of sin 00 sin 10 sin 20 sin 30….. sin 1790 is: 1 mark
[A] 0 [B] 1 [C] -1 [D] 2
9. If three dice are thrown simultaneously, then the probability of getting a score 1 mark
of 4 is:
[A] 5/216 [B] 1/6 [C] 1/36 [D] 1/72
10. The value of sin 50° – sin 70° + sin 10° is equal to: 1 mark
[A] -1 [B] 0 [C] 1 [D] 2
1
11. If z=(2+3𝑖)(1−𝑖), then |z| = 1 mark

1 5
[A] 1 [B] [C] [D] 0
√26 √26

12. If a=1+i, then a2 = 1 mark

[A] 1-i [B] 1+i [C] i [D] 2i
13. If |x|<7, then 1 mark
[A] –x<-7 [B] -x≤-7 [C] –x>-7 [D] -7<x<7
14. If A={1,2,4}, B={2,4,5}, C={2,5}, then (A-B)x(B-C) is: 1 mark
[A] {(1,2),(1,5),(2,5)} [B] {(1,4)}
[C] (1,4) [D] [1,4]
15. For any two sets A and B, A∩(AUB)= 1 mark
[A] A [B] B
[C] ∅ [D] cannot say anything
16. Among 14 players, 5 are bowlers. In how many ways a team of 11 may be 1 mark
formed with atleast 4 bowlers?
[A] 265 [B] 263 [C] 264 [D] 268
17. If ‘a’ is the first term and ‘r’ is the common ratio, then the nth term of a G.P is: 1 mark
[A] arn-1 [B] arn [C] (ar)n-1 [D] (ar)n
18. Distance between the lines 5x+3y-7=0 and 15x+9y+14=0 is: 1 mark
35 1 35 34
[A] [B] 3√34 [C] 3√34 [D] 3√34
√34

Q.P.CODE-M111 Page 2
 19. Assertion: The simplest form of i-35 is –i. 1 mark
Reason: The additive inverse of (1-i) is (-1+i).
[A] Both A and R are true and R is the correct explanation for A.
[B] Both A and R are true and R is not the correct explanation for A.
[C] A is true but R is false.
[D] A is false but R is true.
20. Consider the following statements: 1 mark

Assertion: The figure shows a relationship between the sets A and B, then
the relation in set builder form is {(x,y):y=x2, x,y∈N and -2≤x≤2}.
Reason: The above relation in roster form is {(-1,1),(-2,4),(0,0),(1,1),(2,4)}.
[A] Both A and R are true and R is the correct explanation for A.
[B] Both A and R are true and R is not the correct explanation for A.
[C] A is true but R is false.
[D] A is false but R is true.
SECTION B (10 marks)
VERY SHORT QUESTIONS
21. Let U= {1,2,3,4,5,6,7}, A={2,4,6}, B={3,5} and C={1,2,4,7}. Find: 2x1=
(i) A′U(B∩C′) 2 marks
(ii) (B-A)U(A- C)

22. One card is drawn from a well shuffled deck of 52 cards. If each outcome is 2x1=
equally likely, calculate the probability that the card will be: 2 marks
(i) A black card
(ii) Not a black card
OR
Find the probability that when a hand of 7 cards is dealt from a well shuffled 2x1=
2 marks
deck of 52 cards, it contains:
(i) All 4 kings
(ii) Exactly 3 kings

23 Find the distance between (2,3,5) and (4,3,1). 2 marks

Q.P.CODE-M111 Page 3
 24. Express the following in the form of a+ib: 2 marks
(3 + 𝑖√5)(3 − 𝑖√5)
(√3 + √2𝑖) − (√3 − √2𝑖)

OR
2−√−25
Find the result in the form of a+ib: 1−√−16
2 marks

25. Find (92)5 using Binomial theorem. 2 marks

SECTION C (18 marks)

SHORT QUESTIONS
1
26. If f,g and h are real functions defined by 𝑓(𝑥) = √𝑥 + 1, 𝑔(𝑥) = 𝑥 and ℎ(𝑥) = 2x1½=
3 marks
2𝑥 2 − 3, then find the value of:
(i) (2f+g-h)(1)
(ii) (3f+2g-6h)(0)

27. In the triangle ABC, with vertices A(2,3), B(4,-1) and C(1,2), find the equation 3 marks
and length of altitude from the vertex A.
28. Find the sum of the sequence 7,77,777,7777…. 3 marks
OR
̅̅̅̅?
What is the rational number having the decimal expansion 0.356 3 marks

29. A box contains 5 different red and 6 different white balls. In how many ways 3 marks
can 6 balls be selected so that there are atleast two balls of each colour?

OR
Find the number of arrangements of the letters of the word INDEPENDENCE. 1 mark
In how many of these arrangements:
(i) Do all the vowels occur together? 1 mark
1 mark
(ii) Do the words start with I and end with P?

30. A manufacturer has 600 litres of a 12% solution of acid. How many litres of a 3 marks
30% acid solution must be added to it so that acid content in the resulting
mixture will be more than 15% but less than 18%.
31. Find the derivative of sin x from the first principle. 3 marks

Q.P.CODE-M111 Page 4
 SECTION D (20 marks)
LONG QUESTIONS
𝑥 𝑥 𝑥
32. Find sin2, cos2 and tan2 for the following case: 5 marks
4
tan x= − 3, x lies in Quadrant II
OR
Prove that:
𝜋 9𝜋 3𝜋 5𝜋
2x2½=
(i) 2 cos 13 cos 13 + cos 13 + cos 13 = 0 5marks

(ii) (sin 3𝑥 + sin 𝑥) sin 𝑥 + (cos 3𝑥 − cos 𝑥) cos 𝑥 = 0

33. Calculate the mean, standard deviation and variance of the following table 5 marks
given the age distribution of a group of people:

Age 20-30 30-40 40-50 50-60 60-70 70-80 80-90

No. of 3 51 122 141 130 51 2

person

OR

The mean and standard deviation of 100 observations were calculated as 40

and 5.1, respectively by a student who took by mistake 50 instead of 40 for 5 marks
one observation. What are the correct mean and standard deviation?

34. Find the limits:

𝑥 2 +1
(i) lim (𝑥+100) 1 mark
𝑥→1

𝑥 3 −4𝑥 2 +4𝑥 2 marks

(ii) lim ( )
𝑥→2 𝑥 2 −4

𝑥 15 −1
(iii) lim (𝑥 10 −1) 2 marks
𝑥→1

35. Solve:

3 marks
(i) For the ellipse 16𝑥 2 + 25𝑦 2 = 400, find the length of minor axis,
length of major axis, eccentricity, coordinates of vertices and foci.

(ii) Find the equation of hyperbola, the length of whose latus rectum 2 marks
3√5
is 8 and eccentricity is .
5

Q.P.CODE-M111 Page 5
 SECTION E (12 marks)
CASE-BASED QUESTIONS
36. Vijeta and Rohini are playing cards. Total number of playing cards are 52 in 4x1=
numbers. Following events happen while playing the game of choosing a 4 marks
card out of 52 cards.

Answer the following questions accordingly:

(i) Vijeta draw a card, what is the probability that the drawn card is red?
(ii) Rohini draw a card, what is the probability that the card drawn is an ace ?
(iii) Rohini draw the card again, now what is the probability that the card is
an ace of red colour ?
(iv) Now, Vinay draw a card, what is the probability that the card drawn is a
Jack?

37. A manufacturing company produces certain goods. The company manager 4x1=
used to make a data record on daily basis about the cost and revenue of these 4 marks
goods separately. The cost and revenue function of a product are given by
C(x)= 20x+4000 and R(x)= 60x+2000, respectively, where x is the number of
goods produced and sold.

Choose the correct options accordingly:

(i) How many goods must be sold to release some profit?
(a) x<50 (b) x>50
(c) x≥50 (d) x≤50

(ii) If the cost and revenue functions of a product are given by

C(x)=3x+400 and R(x)= 5x+20 respectively, where x is the number
of items produced by the manufacturer, then how many items must
be sold to realise some profit?
(a) x<190 (b) x>190
(c) x≥190 (d) x≤190

Q.P.CODE-M111 Page 6
 (iii) Let x and b are real numbers. If b>0 and x<b, then:
(a) x is always positive
(b) x is always negative
(c) x=0
(d) x can be positive or negative

(iv) The solution set of 3x-5<x+7, when x is a whole number is given

by:
(a) {0,1,2,3,4,5} (b) (-∞,6)
(c) [0,5] (d) (6, ∞)

38. A submarine is moving in such a way that at a particular moment of time, its 4x1=
angle of elevation for two ships, situated at different positions on the surface 4 marks
1
of water, is 𝛼 and 𝛽 respectively. If 𝑐𝑜𝑠𝑒𝑐 𝛼 = and sec 𝛽 = √2, then answer
√3
the following questions accordingly:

(i) What is the value of sin 𝛼?

(ii) What is the measure of angle 𝛽?
(iii) What is the value of tan 𝛽?
(iv) What is the measure of cos 𝛼?

Q.P.CODE-M111 Page 7
 BAL NIKETAN PUBLIC SCHOOL
END TERM EXAMINATION (2023-24)
MATHEMATICS
CLASS- XI
DATE: 04.03.2024 DURATION: 3 Hours
NO. OF PAGES: 07 M. MARKS: 80

General Instructions:
1. This Question paper contains five sections- A, B, C, D and E. Each section is
compulsory.
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 of 4 marks each with sub-parts.
7. All questions are compulsory. Although internal choices are given in some questions.

SECTION A (20 marks)

MULTIPLE CHOICE QUESTIONS
Q.No. Questions Marks

1. The symmetric difference of A={1,2,3} and B={3,4,5} is: 1 mark

[A] {1,2} [B] {1,2,4,5} [C] {4,3} [D] {2,5,1,4,3}
2. If R is a relation from a finite set A having m elements to a finite set 1 mark
B, having n elements, then the number of relations from A to B is:
[A] 2mn [B] 2mn-1 [C] mn [D] mn
3. The solution set of inequation |x+3|≥10 is 1 mark
[A] [-13,7] [B] [7, ∞)
[C] [(−∞, −13] ∪ [7, ∞) [D] [(−∞, −13] ∩ [7, ∞)
4. If three dice are thrown simultaneously, then the probability of getting 1 mark
a score of 5 is:
[A] 5/216 [B] 1/6 [C] 1/36 [D] 4/216
5. The set of all points in a plane that are equidistant from a fixed line 1 mark
and a fixed point (not on the line) in the plane is called:

Q.P.CODE-M112 Page 1
 [A] circle [B] ellipse [C] parabola [D] hyperbola
1
6. If z=(2+3𝑖)2 , then |z| = 1 mark

1 1 1
[A] 13 [B] 5 [C] 12 [D] 1
𝑥+2
7. The range of the function f(x)=|𝑥+2| , 𝑥 ≠ −2 is: 1 mark

[A] R-{0} [B] R-{-1,1} [C] {-1,1} [D] {1}

8. The number of subsets of a set containing n elements is: 1 mark
[A] n [B] 2n-1 [C] 2n [D] n2
9. The value of cos 00 cos 10 cos 20 cos 30….. cos 1790 is: 1 mark
[A] 0 [B] 1 [C] -1 [D] none of these
10. The value of (1+i) (1+i2)(1+i3) (1+i4) is: 1 mark
[A] 1 [B] -1 [C] i [D] 0
11. If |x|<6, then 1 mark
[A] –x<-6 [B] -x≤-6 [C] –x>-6 [D] -6<x<6
12. Let R be a relation from a set A to set B then, 1 mark
[A] R=AUB [B] R=A∩B [C] R⊆AxB [D] R⊇AxB
13. Number of diagonals that can be drawn by joining the vertices of an 1 mark
octagaon is:
[A] 20 [B] 28 [C] 8 [D] 20
14. If tan A = 1/2 and tan B = 1/3, then the value of A + B is 1 mark
[A] 𝜋/6 [B] 𝜋/4 [C] 0 [D] 𝜋/3
15. If A={1,2,3,4,5}, then number of proper subsets of A is: 1 mark
[A] 120 [B] 30 [C] 32 [D] 31
16. The value of (28)3 is 1 mark

[A] 21952 [B] 21953 [C] 21954 [D] 21958

17. If ‘p’ is the first term and ‘q’ is the common ratio, then the nth term of a 1 mark
G.P is:
[A] pqn-1 [B] pqn [C] qpn-1 [D] (pq)n
18. The angle between the lines 2x-y+3=0 and x+2y+3=0 is 1 mark
[A] 900 [B] 600 [C] 450 [D] 300

Q.P.CODE-M112 Page 2
 19. Assertion: If A = {1, 2, 3}, B = {2, 4}, then the number of relation from 1 mark
A to B is equal to 26.

Reason: The total number of relation from set A to set B is equal to

2n(A).n(B).

[A] Both A and R are true and R is the correct explanation for A.
[B] Both A and R are true and R is not the correct explanation for A.
[C] A is true but R is false.
[D] A is false but R is true.
20. Assertion: If x2+1=0, then solution is ±𝑖. 1 mark

Reason: The value of i-1097 is equal to i.

[A] Both A and R are true and R is the correct explanation for A.
[B] Both A and R are true and R is not the correct explanation for A.
[C] A is true but R is false.
[D] A is false but R is true.
SECTION B (10 marks)
VERY SHORT QUESTIONS
21. Let U= {1,2,3,4,5,6,7}, A={2,4,6}, B={3,5} and C={1,2,4,7}. Find: 2x1=
2 marks
(i) A′ ∩ (B′UC′)
(ii) (A-B)U(C-A)
22. A and B are events such that P(A)=0.42, P(B)=0.48 and
P(A and B)=0.16. Determine:
(i) P(not A) ½ mark
(ii) P(not B) ½ mark
(iii) P(A or B) 1 mark
OR
Find the probability that when a hand of 7 cards is dealt from a well
2x1=
shuffled deck of 52 cards, it contains: 2 marks
(i) All 4 queens
(ii) Atleast 3 queens
23 Find the distance between (-3,7,2) and (2,4,-1). 2 marks

24. 𝑎+𝑖𝑏 2 marks

If x+iy= 𝑎−𝑖𝑏, then prove that x2+y2=1.
OR
3−√−16
Find the result in a+ib: 2 marks
1−√−9

Q.P.CODE-M112 Page 3
 25. Find (96)5 using Binomial Theorem. 2 marks

SECTION C (18 marks)

SHORT QUESTIONS
26. Let f and g be exponential and logarithmic function. Find: 3x1=
(i) (f+g)(1) 3 marks

(ii) (fg)(1)
(iii) (3f)(1)
27. Find the equation of the line passing through (2,2) and cutting off 3 marks
intercepts on the axes whose sum is 9.

28. Find the sum of the sequence 6,66,666,6666…. 3 marks

OR
Use geometric series to express 0.555… = 0. 5̅ as a rational number? 3 marks

29. A committee of 5 is to be formed out of 6 gents and 4 ladies. In how 2x1½=

many ways this can be done, when 3 marks
(i) Atleast two ladies are included
(ii) Atmost two ladies are included
OR
Find the number of arrangements of the letters of the word 1 mark
INDEPENDENCE. In how many of these arrangements:

(i) Do the word start with P? 1 mark

(ii) Do all the vowels never occur together? 1 mark

30. A solution of 8% boric acid is to be diluted by adding a 2% boric acid 3 marks

solution to it. The resulting mixture is to be more than 4% but less than
6% boric acid. If we have 640 litres of the 8% solution, how many litres
of the 2% solution will have to be added?
31. Find the derivative of cos x from the first principle. 3 marks

SECTION D (20 marks)

LONG QUESTIONS
𝑥 𝑥 𝑥
32. Find sin2, cos2 and tan2 for the following case: 5 marks
1
cos x= − 3, x lies in Quadrant III.

OR

Q.P.CODE-M112 Page 4
 Prove that: 2x2½=
2 𝑥+𝑦 5 marks
(i) (cos 𝑥 + cos 𝑦)2 + (sin 𝑥 − sin 𝑦)2 = 4𝑐𝑜𝑠 2
2 2 2 𝑥−𝑦
(ii) (cos 𝑥 − cos 𝑦) + (sin 𝑥 − sin 𝑦) = 4𝑠𝑖𝑛 2
33. The measurements of the diameters (in mm) of the heads of 107 5 marks
screws are given below:

Diameter 33-35 36-38 39-41 42-44 45-47

(in mm)

No. of 17 19 23 21 27
screws

Calculate the mean, standard deviation and variance.

OR

The mean of 5 observations is 4.4 and their variance is 8.24. If three

of the observations are 1,2 and 6, find the other two observations. 5 marks

34. Evaluate:
(i) lim 𝑥 + 3
𝑥→3 1 mark
𝑥 2 −4
(ii) lim ( ) 2 marks
𝑥→2 𝑥 3 −4𝑥 2 +4𝑥
𝑥 10 −1
(iii) lim ( 𝑥 8 −1 ) 2 marks
𝑥→1

35. Solve:

(i) For the ellipse 3𝑥 2 + 2𝑦 2 = 6, find the length of minor axis,

3 marks
length of major axis, eccentricity, coordinates of vertices
and foci.

(ii) Find the equation of hyperbola, whose conjugate axis is 5 2 marks

and the distance between the foci is 13.

SECTION E (12 marks)

CASE-BASED QUESTIONS
36. Rahul and Ravi planned to play Business Board game with two dice. 4x1=
In which they will get the turn one by one, they roll the dice and 4 marks
continue the game.

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 Answer the following questions accordingly:

(i) Ravi got first chance to roll the dice. What is the probability that
he got the sum of the two numbers appearing on the top face of
the dice as 8 ?

(ii) Rahul got next chance, what is the probability that he got the sum
of the two number appearing on the top face of the dice as 13 ?

(iii) Now it was Ravi's turn. He rolled the dice. What is the probability
that he got the sum of the two numbers appearing on the top
face of the dice is less than or equal to 12 ?

(iv) Rahul got next change. What is the probability that he got the
sum of the two numbers appearing on the top face of the dice is
equals to 7 ?

37. Shweta was teaching “method to solve a linear inequality in one 4x1=
variable” to her daughter. 4 marks

Step 1: Collect all terms involving the variable (x) on one side and
constant terms on other side. With the help of this rules reduce it in
the form of ax<b or ax≤b or ax>b or ax≥b.
Step 2: Divide this inequality by the coefficient of variable (x). This
gives the solution set of given inequality.
Step 3: Write the solution set.

Choose the correct options accordingly:

(i) The solution set 24x<100, when x is a natural number is:
(a) {1,2,3,4} (b) (1,4)

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 (c) [1,4] (d) {1,2,3}

𝑥 𝑥
(ii) The solution set of 𝑥 + 2 + 3 < 11 is:
(a) (−∞, 6] (b) (−∞, 6)
(c) (6, ∞) (d) [6, ∞)

(iii) The solution set of −5𝑥 + 25 > 0 is:

(a) (−5, ∞) (b) (−∞, −5)
(c) (5, ∞) (d) (−∞, 5)

(iv) The solution set of 3𝑥 − 5 < 𝑥 + 7 is:

(a) (−6, ∞) (b) (−∞, −6)
(c) (6, ∞) (d) (−∞, 6)

38. A submarine is moving in such a way that at a particular moment of 4x1=

time, its angle of elevation for two ships, situated at different positions 4 marks
on the surface of water, is 𝛼 and 𝛽 respectively. If 𝑐𝑜𝑠𝑒𝑐 𝛼 = √3 and
sec 𝛽 = 2, then answer the following questions accordingly:

(i) What is the value of sin 𝛼?

(ii) What is the measure of angle 𝛽?
(iii) What is the value of tan 𝛽?
(iv) What is the measure of cos 𝛼?

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