Fahaheel Al-Watanieh Indian Private School, Ahmadi, Kuwait

Model Examination 2023-24 Set A

Mathematics (041)
Class XII
Time: 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 of 4 marks
each with sub-parts.

SECTION A

1 2 1
1) If 𝐴 = 2 3 1 is a non-singular matrix and a∈ 𝐴 then set A is
3 𝑎 1
a) R b) {0} c) {4} d) R-{4}

2) If |A| = |kA| where A is a square matrix of order 2, then the sum of all possible values of
k is

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

, 𝑖𝑓 𝑥 < 0
3) If 𝑓(𝑥) = | | is continuous at x = 0, then the value of k is
3, 𝑖𝑓 𝑥 ≥ 0
a) -3 b) 0 c) 3 d) any real number
4) For a square matrix A, 𝐴 − 𝐴 + 𝐼 = 0, then 𝐴 equals
a) A b) A+I c) I-A d) A -I
5) The integrating factor of the differential equation 𝑥 − 𝑦 = 2𝑥 is

a) 𝑒 b) 𝑒 c) x d)

6) The function 𝑓(𝑥) = 𝑥|𝑥| is a

a) Continuous and differentiable at x = 0.
b) continuous but not differentiable at x = 0
c) differentiable but not continuous at x= 0
d) neither differentiable nor continuous at x = 0

XII Model Exam 2023-24 [Set A] Page 1 of 7 Mathema cs 041

 7) If A is a square matrix of order 3 and |A|= -2 then |adj(2A)| is equal to
a) -26 b) 4 c) -28 d) 28
8) ∫ 𝑑𝑥 is equal to

a) 𝑠𝑒𝑐 −𝑥 +𝑐 b) −𝑠𝑒𝑐 −𝑥 +𝑐 c) log | sec −𝑥 +𝑐

𝑑) −log | sec −𝑥 +𝑐

9) If A and B are invertible matrices, then which of the following is incorrect

| |
a) |𝐴𝐵 | = | |
b) |𝐴𝐵 |=
| || |
c) (𝐴𝐵) =𝐵 𝐴

d) (𝐴 + 𝐵) =𝐵 +𝐴

10) Find the sum of degree and order of the differential equation [1 + (𝑦′) ] = (𝑦 ) is
a) 5 b) 4 c) 3 d) 6
11) If 𝜃 is the angle between the vectors 𝑎⃗ and 𝑏⃗, then 𝑎⃗. 𝑏⃗ ≥ 0 only when
a) 0 < 𝜃 < b) 0 ≤ 𝜃 ≤ c) 0 < 𝜃 < 𝜋 d) 0 ≤ 𝜃 ≤ 𝜋

12) The value of 𝜆 for which the vectors 3𝚤⃗ − 6𝚥⃗ + 𝑘⃗ and 2𝚤⃗ − 4𝚥⃗ + 𝜆𝑘⃗ are parallel is

a) b) c) d)

13) The lines 𝑟⃗ = 𝚤⃗ + 𝚥⃗ − 𝑘⃗ + 𝜆(2𝚤⃗ + 3𝚥⃗ − 6𝑘⃗ ) and 𝑟⃗ = 2𝚤⃗ − 𝚥⃗ − 𝑘⃗ + 𝜆(6𝚤⃗ + 9𝚥⃗ − 18𝑘⃗ )
a) Coincident b) skew c) intersecting d) parallel
14) If a line makes angles 𝛼, 𝛽, 𝛾 with the coordinate axes, then find the value of
cos 2𝛼 + cos 2𝛽 + cos 2𝛾.
a) 1 b) -1 c) 0 d) none of these
15) The probability that A speaks truth is and that of B speaks truth is . The probability that

they contradict each other in stating the same fact is

a) b) c) d)

16) If 𝑎⃗ and 𝑏⃗ are such that 𝑎⃗ + 𝑏⃗ = |𝑎⃗ − 𝑏⃗ | then

a) 𝑎⃗ ⊥ 𝑏⃗ b) 𝑎⃗ ∥ 𝑏⃗ c) 𝑎⃗ = 𝑏⃗ d) none of these
17) The objective function 𝑍 = 𝑎𝑥 + 𝑏𝑦 has maximum value 42 at (4,6) and minimum value at
19 at (3,2). Which of the following is true
a) a = 9, b = 1 b) a = 5, b = 2 c) a = 3, b = 5 d) a = 5 , b = 3

XII Model Exam 2023-24 [Set A] Page 2 of 7 Mathema cs 041

 18) The shape of the feasible region formed by the constraints 𝑥 + 𝑦 ≤ 2, 𝑥 + 𝑦 ≥ 5, 𝑥, 𝑦 ≥ 0
is
a) No feasible region b) Triangular region c) Unbounded region d) Trapezium

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.

19) Assertion(A): The function 𝑓(𝑥) = [𝑥(𝑥 − 2)] is increasing in (0,1) ∪ (2, ∞).

Reason (R): 𝑓 (𝑥) = 0 𝑤ℎ𝑒𝑛 𝑥 = 0, 1, 2

20) Assertion (A): The relation 𝑓: {1,2,3,4} → {𝑥, 𝑦, 𝑧, 𝑝} defined by {(1, 𝑥), (2, 𝑦), (3, 𝑧)} is a
bijective function.
Reason (R): The function 𝑓: {1,2,3} → {𝑥, 𝑦, 𝑧, 𝑝} such that 𝑓 = {(1, 𝑥), (2, 𝑦), (3, 𝑧) is one-
one.
SECTION B
21) a) Find the domain of 𝑦 = sin (𝑥 − 4)

OR

b) sin 𝑠𝑖𝑛 + cos (cos 𝜋) + tan (1)

22) a) The median of an equilateral triangle is increasing at the rate of 2√3𝑐𝑚/𝑠.

Find the rate at which its side is increasing.

OR
b) A particle moves along the curve 3𝑦 = 𝑎𝑥 + 1, such that at a point with x -coordinate 1
and y -coordinate is changing twice as fast as x -coordinate. Find the value of a.

23) Find the intervals in which the function 𝑓: 𝑅 → 𝑅 𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑏𝑦 𝑓(𝑥) = 𝑥𝑒 , is increasing.

XII Model Exam 2023-24 [Set A] Page 3 of 7 Mathema cs 041

 24) Find the maximum value of the function 𝑓(𝑥) = 𝑠𝑖𝑛𝑥 + √3𝑐𝑜𝑠𝑥 in the interval [0,𝜋].

25) Evaluate ∫ 𝑑𝑥

SECTION C

26) If (𝑎 + 𝑏𝑥)𝑒 = 𝑥 then prove that 𝑥 =( ) .

27) a) Find ∫ 𝑒 𝑑𝑥

OR

b) Find ∫ ( )( )
𝑑𝑥

28) Evaluate ∫ log(1 + 𝑡𝑎𝑛𝑥) 𝑑𝑥

29) a) Find the general solution of the differential equation:

(𝑥𝑦 − 𝑥 )𝑑𝑦 = 𝑦 𝑑𝑥
OR

b) Find the general solution of the differential equation:

(𝑥 + 1) + 2𝑥𝑦 = √𝑥 + 4

30) Solve the following linear programming problem graphically:

Minimize 𝑍 = 5𝑥 + 10𝑦
Subject to the constraints: 𝑥 + 2𝑦 ≤ 120 , 𝑥 + 𝑦 ≥ 60 , 𝑥 − 2𝑦 ≥ 0 , 𝑥 ≥ 0 , 𝑦 ≥ 0

31) From a lot of 30 bulbs which include 6 defective bulbs, a sample of 2 bulbs is drawn at random
one by one with replacement. Find the probability distribution of number of defective bulbs
and hence find the mean number of defective bulbs.

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 SECTION D
1 1 2 
32) Find the inverse of the matrix A  0 2 3 Using the inverse, A 1 , solve the system of
 3 2 4 
linear equations x  y  2z  1; 2y  3z  1;3x  2y  4z  3

33) Using integration, find the area of the region bounded by the circle x 2 +y 2 =16, line y = x and y-
axis, but lying in the 1st quadrant.

34) a) A function 𝑓: [−4,4] → [0,4] is given by 𝑓(𝑥) = √16 − 𝑥 . Show that f is an onto function
but not a one-one function. Further, find all possible values of a for which f(a) = √7.
OR

b) Let N be the set of all natural numbers and R be the relation on 𝑁 × 𝑁 defined by
(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. Also, find the equivalence class of (2,6).

x  11 y  2 z  8
35) Find the image of the point (2,  1, 5) in the line   .
10 4 11

SECTION E

CASE STUDY-1

36) 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.

XII Model Exam 2023-24 [Set A] Page 5 of 7 Mathema cs 041

 (i) Based on above information find out how much amount can be increased
for maximum revenue.
(ii) Find out maximum revenue received by the telephone company

CASE STUDY -2
37) Gitika house is situated at Shalimar Bag at O, going to Alok’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 school she travels by bus to reach
Alok’s house which is 300 of east and 6 km from point B.

(i) What is vector distance from Gitika’s house to school?

(ii) What is vector distance from school to Alok’s house?
(iii) What is vector distance from Gitika’s house to Alok’s house?

XII Model Exam 2023-24 [Set A] Page 6 of 7 Mathema cs 041

 CASE STUDY-3
38) Recent studies suggest that roughly 12% of the world population is left handed.

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