Roll No.

D.A.V. PUBLIC SCHOOL, AUNDH, PUNE.

UNIT TEST –I (2023-24)
STD: XII SUBJECT – MATHEMATICS MAX MARKS: 50
DATE: 19.07.23 TIME: 2 HRS
__________________________________________________________________________

GENERAL INSTRUCTIONS:
• This question paper contains five sections A, B, C, D and E. Each section is compulsory.
However, there are internal choices in some questions.
• Section A consists of 12 questions (10 MCQs and 02 Assertion – Reason based questions) of 1
mark each.
• Section B consists of 4 Short answers questions of 2 marks each.
• Section C consists of 4 Short answer questions of 3 marks each.
• Section D consists of 2 long answer questions of 5 marks each.
• Section E consists of 2 source based/ Case Study based questions/passage based questions (4
marks each) with sub – parts.
____________________________________________________________________________________

SECTION – A (12 X 1 = 12 marks)

1. If A is a 3 × 3 matrix such that A(adj A) = 10 I, then |𝑎𝑑𝑗 𝐴| is equal to

a)1 b) 10 c) 100 d) 101
𝑥 5 𝜋
2. If sin−1 5 + 𝑐𝑜𝑠𝑒𝑐 −1 4 = 2 , then the value of x is
a) 1 b) 3 c) 4 d) 5
3. If A is a square matrix of order 3, such that |𝐴| = 8, then |3𝐴| equals
a) 8 b) 24 c) 72 d) 216
√4+𝑥−2
4. If𝑓(𝑥) = , 𝑥 ≠ 0 be continuous at x = 0, then 4𝑓(0) is equal to
𝑥
1 1 3
a)2 b)4 c) 1 d)2
5𝜋
5. tan−1 (√2 sin ) is equal to
4
𝜋 𝜋 𝜋 𝜋
a) b) − 4 c) 6 d) − 6
4
cos 𝜃 − sin 𝜃 𝜋
6. If𝐴 = [ ], then find the value of 𝜃 (𝜃 ∈ (0, ))satisfying the equation 𝐴𝑇 + 𝐴 = 𝐼2
sin 𝜃 cos 𝜃 2
𝜋 𝜋 𝜋
a) 2 b) 6 c) 3 d) 𝜋
0 2 0 3𝑎
7. If 𝐴 = [ ] 𝑎𝑛𝑑 𝑘𝐴 = [ ], then the value of k, a and b respectively are
3 −4 2𝑏 24
a) -6, -12, -18 b) -6, -4, -9 c) -6, 4, 9 d) -6, 12, 18
𝑑𝑦
8. If𝑓 ′ (1) = 2 𝑎𝑛𝑑 𝑦 = 𝑓(log 𝑒 𝑥), find 𝑎𝑡 𝑥 = 𝑒.
𝑑𝑥
2 3 1 2
a) b) c) d)
𝑒 𝑒 𝑒 5𝑒
9. If A is skew – symmetric matrix, then A2 is
a)symmetric matrix b) null matrix c)skew symmetric matrix d)none

𝑑𝑦
10. If 2 𝑥 + 2𝑦 = 2 𝑥+𝑦 , then the value of 𝑎𝑡 𝑥 = 𝑦 = 1 is
𝑑𝑥
a) -1 b) 0 c) 1 d) 2

XII|MATHEMATICS 1|Page
 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.

11. Assertion (A): If A is a 3 × 3 non – singular matrix, then|𝐴−1 𝑎𝑑𝑗 𝐴| = |𝐴|.

Reason (R): If A and B both are invertible matrices such that B is inverse of A, then AB = BA = I.

33 𝜋 𝜋
12. Assertion (A): The value of sin−1 (cos ) 𝑖𝑠 − 10 .
5
Reason(R): The principal value branch of cos −1 𝑥 is [0, 𝜋].

SECTION – B(4 X 2 = 8 Marks)

1 2 0 0
13. For what value of x: [1 2 ]
1 2[ 0 1] [2] = 0 ?
1 0 2 𝑥

14. If the area of the triangle with vertices A(x, 4), B (-2, 4) and C(2, -6) is 35 square units, find x.
OR
𝑥 sin 𝜃 cos 𝜃
Prove that the determinant |− sin 𝜃 −𝑥 1 | is independent of 𝜃.
cos 𝜃 1 𝑥

√1+𝑥 2 −1
15. Write the function: tan−1 ( ) , 𝑥 ≠ 0 in the simplest form.
𝑥
−1
16. Draw the graph of 𝑓(𝑥) = sin 𝑥 , 𝑥 ∈ [−1, 1]. Also write the range of f (x).

SECTION – C (4 X 3 = 12 Marks)

√1+sin 𝑥 + √1−sin 𝑥 𝑥 𝜋
17. Prove that cot −1 [ ] = 2 , 𝑥 ∈ (0, 4 )
√1+sin 𝑥 − √1−sin 𝑥
OR
−1 3
Solve for x: cos(tan 𝑥) = sin (cot −1 4)

2 −1 −1 −8 −10
18. Find the matrix X so that [ 1 0 ] 𝑋 = [ 1 −2 −5 ]
−3 4 9 22 15

19. Examine the differentiability of f, where f is defined by

𝑥[𝑥], 𝑖𝑓 0 ≤ 𝑥 < 2
𝑓(𝑥) = { at x = 2.
(𝑥 − 1)𝑥 , 𝑖𝑓 2 ≤ 𝑥 < 3

4 5
20. If 𝐴 = [ ], show that 𝐴 − 3𝐼 = 2(𝐼 + 3𝐴−1 )
2 1

XII|MATHEMATICS 2|Page
 SECTION – D(2 X 5 = 10 Marks)
0 1 1
𝐴2 −3𝐼
21. Find 𝐴−1 if 𝐴 = [1 0 1] and show that 𝐴−1 = 2
1 1 0
OR
2 2 −4 1 −1 0
Given 𝐴 = [−4 2 −4] and 𝐵 = [2 3 4], then find BA and use this product to solve the
2 −1 5 0 1 2
system of equations 𝑦 + 2𝑧 = 7, 𝑥 − 𝑦 = 3 𝑎𝑛𝑑 2𝑥 + 3𝑦 + 4𝑧 = 17.
22. (i) Find the value of a and b for which the function
𝑥+sin 𝑥
, 𝑖𝑓 − 𝜋 < 𝑥 < 0
sin{(𝑎+1)𝑥}
𝑓(𝑥) = 2 , 𝑖𝑓 𝑥 = 0 is continuous at x = 0. (3)
𝑒 sin 𝑏𝑥 −1
2 𝑏𝑥 , 𝑖𝑓 𝑥 > 0
{
𝜋 3𝜋
(ii) Differentiate w.r.t. x: (sin 𝑥 − cos 𝑥)(sin 𝑥−cos 𝑥) , <𝑥< . (2)
4 4
SECTION – E (2X 4 = 8 marks)
Case-based questions (Q23 to Q24)
This section comprises of 2 case study / passage based questions of 4 marks each. First case
study question has two subparts of 2 marks each. Second case study question has three
subparts of marks 1,1, 2 respectively
𝑑𝑥 𝑑𝑦
23. Suppose 𝑥 = 𝑓(𝑡) and 𝑦 = 𝑔(𝑡). Then 𝑑𝑡 = 𝑓′(𝑡) and 𝑑𝑡 = 𝑔′(𝑡).
𝑑𝑦
𝑑𝑦 𝑑𝑦 𝑑𝑡 𝑔′(𝑡)
Therefore, 𝑑𝑥 = 𝑑𝑡
𝑑𝑥 = × 𝑑𝑥 = 𝑓′(𝑡) 𝑤ℎ𝑒𝑟𝑒 𝑓′(𝑡) ≠ 0
𝑑𝑡
𝑑𝑡

Based on the above information, answer the following questions:

𝜋
(i) Find the derivative of 𝑓(tan 𝑥) 𝑤. 𝑟. 𝑡. 𝑔(sec 𝑥)at 𝑥 = 4 where 𝑓 ′ (1) = 2 and 𝑔′ (√2) = 4.
𝑑2𝑦
(ii) Let 𝑥 = 10(𝑡 − sin 𝑡) and 𝑦 = 12(1 − cos 𝑡). Then find 𝑑𝑥 2
24. Manjit wants to donate a rectangular plot of land for a school in his village. When he asked to give
dimensions of the plot, he told that if its length is decreased by 50 m and breadth is increased by 50 m,
then its area will remain same, but if length is decreased by10 m and breadth is decreased by 20 m, then
its area will decrease by 5300 m2.

Based on the above information, answer the following questions

(i) Write the equations in terms of x and y.
(ii) Write matrix equation (in two variables) represented by those equations.
(iii) Find the values of x and y using Matrix method.

XII|MATHEMATICS 3|Page