THE INDIAN SCHOOL, BAHRAIN

FIRST TERM EXAMINATION – JUNE 2023

CLASS XII SUB: MATHEMATICS

TIME : 3 HRS MAX. MARKS: 80
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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.
7. This question paper contains 6 printed pages.
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SECTION A (Multiple Choice Questions)
Each question carries 1 mark
1. A function f : R → R given by f(x) = 2 + x2 is
a)one-one but not onto b) one-one
c) onto d) neither one-one nor onto
2. An equivalence relation R on the set X divides it into equivalence classes A, B and C. then
A  B  C is
a) R b) X c)  d) A
3. If the set A contains 5 elements and the set B contains 6 elements, then the number of
mappings from A to B which are both one – one and onto is:
a) 0 b) 120 c) 720 d) None of these
4. Consider the nonempty set consisting of children in a family and the relation R defined as aRb
if a is a brother of b. Then R is
a) Symmetric but not transitive b) transitive but not symmetric

c) Neither symmetric nor transitive d) both symmetric and transitive

3
5. The value of tan −1 (tan ) is
4
3  − 3
a) b) c) d) −
4 4 4 4

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  −1  −1
6. The value of sin  − sin    is
3  2 
1 1 1
a) b) c) d) 1
2 3 4
7. The Domain of cos-1(2x – 1) is
a) R b) (-1,1) c) [-1 ,1] d) [0 , 1 ]
8. Given that A = [𝑎𝑖𝑗] is a square matrix of order 3×3 and |A| = 5, then the value of
∑ ai 2 Ai 3 , 𝑖=1, 2 and 3 , where 𝐴𝑖𝑗 denotes the cofactor of element 𝑎𝑖𝑗 is:
a) 0 b) 5 c) -5 d) 10
3x 4 
9. The value of x, for which A =   is a singular matrix is:
 6 2 x
a) 2 b) -2 c) ±2 d) 0
2
10. Given that A is a non-singular matrix of order 2 such that A = 3A, then value of |2A| is:
a) 6 b) 36 c) 9 d) 18

11. If A and B are square matrices of order 2 such that A = 3 and B = 2, find 2 AB

a) 6 b) 24 c) 12 d) 48
𝛼2
12. If 𝐴 = [ ] 𝑎𝑛𝑑 |𝐴|3 = 125, then the possible value(s) of 𝛼 is/are
2𝛼
a) 5 b) 125 c)  3 d)  5
 k sin 3x
 + cos x if x  0
 x
13. The function f(x)=  if x = 0 is continuous at x = 0, then the value of k is
 4

a) 1 b) 4 c) -1 d) -4
5 0 0 
14. If A. (adjA) = 0 5 0 , then A + adjA
0 0 5
a) 5 b) 25 c) 30 d) 10
15. If A and B are skew symmetric matrices of same order, AB is symmetric if
a) AB = O b) AB = - BA c) AB = BA d) BA = O
x x  0
2
16. The value of k for which the function f(x) =  is differentiable at x = 0 is:
 kx x  0
a) 1 b) 2 c) any real number d) 0
17. If A and B are square matrices of the same order, and If ( A + B) 2 = A2 + B2, then:
a) AB = BA b) AB = - BA c) AB = O d) BA = O
  cos x − sin x 
18. For what value of x  0,  , is A + AT = 3 I , where A =  
 2  sin x cos x 
  
a) b) c) 0 d)
2 6 2

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 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) : If n(A) = 5, n(B) = 5 then f : A → B ,then f is one –one and onto

Reason (R) : If n(A) = n(B), then every one – one function from A to B is onto
20. Assertion (A) : All continuous functions are differentiable

Reason (R): The function f(x) = x is continuous in R

SECTION B
This section comprises of very short answer type-questions (VSA) of 2 marks each
21. Prove that the diagonal elements of a skew symmetric matrix are 0
22. If A and B are skew symmetric matrices, prove that AB + BA is symmetric
OR

 0 2b − 2
The matrix A =  3 1 3  is given to be a symmetric matrix, Find a and b

3a 3 − 1

(
23. Simplify sin −1 2 x 1 − x 2 ) 1
2
 x 1

 3  3
24. Evaluate 3 sin   + 2 cos −1   + cos −1 (0)
−1
  
 2   2 
OR

Draw the graph of f(x) = cos-1x , x  [−1, 1] . Also write the Range of f

1 − cos 4 x
 if x  0
25. If the function f(x) defined as f(x) =  x2 be continuous, find a
 a if x = 0

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 SECTION C
(This section comprises of short answer type questions (SA) of 3 marks each)
 4 −3 4 
26. Express the matrix  5 1 − 1 as a sum of a symmetric and skew symmetric matrix

− 3 − 3 1 
OR
1 5 
If A is a symmetric matrix and B is a skew symmetric matrix such that A + B =  
3 − 2
then find AB

 3 1  2 − 1 -1 -1 -1
27. If A =   B = − 1 4  verify that (AB) = B A
 − 1 2   
3 2 
28. If A =   , then find A-1 and use it to solve the following system of equations
5 − 7
3x + 5y = 11; 2x – 7y = -3
29. Prove that the relation R on the set N X N defined by (a, b) R (c, d)  a+ d = b + c is an equivalence
relation for (a, b), (c, d) in N  N

 1 + cos x + 1 − cos x  3
30. Find the value of tan −1    x
 1 + cos x − 1 − cos x  ; 2
 
 2k cos x 
  − 2 x if x  2 
31. If f(x) =  is continuous at ,find k
5
 2
if x =
 2
OR
Check if the function f(x) = 3x - x is differentiable at x = 0

SECTION D

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

1 − 2 0   7 2 − 6
   
32. If A 2 1 3 and B = − 2 1 − 3 find AB , Hence, solve the system of equations
   
0 − 2 1 − 4 2 5 

x – 2y = 10; 2x + y + 3z = 8 ; -2y + z = 7,

OR

1 0 2 
If A = 0 2 1, then show that A3 – 6A2 + 7A + 2 I = O, and hence find A-1
2 0 3

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  0 − tan   cos 2 − sin 2 
33. If A =  Prove that I + A =( I – A) where I is a unit matrix of
 tan  0   sin 2
 cos 2 

order 2
x
34. Check if the function f : R → R defined by f(x) = x  R is one- one , onto or not
x +1
2

OR
Let f : R → [7 , ) given by f(x) = 16 x 2 + 24 x + 7 , where R + is the set of positive real numbers.
+

Prove that f is one – one and onto

2ax + b if x  2

35. If the function f(x) =  7 if x = 2 Is continuous at x =1 find the values of a and b
5ax − b if x  2


SECTION E

(This section comprises of 3 case-study/passage-based questions of 4 marks each 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. Case – Study 1: Read the following passage and answer the questions given below:

A manufacturer produces three stationery products Pencil, Eraser, and Sharpener which he sells in two
markets. Annual sales are indicated below:
Market Pencil Eraser Sharpener
A 10,000 2,000 18,000
B 6000 20,000 8,000

The unit sale price of Pencil, Eraser and Sharpener are ₹2.50, ₹ 1.50, ₹1.00 the unit cost of the above
three commodities are ₹2.00, ₹1.00, ₹0.50 respectively. Using matrices, Find,
a) Total Revenue in Market A (1)
b) Total Revenue in Market B (1)
c) Total Profit In Market A (2)
OR
Total Profit in Market B (2)

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 37. Case- Study 2:

The general election of 2019 was a gigantic exercise, About 911 million people were eligible to vote
and the voter turnout was 67%, the highest ever. Let I be the set of Indian citizens who were eligible
to vote. Let R be a relation on I defined as R = { (a, b): a, b  I and both use their voting rights in
General election 2019}

a) Check if R a symmetric relation (1)

b) Check of R a transitive relation (1)
c) Check if R an equivalence relation (2)

38. Case- Study 3:

Two People Ram and Rahim, buy shirts and trousers from a shop to distribute to poor children
during festival season. Ram buys 20 shirts and 5 trousers for a total of ₹9,000 and Rahim buys, 25
shirts and 5 trousers for a total of ₹10,500.

Based on the above information answer the following questions

i) If ₹x and ₹y are the unit price of shirts and trousers respectively, then express the
above information as a system of linear equations
ii) Use matrix method to find the unit price of shirts and trousers.

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