Class XII

Session 2022-23
Mathematics
Test Paper - 7
Time Allowed: 3 Hours Maximum Marks: 70

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 20 MCQ’s 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)

Q1. Roots of 3x + 3 – x = 10/3 are-

(A) 0, 1 (B) 1, –1 (C) 0, –1 (D) None of these

Q2. If |z – 2|  |z – 4| then correct statement is-

(A) R (z)  3 (B) R(z)  3 (C) R(z)  2 (D) R(z)  2

Q3. If in a ABC, cosA / a = cosB / b = cosC / c , then the triangle is ,

(A) right angled (B) isosceles (C) equilateral (D) obtuse

Q4. If the line x – 1 = 0 is the directrix of the parabola y2 – kx + 8 = 0, then one of the values of ‘k’
is

(A) 1/8 (B) 8 (C) 4 (D) 1/4

Q5. The length of intercept on y-axis, by a circle whose diameter is the line joining the points (–
4,3) and (12, –1) is

(A) 3 √2 (B) √13 (C) 4 √13 (D) 3 √3

Page 1
Q6. Number of 3 digit numbers in which the digit at hundreth’s place is greater than the other two
digit is.

(A) 285 (B) 281 (C) 240 (D) 204

Q7. limx→ (1 – x + [x – 1] + [1 – x]) is (where [ *] denotes greatest integer function)

(A) 0 (B) 1 (C) –1 (D) does not exist

Q8. Range of f(x) = 4 x + 2x + 1 is

(A) (0, ) (B) (1, ) (C) (2, ) (D) (3, )

Q9. Which of the following statement is true?

a)3+7=4 or 3-7=4
b) if pune is in Maharashtra, then Hyderabad is in Kerala.
c) it is false that 12 is not divisible by 3.
d) the square of any odd integer is even.

1 2 
Q10. A=   and a (adj a) = ki,then the value of ‘k’is
3 4 
a)2 b) -2 c)10 d)-10

Q11. The principal solution of the equation cot x = − 3 is

   
a) b) c) d) −
6 3 6 6

Q12. Which of the following is a contradiction

(a) (p  q) ~ (p  q) (b) p  (~ p  q)
(c) (p  q)  p (d) None of these

 4 x + 2
Q13. If A =   is symmetric, then x =
 −3
2 x x + 1
(a) 3 (b) 5 (c) 2 (d) 4

Q14. If A and B are square matrices of order n×n, then ( A − B)2 is equal to
(a) A2 − B2 (b) A 2 − 2 AB + B 2
(c) A + 2 AB + B
2 2
(d) A2 − AB − BA + B2

Q15. Matrix A is such that where I is the identity matrix. Then for

(a) nA − (n − 1)I (b) nA − I

(c) 2n −1 A − (n − 1)I (d) 2 n−1 A − I

Page 2
  0 0 − 1
 
Q16. Let A =  0 − 1 0 , the only correct statement about the matrix A is
− 1 0 0 
(a) A =I
2
(b) A = (−1)I , where I is unit matrix
(c) A −1 does not exist (d) A is zero matrix

Q.17 If 3, – 2 are the Eigen values of non-singular matrix A and

|A| = 4. Then Eigen values of adj(a) are
(a) 3/4, –1/2 (b) 4/3, –2 (c) 12, –8 (d) –12, 8

1
Q18. If x + = 2 , the principal value of sin–1 x is
x
  3
a) b) c)  d)
4 2 2

 
Q19. The general value of  satisfying the equation tan  + tan  −   = 2, is
2 
 
a) n  b) n +
4 4
 
c) 2n  d) n + (−1) n
4 4
Q20. The solution of the equation
4cos2 x + 6sin2 x = 5,
 
a) x = n  b) x = n 
2 4
3 3
c) x = n  d) x = n 
2 4
SECTION – B

This section comprises of very short answer type-questions (VSA) of 2 marks each
 x 2 + 3x + k
 ; x 1
 2(x − 1)
2
Q21. If f(x) = is continuous at
 5
 ; x =1
4
x = 1, then find k?

Q22. If the function

 x 2 − (A + 2)x + A
 , for x  2
f(x) =  x−2 is
 =2 forx = 2

is continuous at x = 2, then find A?

Page 3
 x −a
Q23. If f(x) = is continuous at x = a, then f(a) is equal to:
x− a

dy
Q24. If y = sec (tan–1 x), then find ?
dx
xf (2) − 2f (x)
Q25. If f (2) = 4, f’(2) = 1, then =
x−2

SECTION – C

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

Q26. The perimeter of a triangle with sides 3i + 4 j + 5 k, 4 i − 3 j − 5k and 7i + j is.

Q27. If one side of a square be represented by the vector 3i + 4 j + 5 k, then the area of the square is.

OR

If | a | = 3, | b | = 4 and | a + b | = 5, then | a − b | = .

Q28. If OP = 8 and OP makes angles 45 o and 60 o with OX-axis and OY-axis respectively, then
OP = .

Q29. Equation of XY-plane is

x y z
Q30. The angle between the line = = and the plane 3x + 2y – 3z = 4 is
2 3 4

Q31. If the distance of the point (1 1 1) from the origin is half its distance from the plane x + y + z + k
= 0, then k =

SECTION – D

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

x2
Q32. Integrate the following w.r.t x dx
( x 2 + 2)(2 x 2 + 1)
 2
 2 − sin x 
Q33. Evaluate the following
−

 2
log  dx
 2 + sin x 
Q34. Find the area of the sector of the circle bounded by x + y = 16 and the line y = x in the first
2 2

quadrant.
Q35. Find the area of the region lying between the parabolas y = 4ax and x = 4ay
2 2

Page 4
 SECTION - E

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

Q36. The fuel cost per hour for running a train is proportional to the square of the speed it generates in
km per hour. If the fuel costs ₹ 48 per hour at speed 16 km per hour and the fixed charges to run
the train amount to ₹ 1200 per hour. Assume the speed of the train as 𝑣 km/h.
Based on the given information, answer the following questions

(i) Given that the fuel cost per hour is 𝑘 times the square of the speed the train generates in km/h, the
value of 𝑘 is:
a) 16 / 3 b) 1 / 3 c) 3 d) 3 / 16

(ii) If the train has travelled a distance of 500km, then the total cost of running the train is given by
function:
(a) (15 / 16) 𝑣 + 600000 / 𝑣 (b) (375 / 4) 𝑣 + 600000 / 𝑣
(c) (5 / 16) 𝑣 2 + 150000 / 𝑣 (d) (3 / 16) 𝑣 + 6000 /v

(iii) The most economical speed to run the train is:

(a) 18km/h (b) 5km/h (c) 80km/h (d) 40km/h

Q37. An insurance company believes that people can be divided into two classes: those who
are accident prone and those who are not. The company’s statistics show that an
accident-prone person will have an accident at sometime within a fixed one-year period
with probability 0.6, whereas this probability is 0.2 for a person who is not accident
prone. The company knows that 20 percent of the population is accident prone.

Page 5
Based on the given information, answer the following questions.

(i)what is the probability that a new policyholder will have an accident within a year of purchasing a
policy?

(ii) Suppose that a new policyholder has an accident within a year of purchasing a policy. What is the
probability that he or she is accident prone?

Q38. In linear algebra, the determinant is a scalar value that can be computed from the elements of a
square matrix and encodes certain properties of the linear transformation described by the
matrix. The determinant of a matrix A is denoted det(A) or |A|. Using determinants/ Matrices
calculate the following:
Ram purchases 3 pens, 2 bags, and 1 instrument box and pays ₹ 41. From the same shop,
Dheeraj purchases 2 pens, 1 bag, and 2 instrument boxes and pays ₹29, while Ankur purchases 2
pens, 2 bags, and 2 instrument boxes and pays ₹44.

i. Read the above information and answer the following questions: Find the cost of one pen.
a. ₹ 2
b. ₹ 5
c. ₹ 10
d. ₹15
ii. What are the cost of one pen and one bag?
a) ₹ 12
b) ₹ 15
c) ₹ 17
d) ₹25

iii. What is the cost of one pen & one instrument box?
a) ₹ 7
b) ₹ 12
c) ₹ 17
d) ₹25

Page 6