Class XI Session 2023-24

Subject - Mathematics
Sample Question Paper - 3

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
1. If cot θ = and sec ϕ = , where θ lies in quadrant III and ϕ lies in quadrant II, then tan (θ + ϕ) = ? [1]
1 −5

2 3

a) b)
−6 5

11 11

c) d)
2 10

11 11

2. Number of relations that can be defined on the set A = {a, b, c, d} is [1]

a) 24 b) 44

c) 16 d) 216

3. The number of telephone calls received in 245 successive, one-minute intervals at an exchange is given below in [1]
the following frequency distribution.

Number 0 1 2 3 4 5 6 7

Frequency 14 21 25 43 51 40 39 12

What is the median of the distribution?

a) 3.5 b) 5

c) 4 d) 4.5
4. lim
sin x

x−π
is equal to [1]
x→π

a) 1 b) -1

c) 2 d) -2
5. A line cutting off intercept – 3 from the y-axis and the tengent at angle to the x-axis is 3
, its equation is [1]
5

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 a) 5y – 3x + 15 = 0 b) 5y – 3x – 15 = 0

c) None of these d) 3y – 5x + 15 = 0
6. The distance of point P(3, 4, 5) from the yz-plane is [1]

a) 550 b) 5 units

c) 3 units d) 4 units
592 590 588 586 584

7. The value of i +i +i +i +i
− 1 is [1]
582 580 578 576 574
i +i +i +i +i

a) -1 b) -4

c) -3 d) -2
8. 4 boys and 4 girls are to be seated in a row. The number of ways in which this can be done, if the boys and girls [1]
sit alternately, is

a) 4! × 4! b) P(8, 8)

c) none of these d) 2 × 4! × 4!
9. If f(x) = x−4

2√x
, then f'(1) is equal to: [1]

a) 1 b) 4

c) d) 0
5

4
−−−−−
10. √
1+sin x
=? [1]
1−sin x

a) cot b)
x x
tan
2 2

c) tan( d)
π x π x
+ ) cot( + )
4 2 4 2

11. Let A = {a, b, c}, B = {a, b}, C = {a, b, d}, D = {c, d} and E = {d}. Then which of the following statement is not [1]
correct?

a) D ⊇ E b) C - B = E

c) B ∪ E = C d) C - D = E
–
12. The integral part of (√2 + 1) is 6
[1]

a) 98 b) 96

c) 99 d) 100

13. If Cr denotes nCr in the expansion of (1 + x)n, then C0 + C1 + C2 + ... + Cn = ? [1]

a) 2n b) 2n

c) n(2n + 1) d)
1 n
2
3

14. solution set of the inequations x ≥ 2 , x ≤ − 3 is [1]

a) { } b) [ -3, 2 ]

c) ( -3, 2 ) d) [2 , -3 ]
15. If Q = {x : x = , where y ∈ N}, then [1]
1

a) 1 ∈ Q b) 1

2
∉ Q

c) 2 ∈ Q d) 0 ∈ Q

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16. (4 cos315° - 3 cos 15°) = ? [1]

a) 0 b) 1

c) -1 d) 1

√2

17. If f(x) = x100 + x99 ... + x + 1, then f'(1) is equal to: [1]

a) 5049 b) 50051

c) 5050 d) 5051
18. How many 3-digit even numbers can be formed with no digit repeated by using the digits 0, 1, 2, 3, 4 and 5? [1]

a) 56 b) 52

c) 50 d) 54
19. Assertion (A): Let A = {a, b} and B = {a, b, c}. Then, A ⊄ B. [1]
Reason (R): If A ⊂ B, then A ∪ B = B.

a) Both A and R are true and R is the correct b) Both A and R are true but R is not the
explanation of A. correct explanation of A.

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

20. Assertion (A): If the numbers , K, are in GP, then k = ±1. [1]
−2 −7

7 2
a2 a3
Reason (R): If a1, a2, a3 are in GP, then a1
=
a2
.

a) Both A and R are true and R is the correct b) Both A and R are true but R is not the
explanation of A. correct explanation of A.

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

Section B

21. Let R be a relation from N to N defined by R = {(a, b) : a, b ∈ N and a = b2}. Are the following true? [2]

(i) (a, a) ∈ R for all a ∈ N

(ii) (a, b) ∈ Rimplies (b, a) ∈ R
(iii) (a, b) ∈ R, (b, c) ∈ R implies (a, c) ∈ R
Justify your answer in each case.
OR
Find the range of the function given by f(x) .
3
=
2
2−x

22. Evaluate: lim . [2]

sec 5x−sec 3x

sec 3x−sec x
x→0

23. If A and B are two events associated with a random experiment such that P(A) = 0.3, P(B) = 0.4 and P(A ∪ B) = [2]
05, find P(A ∩ B) .
OR
A die is tossed once. What is the probability of getting an even number?
24. Using properties of set, show that: A ∪ (A ∩ B) = A [2]
25. Find the locus of a point, so that the join of (-5, 1) and (3, 2) subtends a right angle at the moving point. [2]
Section C
26. Determine n if 2n
C3 :
n
C2 = 12 : 1 [3]
27. Show that the points A(4, 6, - 3), B(0, 2, 3) and C(-4, - 4, -1) form the vertices of an isosceles triangle. [3]

28. Find an approximation of (0.99)5 using the first three terms of its expansion. [3]

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 OR
7
Using binomial theorem, expand: (x .
2 2
− )
x
1 1
+
[3]
29. Evaluate
x
2
lim
x+2
x→−2

OR
k cos x π
If f(x) ={ , when x ≠ , and if lim f(x) = 3
π
π − 2x 2 x→
2

Find the value of k.

30. Each side of an equilateral triangle is 18 cm. The midpoints of its sides are joined to form another triangle whose [3]
midpoints, in turn, are joined to form still another triangle. The process is continued indefinitely. Find the sum of
the areas of all the triangles.
OR
If a, b, c are the pth, qth and rth terms of a GP, show that (q - r)log a + (r - p)log b+ (p - q)log c = 0.
31. In a class, 18 students took Physics, 23 students took Chemistry and 24 students took [3]
Mathematics of these 13 took both Chemistry and Mathematics, 12 took both Physics and Chemistry and 11
took both Physics an Mathematics. If 6 students offered all the three subjects, find:
i. The total number of students.
ii. How many took Maths but not Chemistry.
iii. How many took exactly one of the three subjects.
Section D
32. In a survey of 44 villages of a state, about the use of LPG as a cooking mode, the following information about [5]
the families using LPG was obtained.

Number of families 0-10 10-20 20-30 30-40 40-50 50-60

Number of villages 6 8 16 8 4 2

i. Find the mean deviation about median for the following data.
ii. Do you think more awareness was needed for the villagers to use LPG as a mode of cooking?
33. Find the equation of the ellipse whose foci are (4, 0) and (- 4, 0), eccentricity = 1/3. [5]
OR
Find the (i) lengths of major and minor axes, (ii) coordinates of the vertices, (iii) coordinates of the foci, (iv)
2
2
y
eccentricity, and (v) length of the latus rectum of ellipes: = 1.
x
+
25 9

34. Solve for x, |x + 1| + |x| > 3 [5]

35. Prove that cos 12o + cos 60o + cos 84o = cos 24o + cos 48o [5]
OR
Prove that: cos 20° cos 40° cos 60° cos 80° = .
1

16

Section E
36. Read the text carefully and answer the questions: [4]
Function as a Relation A relation f from a non-empty set A to a non-empty set B is said to be a function, if
every element of set A has one and only one image in set B.
In other words, we can say that a function f is a relation from a non-empty set A to a non-empty set B such that
the domain of f is A and no two distinct ordered pairs in f have the same first element or component.
If f is a function from a set A to a set B, then we write

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 f

f : A → B or A → B.
and it is read as f is a function from A to B or f maps A to B.
(i) If f(x) = 1

2−sin 3x
, then find the range (f).​​
(ii) If f(1 + x) = x2 + 1, then find the f(2 - h).
(iii) If f(x) = x2 + 2x + 3, then find the value of among f(1), f(2) and f(3).
OR
What is the equation of a given figure?

37. Read the text carefully and answer the questions: [4]
In a hostel 60% of the students read Hindi newspapers, 40% read English newspapers and 20% read both Hindi
and English newspapers.

(i) A student is selected at random. She reads Hindi or English newspaper?

(ii) A student is selected at random. Did she read neither Hindi nor English newspapers?
(iii) A student is selected at random. She reads Hindi but not English Newspaper?
OR
A student is selected at random. She reads English but not Hindi Newspaper?
38. Read the text carefully and answer the questions: [4]
The conjugate of a complex number z, is the complex number, obtained by changing the sign of imaginary part
of z. It is denoted by z̄ .
The modulus (or absolute value) of a complex number, z = a + ib is defined as the non-negative real number
−−−−−−
√a2 + b2 . It is denoted by |z|. i.e.
−−−−−−
2 2
|z| = √a + b

Multiplicative inverse of z is z̄

2
. It is also called reciprocal of z.
|z|

2
z z̄ = |z|

(i) If f(z) = , where z = 1 + 2i, then find |f(z)|.

7−z

2
1−z

(ii) Find the value of(z + 3)(z̄ + 3).

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