CHANDA

South End Model School

Half Yearly Examination 2025-26
Subject : Mathematics(041)
Full Marks : 80 Standard : XI Duration: 3 Hours

GENERAL INSTRUCTION :-

I. This Question paper contains – five sections A, B, C, D and E. Each section is compulsory.

II. Section A has 18 MCQs and 02 Assertion-Reason based questions of 1 mark each.

III. Section B has 5 Very Short Answer (VSA)-type questions of 2 marks each.

IV. Section C has 6 Short Answer (SA)-type questions of 3 marks each.

V. Section D has 4 Long Answer (LA)-type questions of 5 marks each.

VI. Section E has 3 case based units of assessment (4 Marks each) with sub parts.

Section-A

1. Which of the following are sets

a) The collection of all the whole numbers less than 10

b) The collection of good hockey players in India
c) The collection of all difficult chapters in this book
d) None of these

2. Value of ⅈ19

a) 1 b) -1 c) -i d) i
31𝜋
3. Value of sin ( ) is
3

√3 1
a) b) c) -1 d) none of these
2 2

4.Which of the following are examples of the singleton set

a) {𝑥: 𝑥 ∈ 𝑧, 𝑥 2 = 4} b) {𝑥: 𝑥 ∈ 𝑧, |𝑥| = 1} c) {𝑥: 𝑥 ∈ 𝑁, 𝑥 2 = 16} d) none of these

5. Equation of y-axis is

a) x=0 b) y=0 c) x=a d) y=a

6. The value of sec 𝜃 can

a) never be greater than 1 b) never be less than 1

c) never be equal to 1 d) never lies between -1 & 1

7. Multiplicative identity of the complex number is

a) 0 b) 1 c) -i d) i
sin 4𝑥
8. lim
𝑥→0 6𝑥
2 4 3
a) b) c) d) none of these
3 3 2

9. The distance between the points A(2,-3) & B(-6,3) is

 a) a) 4 units b) 6 units c) 8 units d) 10 units

10. Which of the following are examples of the null set?

b) a) set of even prime numbers b) 𝐴 = {𝑥: 𝑥 ∈ 𝑁, 1 < x ≤ 2}

c) set of odd natural numbers divisible by 2
d) none of these

11. The value of cosec[−1110°]

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

12. Value of 0!

a) 0 b) 1 c) 10 d) none of these
√5
13. sin 𝑥 = & 0 < x < π/2, the value of sin 2𝑥
3

2√5 4√5 9√5 4√5

a) b) c) d)
9 9 2 2

𝑥 3 −1
14. lim
𝑥→1 𝑥−1

a) 2 b) 3 c) 5 d) 4

15. The inclination of a line parallel to the y-axis or the y-axis itself

a) 0° b) 90° c) 45° d) 30°

16. The slope of a vertical line is

a) 90° b) 0° c) not defined d) none of these

17. The points A(-2,3), B(1,2) & C(K,0) are collinear, then K=?

a) 5 b) 6 c) 7 d) 8

18. If cos 𝜃 = −1/2 & 𝜃 lies in quadrant II, then (2 sin 𝜃 + tan 𝜃) =?

a) 0 b) 1 c) 2 d) none of these

In the following questions a statement of assertion is followed by a statement of

Reason. 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. ASSERSTION : R= {(1, 2), (3, 4), (4, 5), (5, 6), (6, 6)} is a function.

REASON : The domain of a real valued function is a subset of real numbers.

20. ASSERTION : .f(x) = x², f: N to N is a function.

REASON: All relations are functions.

Section-B
𝑎𝑥 −𝑏𝑥
21. Evaluate lim
𝑥→0 𝑥
 1+cos 2𝑥 ⅆ𝑦
22. Differentiate:- 𝑦 = √ , find .
1−cos 2𝑥 ⅆ𝑥

2+3𝑖
23. Find the multiplicative inverse of
1+𝑖
√6+√2
24. Prove that sin 75° = .
4

25. Find a point on the y-axis which is equidistant from A(-4,3) & B(5,2).

Section-C
4𝑥 − 5, 𝑥 ≤ 2
26. Let 𝑓(𝑥) = {
𝑥 − 𝑎, 𝑥 > 2

If, lim 𝑓(𝑥) exists then find the value of a.

𝑥→2
𝑐𝑜𝑠𝑥
27. Differentiate
𝑙𝑜𝑔𝑥

or
cos 9°+𝑠𝑖𝑛9°
Prove that, = 𝑡𝑎𝑛54°
cos 9°−𝑠𝑖𝑛9°

28. Show that the points A(2,-1), B(3,4), C(-2,3) & D(-3,-2) are the vertices of a rhombus.

29. Prove that

(𝑥+1+i) (𝑥+1-i) (𝑥-1+i) (𝑥-1-i) = (𝑥 4 + 4)

30. Let R= {(𝑥, 𝑥 3 ): 𝑥 is a prime number less than 10}

i) Write R in roster form

ii) Find dom(R) & range (R).

1 1 𝑥
31. If + 7! = 8!, find the value of 𝑥.
6!

Section-D

32. Show that, 𝑐𝑜𝑡2𝑥 𝑐𝑜𝑡𝑥 − 𝑐𝑜𝑡3𝑥𝑐𝑜𝑡2𝑥 − 𝑐𝑜𝑡3𝑥𝑐𝑜𝑡𝑥 = 1

√2−√1+𝑐𝑜𝑠𝑥
33. lim =?
𝑥→0 sin2 𝑥

34. Prove that, 𝑐𝑜𝑠20°𝑐𝑜𝑠40°𝑐𝑜𝑠60°𝑐𝑜𝑠80° = 1/16

OR

1+sin 𝑥 𝜋 𝑥
Prove that, √ = 𝑡𝑎𝑛 ( 4 + 2)
1−sin 𝑥

5 5
(𝑥+2)3 −(𝑎+2)3
35. lim
𝑥→𝑎 𝑥−𝑎

OR

Using of 1st principal of derivatives find f'(x) when f(x) = log(cos x)

Section-E

36. In a city of 56,000 people, following is the number of fans of players Rohit (R), Virat (V) and
Dhoni (D):

Based on the above information, answer the following:

I. How many people are fans of at least 2 players?

II. How many people are fans of exactly 1 player?
III. How many people are fans of exactly 2 players?
IV. How many people follow R or V but not D?

37.A conjugate of a complex number is another complex number that has the same real part as the

Original complex number, and the imaginary part has the same magnitude but opposite sign. If we

Multiply a complex number with its conjugate, we get a real number.

Based on the above information, answer the following questions.

(a) Find the conjugate of the following : (2 + i)² (1)

(b) Find the multiplicative inverse of (4 – 3i). (1)
(c) Express (3 + 4i) (6 – 3i) (5 + i) in the form (a + ib). (2)

38. Raj works at a book store. While arranging some books on the book shelf, he observed that there are

5 History books, 3 Mathematics books and 4 Science books which are to be arranged on the shelf.

(i) In how many ways can he select either a History book or a Maths book? (1)

(ii) If he selects 2 History books, 1 Maths book and 1 Science book to arrange them, then find the

number of ways in which selection can be made. (1)

(iii) Find the number of ways, if the books of same subject are put together. (1)

(iv) Find the number of arrangements, if he selects 3 History books, 2 Maths Books, 2 Science books. (1)