Class 11 Question Paper 2023-24 Maths (Half Yearly)

2023-24
HALF YEARLY
EXAM
NCERT BASED
SYLLABUS
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FOR
CBSE BOARD
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Half Yearly Exam 2023-24
Question Paper

Class - XI Subject – Mathematics (041)
Time – 3 Hours Maximum Marks- 80

General instructions:

Read the following instructions very carefully and follow them:

a) This question paper contains 38 questions. All questions are compulsory.
b) Question paper is divided into 5 sections section A,B,C,D and E
c) In section A, Question number 1 to 18 are Multiple Choice Questions (MCQ) type and
Question number 19 and 20 are Assertion- Reason based questions of 1 mark each.
d) In section B Question number 21 to 25 are Very Short Answer (VSA) type questions of 2
marks each
e) In section C Question number 26 to 31 are Short Answer (SA) type questions carrying 3
marks each.
f) In section D Question Number 32 to 35 are Long Answer a (LA) type questions carrying 5
marks each.
g) In section E Question number 36 to 38 are case study-based questions carrying 4 marks
h) There is no overall choice. How ever there internal choices in some questions.

SECTION – A (1x20=20)

1. Two finite sets have m and n elements. Then number of subsets of the first
set is 112 more than that of second set. The value of m and n are
respectively.
(a) 4, 7 (b) 7, 4 (c) 4, 4 (d) 7, 7

2. If A = {x:x is a multiple of 3} and B = {x:x is a multiple of 5}, then A – B is
(a) 𝐴 ∩ 𝐵 (b) 𝐴 ∩ 𝐵 (c) 𝐴 ∩ 𝐵 (d) 𝐴 ∪ 𝐵

3. If set A and set B are defined as
1
A = {(x,y) : y = , 𝑥 ≠ 0, 𝑋 ∈ 𝑅}
𝑥

B = {(𝑥, 𝑦): 𝑦 = −𝑥, 𝑋 ∈ 𝑅}, then
(a) 𝐴 ∩ 𝐵 = 𝐴 (b) 𝐴 ∩ 𝐵 = 𝐵 (c) 𝐴 ∩ 𝐵 = ∅ (d) 𝐴 ∪ 𝐵 = 𝐴

𝑥2
4. Range of the function f(x) = is
𝑥 2 +1

(a) (-1, 0) (b) (-1, 1) (c) [0, 1) (d) (1, 1)

1

5. If x = {a, b, c}, y = {1, 2, 3} then which of the following is not a relation from x to y ?
(a) 𝑅1 = {(𝑎, 1), (𝑎, 2), (𝑎, 3)} (b) 𝑅2 = {(𝑎, 2), (𝑏, 1), (𝑐, 3)}
(c) 𝑅3 = {(3, 𝑐 ), (𝑏, 3), (𝑐, 3), (𝑏, 1), (3, 𝑎)} (d) None of these

6. If A lies in the second quadrant and 3 tanA+4 = 0 then the value of
2cotA – 5cosA + sin A is
−53 23 37 7
(a) (b) (c) (d)
10 10 10 10

𝜋
7. If 𝛼 + 𝛽 = , then the value of (1+tan𝛼)(1+tan𝛽)
4

(a) 1 (b) 2 (c) -2 (d) None of these

8. The value of sin 50˚ - sin 70˚ + sin 10˚ is
1
(a) 1 (b) 0 (c) (d) 2
2

9. The value of cos 10 cos 20 cos 30 _ _ _ _ _ _ _ cos 1790 is
1
(a) (b) 0 (c) 1 (d) -1
√2

1−𝑡𝑎𝑛2 15˚
10. The value of
1+𝑡𝑎𝑛2 15˚

√3
(a) 1 (b) √3 (c) (d) 2
2

1 1
11. If tan A = and tanB = then the value of A+B is
2 3
𝜋 𝜋
(a) (b) 𝜋 (c) 0 (d)
6 4

−4 𝜃
12. If sin𝜃 = and 𝜃 lies in third quadrant the value of cos is
5 2
1 1 1 1
(a) (b) − (c) − (d) −
5 √10 √5 10

1
13. If z = then |z| is equal to
(2+3𝑖)2
1 1 1
(a) (b) (c) (d) None of these
13 5 12

−2
14. If z = , then the value of arg (z) is
1+𝑖√3
𝜋 2𝜋 𝜋
(a) 𝜋 (b) (c) (d)

3 3 4

15. How many 10 – digit numbers can be formed by using the digits 1 and 2
(a) 10𝑝2 (b) 10𝑐2 (c) 210 (d) 10!

2

16. The product of r consecutive positive integers is divisible by
(a) 𝑟! (b) 𝑟! + 1 (c) (𝑟 + 1)! (d) None of these

17. There are 6 periods on each working day of a school. In how many ways can one
arrange 5 subjects such that each subject is allowed at least one period
(a) 360 (b) 720 (c) 3600 (d) 1800

18. The value of (7𝑐0 + 7𝑐1 ) + (7𝑐1 + 7𝑐2 ) +_ _ _ _ _ _ +(7𝑐6 + 7𝑐7 )
(a) 27 − 1 (b) 28 − 2 (c) 28 − 1 (d) 28

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.
−1±√19𝑖
19. Assertion (A) : √5𝑥 2 + 𝑥 + √5 = 0 then 𝑥 =
2√5

−𝑏±√𝑏 2 −4𝑎𝑐
Reason (R) : If 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 then 𝑥 =
2𝑎

20. Assertion (A) : If A = {1, 2, 3, 4, 5}, B = {2, 4, 6} and C = {3, 4, 6} then
(A∪B)∩C = {3, 4, 6}
Reason (R) : (A∪B)' = A'∩B'

SECTION – B (2x5=10)

21. If ∪ = {𝑥: 𝑥 ≤ 10, 𝑋 ∈ 𝑁}
A = {𝑥: 𝑋 ∈ 𝑁, 𝑥 𝑖𝑠 a prime ≤ 10}
B = {𝑥: 𝑋 ∈ 𝑁, 𝑥 𝑖𝑠 even ≤ 10} write A∩B' in roster form
OR
If P(A) = P(B), prove that A = B

22. If A = {2,4,6,9} and B = {4,6,18,27,54} let R be a relation from set A to set B defined
as R = {(a,b) : a<b and a is a factor of b} Find domain and range of relation.
OR
𝑎𝑥−𝑏
If 𝑦 = 𝑓(𝑥 ) = , show that 𝑥 = 𝑓(𝑦)

𝑏𝑥−𝑎

3

𝑎+𝑖𝑏
23. If 𝑥 + 𝑖𝑦 = , prove that 𝑥 2 + 𝑦 2 = 1
𝑐+𝑖𝑑

OR
(3−2𝑖)(2+3𝑖)
Find the conjugate of
(1+2𝑖)(2−𝑖)

24. Solve for x and show the graph of solution on the number line
4 6
≤3≤
𝑥−1 𝑥+1

25. if 𝑛𝑝𝑟 = 720 𝑎𝑛𝑑 𝑛𝑐𝑟 = 120 find the value of r

SECTION – C (3x6=18)

26. Find the domain and range of the function
If A = {1,2,3}, B = {3,4} and C = {4,5,6}, find (i) A x (B ∩ C) (ii) (A x B) ∩ (A x C)

Sin5x−2sin3x+Sinx
27. Prove that = tan 𝑥
Cos5x−cosx

3 x x x
28. If tan x = and x lies in third quadrant, find the value of Sin , Cos and tan
4 2 2 2

OR

Show that √2 + √2 + √2 + 2𝑐𝑜𝑠8𝜃 = 2𝑐𝑜𝑠𝜃

29. Prove that 𝑐𝑜𝑠6𝑥 = 32𝑐𝑜𝑠 6 𝑥 − 48𝑐𝑜𝑠 4 𝑥 + 18𝑐𝑜𝑠 2 𝑥 − 1
OR
𝑠𝑖𝑛𝑥 𝑠𝑖𝑛3𝑥 𝑠𝑖𝑛9𝑥 1
Prove that + + = (𝑡𝑎𝑛27𝑥 − 𝑡𝑎𝑛𝑥)
𝑐𝑜𝑠3𝑥 𝑐𝑜𝑠9𝑥 𝑐𝑜𝑠27𝑥 2

𝑍 𝑍
30. Let 𝑍1 = 2 − 𝑖, 𝑍2 = −2 + 𝑖 𝑓𝑖𝑛𝑑 𝑅𝑒 ( 1 2)
𝑧1

31. A solution of 8% boric is to be diluted by adding 2% boric acid solution to it. The
resulting mixture is to be more than 4% but less than 6% boric acid. If we have 640
litres of 8% solution, how many litres of the 2% solution will have to be added.

SECTION – D (5x4=20)

1 1
32. If 𝑎𝑓(𝑥 ) + 𝑏𝑓 ( ) = , 𝑤ℎ𝑒𝑟𝑒 𝑎 ≠ 𝑏, 𝑎𝑛𝑑 𝑥 ≠ 𝑜 𝑓𝑖𝑛𝑑 𝑓(𝑥)
𝑥 𝑥
OR

(a) Find the domain and range of the function𝑓(𝑥 ) = √2 − 𝑥
(b) Write the signum function with domain and range also draw the graph of the
function.

4

2𝑏
33. If 𝑎 𝑐𝑜𝑠2𝜃 + 𝑏𝑠𝑖𝑛2𝜃 = 𝑐 ℎ𝑎𝑠 𝛼 𝑎𝑛𝑑 𝛽 as its roots then prove that 𝑡𝑎𝑛𝛼 + 𝑡𝑎𝑛𝛽 =
𝑎+𝑐

OR
3
𝑠𝑖𝑛20. 𝑠𝑖𝑛40. 𝑠𝑖𝑛60. 𝑠𝑖𝑛80 =
16
z−1
34. If |z| = 1, then prove that : (z ≠ 1)is a purely imaginary number. Type equation here.
z+1

35. Find the number of arrangements of the letters of the word INDEPENDENCE. In
how many of these arrangements
(i) Do the words start with P?
(ii) Do all the vowels always occur together?
(iii) Do all the vowels never occur together?
(iv) Do the words begin with I and end with P ?

SECTION – E (4x3=12)

36. A mathematics teacher of class XI write two things about Mohan and Sohan, Which
are below in figure.
Mohan Sohan
n n
Age Pr Pr+1
n n
Height (in feet) Cr Cr-1

Now answer the following question which are based on above data.
(i) If the age of Mohan and Sohan are equal, then find the value of (n-r) [1]
(ii) If the height of Mohan and Sohan are equal, then find the value of [1]
(n-2r).
(iii) Find the age of Mohan and the height of Sohan. [2]
OR
(2𝑛)!
Prove that : = {1.3.5 … (2𝑛 − 1)}2𝑛 .
𝑛!

37. Consider the information given below

Let P (a,b) be any point on the unit circle given below, which has its centre at origin
o. It is given that ∠𝐴𝑂𝑃 = 𝑥 𝑟𝑎𝑑𝑖𝑎𝑛.

Now answer the questions based on figure given below.

5

√3 −1
(i) 𝐼𝑓 𝑎 = 𝑎𝑛𝑑 𝑏 = , 𝑇ℎ𝑒𝑛 𝑓𝑖𝑛𝑑 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑥 𝑖𝑛 𝑟𝑎𝑑𝑖𝑎𝑛. [1]
2 2

(ii) 𝐼𝑓 𝑥 = 250 , 𝑡ℎ𝑒𝑛 𝑓𝑖𝑛𝑑 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑥 𝑖𝑛 𝑟𝑎𝑑𝑖𝑎𝑛 [1]
(iii) 𝐼𝑓 𝑥 = 750 𝑎𝑛𝑑 𝑦 = 150 𝑡ℎ𝑒𝑛 𝑓𝑖𝑛𝑑 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑡𝑎𝑛𝑥 + 𝑡𝑎𝑛𝑦 [2]

38. A quiz completion is very popular now-a-days. A director of an institute organized a
quiz competition includes three subjects Mathematics, Physics and Chemistry for all
his students. He issued form so that student could opt the subject(s) or not
participate. Total 175 students filled the form and opted subject(s) as follows:
Mathematics 100, Physics 70, Chemistry46, Mathematics and Physics 30,
Mathematics and Chemistry 28, Physics and Chemistry 23, Mathematics, Physics
and Chemistry 18
On the basics of above information, answer the following
(i) Find the number of students who participated in Mathematics only [1]
(ii) Find the number of students who did not participate in the quiz competition [1]
(iii) Find the number of students who participated in [2]
(a) Physics only
(b) Chemistry only

6

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