Class 11 Sample Paper Maths (Half Yearly)

SAMPLE
PAPER
Half Yearly Examination

Prepared For
NCERT BASED SYLLABUS
Applicable For
CBSE BOARD AND STATE BOARD USING NCERT

WWW.

HALF YEARLY EXAMINATION
SAMPLE QUESTION PAPER

CLASS XI SUBJECT: MATHEMATICS
Time: 3 Hrs. M.M: 80

General Instructions:-
 This question paper contains two parts A and B.
 Each part is compulsory.
 Part A carries 24marks and Part B carries 56 marks
 Part-A has Objective Type Questions.
 Part -B has Descriptive Type Questions.
 Both Part A and Part B have choices.
Part – A:
 It consists of two sections- I and II.
 Section I comprises of 16 questions of very short answer type.
 Section II contains 2 case studies.
 Each case study comprises of 5 case-based MCQs. An examinee
is to attempt any 4 out of 5 MCQs.
 Internal choice is provided in 2 questions of section – I.
Part – B:
 It consists of three sections- III, IV and V.
 Section III comprises of 10 questions of 2 marks each.
 Section IV comprises of 7 questions of 3 marks each.
 Section V comprises of 3 questions of 5 marks each.
 Internal choice is provided in 2 questions of Section –III, 2
questions of Section- IV and 1 question of Section-V. You have
to attempt only one of the alternatives in all such questions.
PART-A (SECTION – I) (16 x 1 =16)
1. Write the following set in the roster form. A = {x | x is a positive integer less than 10 and
2x – 1 is an odd number}.
2. If B′ ⊂ A′, show that A ⊂ B.
3. If X and Y are 2 sets such that n(X) = 17, n(Y) = 23 and n(X⋃ 𝑌) = 38, find n(X∩ 𝑌).
OR
Given U = [–5, 5] and A is (–3, 5], then find A′
𝒙𝟐 𝟐𝒙 𝟏
4. Find the domain of the function f(x) = 𝒙𝟐

𝟖𝒙 𝟏𝟐
5. Is the given relation a function? Give reasons for your answer if S = {(n, n2): n is a
positive integer}
6. Let f and g be real functions defined by f (x) = 2x + 1 and g(x) = 4x – 7. For which real
number x, f(x) = g(x)?

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7. Write in degrees.
OR
Convert 40° 20′45’’ into radian measure.

8. Find the value of sin 75° cos 15° + cos 75° sin 15°
𝐭𝐚𝐧 𝟔𝟗𝟎 𝒕𝒂𝒏𝟔𝟔𝟎
9. Find the value of 𝟏 𝐭𝐚𝐧 𝟔𝟗𝟎𝒕𝒂𝒏𝟔𝟔𝟎
10. Express the following as sum or difference: 2 sin 5θ sin 3θ
𝐬𝐢𝐧 𝒙 𝐬𝐢𝐧 𝟑𝒙
11. Prove that 𝒔𝒊𝒏𝟐𝒙 𝒄𝒐𝒔𝟐𝒙 = 2 sin x
12. Find the Range of the polynomial function f (x) = 3x2 – 4x + 9?
13. Give a rough sketch of the function 𝑓(𝑥) = 𝑥 − [𝑥] .
14. Solve the inequality, 3x – 5 < x + 7, when x is a whole number
15. Solve the inequality 𝑥 + < 11 −
16. If | x – 1| > 5, then find the value of x.

PART – A (SECTION- II) (2 x 4 =8)
Case study-based questions are compulsory. Attempt any 4 sub-parts
of each question. Each sub-part carries 1 mark.
17. Method to find the sets when Cartesian Product is given.
For finding these two sets, we write first element of each ordered pair in first set say
A and corresponding second element in second set B(say).
If there are p elements in set A and q elements in set B, then there will be pq elements
in A × 𝐵.
Based on the above topic answer the following questions.
I. If A × 𝐵 = {(a,1),(b,3),(a,3),(b,1),(a,2),(b,2)}. Then A & B are
a) {1,3,2}, {a, b}
b) {a, b}, {1,3}
c) {a, b}, {1,3,2}
d) None of these
II. If the set A has 3 elements and set B has 4 elements, then find the number of
elements of A × 𝐵.
III. A & B are two sets given in such a way that A × 𝐵 contains 6 elements. If
three elements of A × 𝐵 are (1,3), (2,5) & (3,3) then find A & B.
IV. Find the remaining elements of A × 𝐵 in the above question III.
V. The cartesian product P × 𝑃 has 16 elements among which are found (a,1) &
(b,2) then find P.
18. The school organized a cultural event for 100 students. In the event, 15 students
participated in dance, drama & singing. 25 students participated in dance & drama; 20

students participated in drama & singing; 30 students participated in dance & singing. 8
students participated in dance only; 5 students in drama only and 12 students in singing
only.
Based on the above information, answer the following questions.

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I. Find the number of students who participated in dance?
II. Find the number of students who participated in drama?
III. Find the number of students who participated in singing?
IV. Find the number of students who participated in dance & drama but not in
singing?
V. Find the number of students who did not participated in any events?

PART – B (SECTION -III) (10 x 2 = 20)
All questions are compulsory. In case of internal choices attempt anyone

19. A and B are two sets such that: n (A – B) = 14 + x, n (B – A) = 3x and n (A ∩ B) = x, draw a
Venn diagram to illustrate the given information and if n(A) = n(B)
then find the value of x.
20. Two finite sets have m and n elements. The total number of subsets of the first set is 56
more than the total number of subsets of the second set.
Find the values of m and n.

OR
Let T = 𝑥: −5 = . Is T an empty set? Justify you answer

21. Find the domain and range for the following function f(x) = √
1−𝑥, 𝑥<0
22. The function f(x) is defined by f(x) = 1, 𝑥 = 0 . Draw the graph of f(x).
1 + 𝑥, 𝑥 > 0
23. A train is travelling on a curve of 700 m radius at 14 km/h, Through what angle will it
turn in one minute?

OR
A wheel makes 270 revolutions in one minute. Through how many radians does
it turn in one second?
24. Find the value of 2 𝑠𝑖𝑛 + 2 𝑐𝑜𝑠 − 2𝑡𝑎𝑛
25. Find the value of tan 720° – cos 270° – sin 150° cos 120°.
26. Express each of the following as a product: sin 320 + sin 540
27. If A = {x: x ∈ W, x < 2}, B = {x: x ∈ N, 1 < x < 5}, C = {3, 5}
find (i) A × (B ∩ C) (ii) A × (B 𝖴 C)
28. Solve | | ≤

PART-B (SECTION- IV) (7 x 3 =21 )
29. Let U = {x ∈ N : x ≤ 8}, A = {x ∈ N : 5 < x2 < 50} and B = {x ∈ N : x is prime number less
than 10}. Draw a Venn diagram to show the relationship between the given sets. Hence

list the elements of the following sets (i) A′ (ii) B′ (iii) A – B (iv) A ∩ B′.

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30. Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}, B = {2, 4, 6, 8} and C = {3, 4, 5, 6}.
find (i) A′ (ii) (A ∩ C)′ (iii) (A′)′ (iv) (B – C)′
31. If sin x = , cos y = and x & y both lie in the second quadrant,
find the value of sin (x +y)
32. Prove that cot x cot 2x – cot 2x cot 3x – cot 3x cot x = 1
𝐬𝐢𝐧 𝑨 𝐬𝐢𝐧 𝟑𝑨 𝐬𝐢𝐧 𝟓𝑨 𝐬𝐢𝐧 𝟕𝑨
33. Prove that = 𝒄𝒐𝒕 𝟐𝑨.
𝐜𝐨𝐬 𝑨 𝐜𝐨𝐬 𝟑𝑨 𝐜𝐨𝐬 𝟓𝑨 𝐜𝐨𝐬 𝟕𝑨
OR
𝐜𝐨𝐬 𝟖𝑨 𝐜𝐨𝐬 𝟓𝑨 𝐜𝐨𝐬 𝟏𝟐 𝑨 𝐜𝐨𝐬 𝟗𝑨
Prove that = 𝐭𝐚𝐧 𝟒 𝑨.
𝐬𝐢𝐧 𝟖𝑨 𝐜𝐨𝐬 𝟓𝑨 𝐜𝐨𝐬 𝟏𝟐𝑨 𝐬𝐢𝐧 𝟗𝑨

34. Show that tan 3x tan 2x tan x = tan 3x – tan 2x – tan x
35. Draw the graph of f (x) = 1 − |𝑥 − 2|, x ∈ R. What are the domain and range of
f (x) = 1 − |𝑥 − 2|?
OR
Reduce the function 𝑓(𝑥) = |𝑥 − 2| + |2 + 𝑥|, −3 ≤ 𝑥 ≤ 3 & draw the graph for the
given function.

PART-B (SECTION-V) (3 x 5 = 15)

36. In a survey it was found that 21 persons liked product P1, 26 liked product P2
and 29 liked product P3. If 14 persons liked products P1 and P2; 12 persons liked
product P1 and P3; 14 persons liked product P2 and P3, and 8 liked all the
three products.

i. Find how many liked only one product?
ii. Find how many liked only 2 products?
iii. Find the number of persons who liked P1 & P2 but not P3.
37. Prove that sin 10 sin 30 sin 50 sin 70 =
38. A manufacturer has 600 litters of a 12% solution of acid. How many liters of a
30% acid solution must be added to it, so that acid content in the resulting
mixture will be more than 15% but less than 18%.
OR
Solve the following system of inequations:
5𝑥 3𝑥 39 2𝑥 − 1 𝑥 − 1 3𝑥 + 1
+ > & − < , 𝑥 𝜀 𝑅.
4 8 8 12 3 4

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