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Class 11 - Mathematics
Sample Paper - 05 (2022-23)

Maximum Marks: 80
Time Allowed: : 3 hours

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 OP makes 4 revolutions in one second, the angular velocity in radians per second is
a) 2π
b) π
c) 8π
d) 4π
2. An aeroplane flies around a square, the sides of which measure 100 miles each. The aeroplane covers at speed of 100
m/h on the first side, at 200 m/h on the second side. At 300 m/h the third side and 400 m/h on the fourth side. The
average speed of the aeroplane around the square is
a) 900 m/h
b) 200 m/h
c) 192 m/h
d) 195 m/h
3. A coin is tossed three times. What is the probability of getting head and tail alternately?
1
a) 8
1
b) 4
3
c) 4
1
d) 2
4. If f (x) = √1 − x 2, x ∈ (0, 1), then f'(x), is equal to

a) √1 − x 2
b) √x 2 − 1
1
c)
√1 − x 2
−x
d)
√1 − x 2

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5. A point moves such that its distance from the point (4, 0) is half that of its distance from the line x = 16. The locus of the
point is
a) 3x2 + 4y2 = 192
b) 4x2 + 3y2 = 192
c) None of these
d) x2 + y2 = 192
6. Let A and B be two sets such that n(A) = 35, n(B) = 42 and n(A ∩ B) = 17, find n(A − B)
a) 25
b) 19
c) 18
d) 17

| |
1 1 1
7. Let ω be a complex number such that 2ω + 1 = z where z = √− 3. If 1 − ω2 − 1 ω 2 = 3k, then k is equal to
1 ω2 ω7

a) 1
b) -z
c) -1
d) z
1
8. The domain of the function f defined as f(x) = is
√ | sin x | + sin x
a) (-2nπ, 2nπ)
b) none of these
c) (2nπ, (2n + 1)π)
π π
d) ((4n - 1) , (4n + 1) )
2 2
9. solution set of the inequations x ≥ 2 , x ≤ − 3 is
a) { }
b) [ -3, 2 ]
c) ( -3, 2 )
d) [2 , -3 ]
−4 θ
10. If sinθ = 5
, and θ lies in third quadrant then the value of cos 2 is
1
a) −
√5
1
b) 5
1
c) −
√10
1
d)
√10
11. If the four-letter words (need not be meaningful) are to be formed using the letters from the word
MEDITERRANEAN such that the first letter is R and the fourth letter is E, then the total number of all such words is:
11 !
a)
( 2 ! )3

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b) 59
c) 110
d) 56
12. If a, 4, b are in A.P.; a, 2, b are in G.P.; then a, 1, b are in
a) A.P.
b) none of these
c) G.P.
d) H.P.
13. The sum of the series 20C0 - 20C1 + 20C2 - 20C3 + ... + 20C10 is
1
a) 20C
2 10

b) 20C10
c) 0
d) 20C11
14. The solution set for |x| > 7
a) (− ∞, − 7) ∩ (7, ∞)
b) none of these
c) (7, ∞)
d) (− ∞, − 7) ∪ (7, ∞)
20 n
15. Each set Xr contains 5 elements and each set Yr contains 2 elements and ⋃ x r = S = ⋃ Y r. If each element of S belong
r=1 r=1
to exactly 10 of the Xr’s and to exactly 4 of the Yr’s, then n is
a) 10
b) 20
c) 50
d) 100

16. Let x = log csc 2π

3 ( ) 148
111
and y is the value of tan 780o, then (x - y) is the tangent of the angle :

π
a) 3
π
b) 6
5π
c)
12
π
d) 12
17. Mark the correct answer for (1 + 2i)-2 = ?

a)
( −3
25
−
4
25
i
)
b) ( −3
25
+
4
25
i )
c) None of these

d)
( 3
25
−
4
25
i
)
18. The number of diagonals of a regular polygon is 35. Then the number of sides of polygon is
a) 11

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b) 9
c) 12
d) 10

19. Assertion (A): Number of terms in the expansion of 2x −

( )
x
4
2
10
is 11.

Reason (R): Number of terms in the expansion of, (x + a)n is n + 1.

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.
20. Assertion (A): Domain and Range of a relation R = {(x, y): x - 3y = 0} defined on the set A = {1, 2, 3} are respectively
{1, 2, 3} and {2, 4, 6}.
Reason (R): Domain and Range of a relation R are respectively the sets {a: a ∈ A and (a, b) ∈ R} and {b: b ∈ A
and (a, b) ∈ R}
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.
Section B
x+1
21. If f(x) = x−1
then show that f{f(x)} = x
sin 3θ
22. Evaluate: lim tan 2θ
.
θ→0
23. Find the coordinates of the focus, axis of the parabola, the equation of the directrix and the length of the latus rectum: x2
= -16y

OR

Find the equation of the circle which touches the lines 4x - 3y +10 = 0 and 4x - 3y - 30 = 0 and whose centre lies on the
line 2x + y = 0.
24. For any sets A and B, prove that: (A - B) ∪ (A ∩ B) = A
1
25. Find the angle between the lines whose slopes are √3 and .
√3
Section C
26. i. Let A = {8,11,12,15,18,23} and f is a function from A → N such that f(x) = highest prime factor of x, find f and its
range.
x 2 + 2x + 1
ii. Find the domain of the function f(x) = .
x 2 − 8x + 12
x−2 y+1 z−2
27. Find the distance between the point (-1, -5, -10) and the point of intersection of the line 3
= 4
= 12
and the plane
x - y + z = 5.

OR

Verify that (0, 7, 10), (-1, 6, 6) and (-4, 9, 6) are the vertices of a right-angled triangle.

28. Expand the given expression

( )
2
x
−
x
2
5

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OR

Which of the following is larger? 9950 + 10050 or 10150

29. Find the square roots: 7 - 24i.

OR

| | | | | |
If z1 = 3 + i and z2 = 1 + 4i, then verify that z 1 − z 2 > z 1 − z 2 .
30. Solve | x − 1 | + | x − 2 | ≥ 4.
31. If 22P r + 1 : 20
P r + 2 = 11 : 52. find r.
Section D
32. A box contains 100 bulbs, 20 of which are defective. 10 bulbs are selected for inspection. Find the probability that:
i. all 10 are defective
ii. all 10 are good
iii. at least one is defective
iv. none is defective
33. Find the derivative of (sinx + cosx) from first principle.

OR

Differentiate log sin x from first principles.

√3
34. Prove that: sin 20o sin 40o sin 80o =
8

OR

tan A + sec A − 1 1 + sin A

Prove that tan A − sec A + 1
= cos A
35. Calculate the mean deviation about the median for the following data:
Height (in cm) 95 - 105 105 - 115 115 - 125 125 - 135 135 - 145 145 - 155

Number of boys 9 13 25 30 13 10
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Section E
36. Read the text carefully and answer the questions:
The girder of a railway bridge is a parabola with its vertex at the highest point, 10 m above the ends. Its span is 100 m.

i. Find the coordinates of the focus of the parabola.

ii. Find the equation of girder of bridge and find the length of latus rectum of girder of bridge.​​

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iii. Find the height of the bridge at 20m from the mid-point.

OR

Find the radius of circle with centre at focus of the parabola and passes through the vertex of parabola.
37. Read the text carefully and answer the questions:
A student of class XI draw a square of side 10 cm. Another student join the mid-point of this square to form new square.
Again, the mid-points of the sides of this new square are joined to form another square by another student. This process
is continued indefinitely.

i. The sum of the perimeter of all the square formed is (in cm)
a) 40
b) 40 + 40√2
c) 80 + 40√2
d) None of these
ii. The sum of areas of all the square formed is (in sq cm)
a) 250
b) None of these
c) 200
d) 150
iii. The perimeter of the 7th square is (in cm)
5
a)
2
b) 10
c) 5
d) 20

OR

The area of the fifth square is (in sq cm)

25
a)
4
b) 25
c) 50
25
d) 2
38. Read the text carefully and answer the questions:
A survey is conducted by a career counsellor in a college to find career choice of students after the Intermediate. There
are 100 students that goes for Engineering Courses, 50 wants to make their career in Medical, 100 students continue their
further study in Arts. There are 10 students that go for both Engineering and Medical, and 3 goes for Medical and Arts.

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There are 3 students that do not go for any further studies.

i. Find the number of students that goes for Engineering or Art.

ii. Find the number of students that goes for both Medical and Art.

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