This document contains a practice test for the NTSE Stage II exam with 46 multiple choice math questions. The questions cover topics like geometry, trigonometry, probability, arithmetic progressions, and algebra. For each question there are 4 possible answer choices labeled A, B, C, or D. The test is presented across 4 pages with around 10-12 questions per page.

1. Page # 1
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PRACTICE SHEET_MATHS
PRACTICE SHEET - 5
MATHS
NTSE [STAGE - II]
1. A cone is 8.4 cm high and the radius of its
base is 2.1 cm. It is method and recast into
a sphere. The radius of the sphere is
(A) 4.2 cm (B) 2.1 cm
(C) 2.4 cm (D) 1.6 cm
2. In a cylinder, radius is doubled and height is
halved, curved surface area will be
(A) halved (B) double
(C) same (D) four times
3. The radii of two cylinders are in the ratio
2:3 and their heights are in the ratio of 5:3.
The ratio of their volumes is
(A) 10:17 (B) 20:27
(C) 17:27 (D) 20:37
4. If sin  and cos are roots of the equation px2
+ qx + r = 0, then:
(A) p2
- q2
+ 2pr = 0 (B) (p + r)2
= q2
- r2
(C) p2
+ q2
- 2pr = 0 (D) (p - r)2
= q2
+ r2
5. 4 years back, A’s age was 4 times that B’s
age. What is A’s present age, if after 3
years, B’s age will be
1
3
rd of A’s age?
(A) 56 (B) 60
(C) 63 (D) 66
6. In the adjacent figure, if AB = 12cm, BC =
8cm and AC = 10cm, then AD =
A D B
E
F
C
(A) 5cm (B) 4cm
(C) 6cm (D) 7cm
7. The number of distinct integers in the collec-
tion
2 2 2 2
10 10 10 10
, , ...........,
1 2 3 20
       
       
       
, where
[x] denotes the largest integer not exceeding
x, is
(A) 20 (B) 18
(C) 17 (D) 15
8. Let Tk
denote the k-th term of an arithmetic
progression. Suppose there are positive inte-
gers m n
 such that Tm
= 1/n and Tn
= 1/m.
Then Tmn
equals
(A)
1
mn
(B)
1 1
m n

(C) 1 (D) 0
9. Which of the following is not a linear
equation:
(A) 3(x+8)= 2x+7 (B) x(x+3)=-x(3-x)+8
(C) x(x-8)=-3x(x-5) (D) x2+4x=x(x+8)+5
10. If a linear equation has solutions (-2,2),
(0,0) and (2,-2) then it is of the form
(A) y=x (B) y=-x
(C) -2x+y=-0 (D) -x+2y=0
11. The system of equation
x 0
x 1
 



  


has solution
as
(A) (0,1) (B) (0,-1)
(C) (-1,0) (D) None of these
12. The sum of 5 numbers in geometric progres-
sion is 24. The sum of their reciprocals is 6.
The product of the terms of the geometric pro-
gression is
(A) 36 (B) 32
(C) 24 (D) 18
13. In a triangle ABC, let AD be the median from
A; let E be a point on AD such that AE:ED =
1:2; and let BE extended meets AC in F. The
ratio of AF/FC is
(A) 1/6 (B) 1/5
(C) 1/4 (D) 1/3
14. In a triangle ABC, a point D on AB is such that
AD:AB = 1:4 and DE is parallel to BC with E on
AC. Let M and N be the mid points of DE and
BC respectively. What is the ratio of the area
of the quadrilateral BNMD to that of triangle
ABC?
(A) 1/4 (B) 9/32
(C) 7/32 (D) 15/32

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PRACTICE SHEET_MATHS
15. Digits a and b are such that the product
4a1 25b
 is divisible by 36 (in base 10). The
number of ordered pairs (a, b) is
(A) 15 (B) 8
(C) 6 (D) 4
16. If x = (7 + 4 3 ), then the value of
1
x
x

is :
(A) 8 (B) 6
(C) 5 (D) 4
17. The value of the expression
1 1 1
....upto 99 terms
2 1 3 2 4 3
  
  
is equal to :
(A) 9 (B) 3
(C) 1 (D) 0
18. In the figure A = CED, CD = 8 cm, CE = 10
cm, BE = 2 cm, AB = 9 cm, AD = b and
DE = a. The value of a + b is :
(A) 13 cm
(B) 15 cm
(C) 12 cm
D
C
A B
9 cm
b
a E
10 cm
2 cm
8 cm
(D) 9 cm
19. ABC is a right angled triangle, where
B = 90°. CD and AE are medians. If AE = x
and CD = y then, correct be the ratio of their
measures ?
(A) x2
+ y2
= AC2
(B) x2
+ y2
= 2AC2
(C) x2
+ y2
=
2
3
AC
2
A
D
B C
E
x
y
(D) x2
+ y2
=
2
5
AC
4
20. If four numbers in A. P. are such that their sum
is 50 and the greatest number is 4 times the
least, then the numbers are :
(A) 5, 10, 15, 20 (B) 4, 10, 16, 22
(C) 3, 7, 11, 15 (D) None of these
21. If the value of I3_23+33 +...+n3= 2025 them
the value of 1+2+3+....+ n is
(A) 45 (B) 81
(C) 285 (D) 675
22. If Sr
denotes the sum of the first r terms of an
A.P. Then, S3n
: ( S2n
-Sn
) IS
(A) n (B) 3n
(C) 3 (D) None of these
23. If
n 1 n 1
n n
a b
a b
 


is the A.M. between a and b.
Then, find the value of n.
(A) 1 (B) 2
(C) 0 (D) 3
24. A circle is inscribe in trapezoid PORS.
If PS = QR = 25 cm, PQ = 18 cm and SR = 32
cm, what is the length of the diameter of the
circle ?
(A) 14 cm
(B) 25 cm
(C) 24 cm
P Q
R
S
(D) 674 cm
25. Solution of inequality
     
2 3
x 1 x 1 x 4 0
    is:
(A) –1<x<3 (B) 2 x 4
  
(C) 1 x 4
   (D) 1 x 2
  
26. ln a rectangle ABCD the lengths of sides AB,BC,
CD, and DA are (5x + 2y +2) cm. (x + y + 4)
cm., (2x +5y – 7) cm and (3x +2y – 11) cm
respectively. Which of the following statements
is /are true?
(A) One of the sides of the rectangel is 15 cm
long.
(B) Each diagonal of the rectnagle is 39 cm
long.
(C) Perimeter of the rectangle is 102 cm.
(D) All of the above
27. If x3
= a + 1 and x + (b/x) = a, then x equals:
(A) 2
a(b 1)
a b


(B) 2
ab 1
a b


(C) 2
ab a 1
a b
 

(D) 2
ab a 1
a b
 

28. If a + b + c = 1, a2
+ b2
+ c2
= 21 and abc = 8
then find the value of (1 – a) (1 – b) (1 – c)
(A) –10 (B) –18
(C) –24 (D) –30
29. The perimeter of a triangle with vertices
(0,4), (0,0) and (3,0) is
(A) 5 (B) 12
(C) 11 (D) 7+ 5
30. The distance between the points (2,k) and
(-4,1) is 2 10 units, then the value of k is
(A) -1 (B) 1
(C) -3 (D) none of these

3. Page # 3
Corporate Office : Motion Education Pvt. Ltd., 394 - Rajeev Gandhi Nagar, Kota
PRACTICE SHEET_MATHS
31. The polynomials ax3
+ 3x2
– 3 and
2x3
– 5x + a when divided by (x – 4) leaves
remainders R1
& R2
respectively then value of
'a' if 2R1
–R2
= 0.
(A)
18
–
127
(B)
18
127
(C)
17
127
(D)
17
–
127
32. If (x + a) is a factor of x2
+ px + q and
x2
+ mx + n then the value of a is :
(A)
m– p
n – q
(B)
n – q
m– p
(C)
n p
m q


(D)
m p
n q


33. In a single throw of two dice what is the
probability of not getting the same number on
both the dice ?
(A)
1
6
(B)
2
3
(C)
5
6
(D)
1
3
34. A card is drawn at random from a pack of 52
cards. What is the probability that the card
drawn is a spade or a king ?
(A)
4
13
(B)
3
13
(C)
2
13
(D)
2
13
35. The probability of occurrence of two events E
and F are 0.25 and 0.30 respectively. The
probability of their si- multaneous occurrence
is 0.14. The probability that either E occurs or
F occurs is :
(A) 0.31 (B) 0.61
(C) 0.69 (D) 0.89
36. If the end points of the diameter of a circle
are (4,6) and (8,4), the radius of the circle
is
(A) 2 5 (B) 5
(C) 10 (D) 2 20
37. A cone is divided into two parts by drawing a
plane through the mid point of its axis parallel
to its base then the ratio of the volume of two
parts is :
(A) 1 : 3 (B) 1 : 7
(C) 1 : 8 (D) 1 : 9
38. If the angle of elevation of a cloud from a
point 200 metres above a lake is 30° and the
angle of depression of its reflection in the lake
is 600, then the height of the cloud (in metres)
above the lake is :
(A) 200 (B) 300
(C) 500 (D) none
39. The first, second and last term of an AP are
a,b and 2a. The number of terms in an AP
is
(A)
b
a a

(B)
a
b a

(C)
b
b a

(D)
a
b a

40. Sum of first 5 terms of an AP is one-fourth
of the sum of next five terms. If the first
term is -2, them its common difference is
(A) 3 (B) 6
(C) -3 (D) -6
41. If cosec  =
2 2
2 2
a b
a b


, then cot2  is
(A)
2 2
2 2 2
4a b
(a b )


(B)
2 2
2 2 2
4a b
(a b )

(C) 2 2
2ab
(a b )

(D) none of these
42. If a cos + b sin m-and b cos  sin =
n, what will be the value of a2+b2 ?
(A) m2+n2 (B) m2-n2
(C) mn (D) m+n
43.
o o
o o
cos 42 sin42
cos 42 sin42


is equal to
(A) sec 84°+ tan 84°
(B) sec 84°-tan 84°
(C) -sec 84°- tan 84°
(D) -sec 84°+tan 84°
44. From the top of a lighthouse, the angle of
depression of two opposite points were found
to be  and 90°-, respectively. If the
distance between the points is 50 m, the
height of the lighthouse is
(A) 25 sin  (B) 50 cos 
(C) 25 cos 2 (D) 25 sin 2
45. A flag which is at the top of 150 m high
building has angles of elevation of its top
and botton at a point on the ground 60°
and 30° respectively, what is the height of
the flag?
(A) 300 m (B) 500 m
(C) 300 3 m (D) 500 3 m
46. The angle of elevation of a bird flying above
an aeroplane, as observed from the
aeroplane is 30°. At same time the angle of
elevation of aeroplane as observed from a
point on the ground and vertically below the

4. Page # 4
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PRACTICE SHEET_MATHS
bird is 60°. If the shortest distance between
the bird and the aeroplane at that moment
is 1500 m, then the height of the aeroplane
above the ground is
(A) 1000 m (B) 1200 m
(C) 4500 m (D) 2250 m
47. A building which is 30 m high was observed
from a point on the ground observer found
the angle of elevation of a point on the
second floor of the building which is 10 m
above the ground same as the angle
subtended by the rest of the building above
the point P. If the height of the observer is
to be, ignored approximate distance between
the observer and the foot of the building
is use ( 3 1.732)

(A) 17.32 (B) 20
(C) 21.21 (D) none of these

5. Page # 5
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PRACTICE SHEET_MATHS
ANSWER KEY
PRACTICE SHEET - 5
MATHEMATICS
NTSE [STAGE - II]
1. B 2. C 3. B 4. A 5. D 6. D 7. D
8. C 9. C 10. B 11. B 12. B 13. C 14. D
15. C 16. D 17. A 18. A 19. D 20. A 21. A
22. C 23. C 24. C 25. B 26. D 27. C 28. B
29. B 30. B 31. B 32. B 33. C 34. A 35. B
36. B 37. C 38. A 39. C 40. D 41. A 42. A
43. B 44. B 45. B 46. C 47. A