MATHS - QUESTION

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This file contains sample questions for three levels:
1. School Level
2. Competition Level
3. Graduate Level
1. School Level
Note: The incorrect options need to be logical. Explanations are needed for the right answers as well as for
the logical distracters.
1. If A:B = 2:3 and B:C = 4:5, then A:B:C is
(1) 2:3:5 (2) 5:4:6 (3) 6:4:5 (4) 8:12:15
2. If (a - b) is 6 more than (c + d) and (a + b) is 3 less than (c - d), then (a - c) is
(1) 0.5 (2) 1 (3) 1.5 (4) 2.5
3. If the altitude of a triangle is increased by 10 per cent while its area remains same, then its
corresponding base will have to be decreased by
(1) 10.02 per cent (2) 9.01 per cent (3) 9.11 per cent (4) 11.9per cent
4. A number
 – is less than 50
 – is multiple of 7
 – has total 3 factors
 What is this number?
 (1) 14 (2) 42 (3) 49 (4) 35
5. Amita started to draw her field at 11 : 55 a.m. and finished at 12 : 05 p.m. How much time did she take
to complete the drawing?
 (1) 50 minutes (2) 1 hour 50 minutes (3) 10 minutes (4) 1 hour 10 minutes
6. Two brands of chocolate are available in the packing of 10 and 12. If I have to take both brands of
chocolate with the same quantity, what is the least number of packets of each brand should I purchase?
 (1) Two pack of 10 and two pack 12 (2) Six pack of 10 and five pack of 12
 (3) Twelve pack of 10 and ten pack of 12 (4) Twenty two pack of each
7. Study the following pattern and find out next term.
2025
45
45
1225
35
35
625
25
25
225
15
15
25
5
5
× × × × ×
 (1) 2825
55
55
× (2) 3225
55
55
× (3) 3025
55
55
× (4) 10025
55
55
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8. If 2x + 3y : 3x + 5y = 18 : 29, then
y
x is equal to
 (1) 4
3 (2) 3
4 (3) 3
2 (4) 6
5
9. A piece of paper in the shape of a sector of a circle (see figure 1) is rolled up to form a right-circular
cone (see figure 2). The value of the angle θ is
 (1) 13
10π (2) 13
9π (3) 13
5π (4) 13
6π
10. Consider an incomplete pyramid of balls on a square base having 18 layers; and having 13 balls on
each side of the top layer. Then the total number N of balls in that pyramid satisfies
 (1) 9000 < N < 10000 (2) 8000 < N < 9000 (3) 7000 < N < 8000 (4) 10000 < N < 12000
11. A frog is presently located at the origin (0, 0) in the xy-plane. It always jumps from a point with integer
coordinates to a point with integer coordinates moving a distance of 5 units in each jump. What is the
minimum number of jumps required for the frog to go from (0, 0) to (0, 1)?
 (1) 2 (2) 3 (3) 4 (4) 9
12. There are 30 questions in a multiple choice test. A student gets 1 mark for each unattempted question.
0 mark for each wrong answer and 4 marks for each correct answer. A student answered x question
correctly and scored 60. The possible value of x can be
 (1) 15 (2) 10 (3) 6 (4) 5
13. Three taps A, B and C fill up a tank independently in 10 hr, 20 hr, 30 hr, respectively. Initially the tank is
empty and exactly one pair of taps is open during each hour and every pair of taps is open at least for
one hour. What is the minimum number of hours required to fill the tank?
 (1) 8 (2) 9 (3) 10 (4) 11 TCYonline.com 3
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2. Competition Level
Note: The incorrect options need to be logical. Explanations are needed for the right answers as well as for
the logical distracters.
Directions for Questions 1 - 2: Consider the following:
Let f(x) = and
0, x 1
1 x, x 1
⎜
⎜
⎝
⎛
>
− <
g(x) = f(x – 1) + f(x + 1) for all x ∈R.
1. What is ( ) ∫
−
0
3
g x dx equal to?
 (1) 0 (2) 1
 (3) 2 (4) None of the above
2. What is ( ) ∫
3
0
g equal to? x dx
 (1) 0 (2) 1
(3) 2 (4) None of the above
3. Consider the following statements:
 1. f(x) is a continuous function for all ak ⊂ R.
 2. f(x) is a differentiable function for all ak ∈ R.
 Which of the statements given above is/are correct?
 (1) 1 only (2) 2 only (3) Both 1 and 2 (4) Neither 1 nor 2
4. Consider the following statements:
 1. f(x) is continuous at x = 1 provided a2k = 0.
 2. f(x) is differentiable at x = 1 provided a2k+1 = 0.
 Which of the statements given above is/are correct?
 (1) 1 only (2) 2 only (3) Both 1 and 2 (4) Neither 1 nor 2
Directions for Questions 5 – 6: Consider the function f(x) = |x2
 – 2| on the interval – 3 < x < 3.
5. Which of the following statements is/are correct?
 1. The function attains relative minimum at only one point.
 2. The function attains relative maximum at only one point.
 (1) 1 only (2) 2 only (3) Both 1 and 2 (4) Neither 1 nor 2
6. Which of the above statements is/are correct?
 1. The absolute maximum value of the function is 2.
 2. The absolute minimum value of the function is 0.
 (1) 1 only (2) 2 only (3) Both 1 and 2 (4) Neither 1 nor 2 TCYonline.com 4
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Directions for Questions 7 - 8:
A line L is perpendicular to the lines having direction ratios <3, 2, 1> and <1, 2, 3>.
7. What is the sum of the direction cosines of the line L?
 (1) 0 (1) 1
 (3) 3 (4) None of the above
8. Consider the following statements:
1. The sum of the direction ratios of the line L is equal to sum of the direction cosines of the line
 L.
 2. The angle made by the line L with the y-axis is twice the angle made by the line L with the x-
 axis.
 Which of the statements given above is/are correct?
 (1) 1 only (2) 2 only (3) Both 1 and 2 (4) Neither 1 nor 2
Directions for questions 9- 11 : Consider the plane containing the line 1
z 2
2
y 3
3
x 1
−
+ = − = −
+ and passing
through the point (1, – 1, 0).
9. What are the direction ratios of the normal to the plane?
 (1) <1, 1, 1> (2) <1, – 1, 1>
 (3) <3, 2, 1> (4) None of the above
10. What is the angle made by the plane with the x-axis?
 (1) tan 2 −1 (2) cot 2 −1
(3) 6
π (4) None of the above
11. What is the equation of the plane?
 (1) x + y – z = 0 (2) – 3x + 2y – z + 6 = 0
 (3) x + y + z = 0 (4) None of the above
12. Two trains, of same length, are running in parallel tracks in opposite directions with speed 65 km/hr and
85 km/hr respectively. They cross each other in 6 seconds. The length of each train is
 (1) 100 metre (2) 115 metre (3) 125 metre (4) 150 metre
13. 3333(333)(333)(334)3333333334++----ΧΧΧ is equal to
(1) 12 (2) 11 (3) 10 (4) 15
14. The value of (12 + 22 + 32…..+ 202) - (1 + 2 + 3 +….+ 20) is
(1) 5320 (2) 4200 (3) 3150 (4) 2660 TCYonline.com 5
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15. In the figure, AKHF, FKDE and HBCK are unit squares and AD and BF intersect at a point X. The ratio
of the areas of triangles AXF and ABF is
(1) ¼ (2) 1/5 (3) 1/8 (4) 1/6
16. A regular octagon is formed by cutting congruent isosceles right-angled triangles from the corners of a
square. If the square has side length 1, then the side-length of the octagon is
 (1) 2
2 − 1 (2) 2 − 1 (3) 4
5 − 1 (4) 3
5 − 1
17. the value ( ) ( ) ∫
−
+ +
2012
2012
3 5 sin x x 1dx is
 (1) 2012 (2) 2013 (3) 0 (4) 4024
18. In the adjacent figures, which has the shortest path?

(1) Fig. 1 (2) Fig. 2 (3) Fig. 3 (4) Fig. 4 TCYonline.com 6
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3. Graduate Level
Note: The incorrect options need to be logical. Explanations are needed for the right answers as well as for
the logical distracters.
1. In a race of 1000 m, A can beat B by 100 m. In a race of 400 m, B beats C by 40 m. In a race of 500 m,
A will beat C by
(1) 95 m (2) 50 m (3) 45 m (4) 60 m
2. 1111...1223341011++++ΧΧΧΧ is equal to
(1) 911 (2) 1011 (3) 1211 (4) 109 110
3. The least one of 4, 23
, 25
 and 32
 is
(1) 23
 (2) 4 (3) 25
 (4) 32
4. The difference between a discount of 40 per cent on Rs. 500 and two successive discounts of 36 per
cent and 4 per cent on the same amount is
(1) zero (2) Rs. 1.93 (3) Rs. 2.00 (4) Rs. 7.20
5.
1.5625 0.0289 72.9 64
0.081 0.324 4.624
× × ×
× × is
 (1) 24 (2) 2.4 (3) 0.024 (4) 2.04
6. If x + y + z = 15, xy + yz + zx = 75, then 3z
x + 4y + z is equal to
 (1) 1 (2) 0 (3) 2 (4) – 1
7. If the difference between an exterior angle of a regular polygon of n sides and an exterior angle of
another regular polygon of (n + 1) sides is equal to 5°, then the value of n is
 (1) 5 (2) 6 (3) 7 (4) 8
8. The value of
6
1 6
1 2
7
21
5
1 2
8 8
−
−
−
− × is
 (1) 0 (2) 4 (3) 3 (4) 2
9. The total number of prime factors in (25)10 × (24)20 × (26)25 is
 (1) 1000 (2) 120 (3) 150 (4) 140
10. Let a, b and c be numbers in the set (1, 2, 3, 4, 5, 6) such that the curves y = 2x3
 + ax + b and y = 2x3
 +
cx + d have no point in common. The maximum possible value of (a – c)2
 + b – d is
 (1) 0 (2) 5 (3) 30 (4) 36
11. Consider the conic ex2
 + π y2
 – 2e2
x – 2 π 2
y + e3
 + 3 π = e. Suppose P is any point on the conic and S1
. S2 are the foci of the conic, then the maximum value of (PS1 + PS2) is
 (1) πe (2) πe (3) 2 π (4) 2 eTCYonline.com 7
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12. The value
( )
( ) ∫
∫
π
+
π
+
/ 2
0
2 1
/ 2
0
2 1
sin x dx
sin x dx
 is
 (1)
2 1
2 1
−
+ (2)
2 1
2 1
+
− (3)
2
2 + 1 (4) 2 − 2
13. Let [x] and {x} be the integer part and fractional part of a real number x respectively. The value of the
integeral ∫
5
0
[ is x] {x} dx
 (1) 5/2 (2) 5 (3) 34.5 (4) 35.5
14. Let f(x) =
x 1
x 1
−
+ for all x ≠ 1. Let f1
(x) = f(x), f2
(x) = f(f(x)) and generally fn
(x) = f(fn–1(x)) for n > 1. If P =
f
1
(2) f2
(3) f3
(4) f4
(5), which of the following is a multiple of P?
 (1) 125 (2) 375 (3) 250 (4) 147