Solved Paper

1. Which of these numbers is rational?
(a) π (b) (0.04)−1
(c) 3 −1 (d) 3 0.8
(e) 4 0.00016
2. For the equation px2 + px + q = 0, the value of the
discriminant is zero. The roots of this equation are:
(a) imaginary
(b) irrational
(c) rational and unequal
(d) rational and equal
(e) real and equal
3. The sum of two times one natural number and three
times another natural number is less than 24. If the first natural
number is less than or equal to eight, the highest value of the
second natural number is:
(a) 5 (b) 6 (c) 7
(d) 8 (e) 9
4. A shopkeeper bought 800 kg rice at Rs 3840. He had
to sell it at a loss of as much as he received for 16 kg. The
selling price (per kg, in Rs) will be:
(a) Rs 40 (b) Rs 100 (c) Rs 50
(d) Rs 80 (e) Rs 65
5. The fractions
9
14
,
8
13
,
5
7
and
7
9
can be arranged in
the descending order as:
(a)
7
9
5
7
9
14
8
13
, , ,
(b)
5
7
9
14
8
13
7
9
, , ,
(c)
9
14
5
7
8
13
7
9
, , ,
(d)
7
9
9
14
5
7
8
13
, , ,
(e)
8
13
9
14
5
7
7
9
, , ,
6. The boys and girls in a school are in the ratio 3 : 7.
25% of the boys and 20% of the girls are scholarship holders.
How much per cent of the total students do not get
scholarships?
(a) 68 (b) 75
(c) 79.8 (d) 82.5
(e) 78.5
7. If the diameter of a wire is increased by 10%, by how
much per cent approximately, will its length be decreased, if
the volume remains the same?
(a) 15 (b) 16 (c) 17
(d) 18 (e) 19
8. There are two numbers R and S, related by the
equation R = S2. Now, if S is increased by 10%, what will
happen to R?
(a) R increases by 10%
(b) R decreases by 10%
(c) R increases by 21%
(d) R decreases by 21%
(e) R remains unchanged
9. Mr X gets a salary of Rs 1,44,000 p.a. Assuming the
salary to be the same every month, what will happen to his
average income per day?
(a) Maximum for (Jan, Feb, March) period
(b) Maximum for (Feb, March, April) period
(c) Minimum for (July, Aug, Sep) period
(d) All three above are true
(e) Both (b) and (c) above
10. The value of log10 16 – 3 log10 2 + log10 5 is:
(a)
1
2
(b) 1
1
2
(c) 0 (d) 1
(e) 2
11. A cistern is two-third full of water. Pipe A can fill
the remaining part in 12 minutes and pipe B in 8 minutes.
Once the cistern is emptied, how much time will they take to
fill it together completely?
(a) 12 minutes
(b) 12 min, 12 sec
(c) 14 min, 24 sec
(d) 10 min, 12 sec
(e) 14 min, 40 sec
12. Which inequations are represented by the shaded
area shown in the graph?
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1140 JULY 2002 THE COMPETITION MASTER
(a) x > 0, y > 0 (b) x > 0, y ≥ 0
(c) x > y, y > 0 (d) x = y, y < 0
(e) x ≥ y, y ≥ 0
13. There are 7 points in a plane, no three of them being
collinear. The number of triangles formed by using these points
is:
(a) 7 (b) 21 (c) 10
(d) 4 (e) 35
14. In how many ways can we arrange 6 books on
different subjects, in a shelf?
(a) 6 (b) 60 (c) infinite
(d) 720 (e) 120
15. The recurring decimal 2.345 can be expressed in the
rational form as:
(a)
2343
990
(b)
2345
999
(c)
2343
999
(d)
2345
99
(e)
2345
1000
16. If a = 4 9 , b = 6 26 , c = 3 5 , then, which of these
statements is true?
(a) 4 9 < 6 26 < 3 5
(b) 4 9 > 6 26 > 3 5
(c) 6 26 > 3 5 > 4 9
(d) 6 26 < 3 5 < 4 9
(e) None of these
17. The value of the expression x 7x 10
x 2x 15
2
2
+ +
+ −
is:
(a)
x 5
x 3
+
− (b)
x 2
x 3
+
− (c)
x 3
x 2
−
+
(d)
x 5
x 2
+
+ (e)
x 2
x 5
+
+
18. If 3
1
2
x + 2
1
2
x = 64 – 2x, then x equals:
(a) 8 (b) 7 (c) 6
(d) 5 (e) 4
19. A 90 kg salt solution has 10% salt in it. How much
water must be evaporated from it so that the solution now
contains 25% salt?
(a) 54 (b) 45 (c) 36
(d) 27 (e) 18
20. Mr Mohit is 7 times as old as his son. 10 years hence
he will be 3 times as old as his son. What are their present
ages (in years)?
(a) 4, 28 (b) 5, 35 (c) 3, 21
(d) 6, 42 (e) 7, 49
Directions (Q. 21-25): Study the following bar-graph and
answer the questions (21-25) given below. (The diagram shows the
sale of six garment companies in three successive financial years.)
21. Which of the following garment companies has a
fluctuating sales figure over the given period?
(a) Lambu Tewari and Brothers
(b) Dabbu Bhai and Sons
(c) Motu Bhai and Brothers
(d) Champa Garments
(e) Shyamu and Company
22. What is the total percentage increase in the sale of
garments in 2000-2001 with respect of 1999-2000?
(a) 4% fall (b) 4% rise
(c) No change (d) 7% increase
(e) 5% decrease
23. For the total 3-year period under consideration, the
nearest competitor of Ramu and Company is:
(a) Champa Garments
(b) Motu Bhai and Brothers
(c) Shyamu and Company
(d) Dabbu Bhai and Sons
(e) None of these
24. For the years 1998-99 and 1999-2000, which company
has the minimum rate of change of sales?
(a) Dabbu Bhai and Sons
(b) Lambu Tewari and Brothers
(c) Ramu and Company
(d) Champa Garments
(e) None of these
25. For the years 1999-2000 and 2000-2001, which
company has the maximum rate of change of sales?
(a) Dabbu Bhai and Sons (b) Ramu and Company
(c) Shyamu and Company (d) Lambu Tewari and Sons
0
10
20
30
40
50
60
70
80
90
100
44
66
80
46
68
52
33
57
63
34
57
60
20
40
43 46
40
30
Ramu & Co. Shyamu
& Co.
Champa
Garments
Motu Bhai
& Brothers
Lambu
Tewari &
Brothers
Dabbu Bhai
& Sons
1998-99 1999-2000 2000-2001
Sales in Rs. (Crores)
1
1 2 3 4
2
3
4
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1141 JULY 2002 THE COMPETITION MASTER
(e) Champa Garments
26. A train running at the speed of 90 km/hr crosses a
platform of length 160 m in 10 seconds. What is the length of
the train (in metres)?
(a) 60 (b) 90 (c) 150
(d) 140 (e) 40
27. The average of 4 consecutive even numbers is 9.
Which of these is the first number?
(a) 6 (b) 4 (c) 8
(d) 10 (e) 12
28. The distance between Charu’s and Mani’s places is
120 km. Charu travelled the whole distance from her place to
Mani’s at 30 km/hr but returned at 40 km/hr. Her average
speed for the whole journey is (approximately):
(a) 36 km/hr (b) 37 km/hr (c) 33 km/hr
(d) 34 km/hr (e) 35 km/hr
29. The average of the 1st 50 natural numbers is:
(a) 25 (b) 25.5 (c) 26.4
(d) 50 (e) 50
30. A certain sum of money doubles in 10 years at simple
interest. What is the rate of interest?
(a) 20% (b) 30% (c) 10%
(d) 5% (e) 12%
Directions (Q. 31-35): Study the pie-chart and table given
below and answer the questions that follow—(The data shows
statistics about 4 schools (A, B, C and D) as in the year 2001.)
31. Which school has the maximum number of girl
students?
(a) A (b) B (c) both C and D
(d) C (e) D
32. If the number of students of school A increase by
12.5% and that of school B decrease by 10%, what is the ratio
of the number of students in the two schools?
(a) 2 : 3 (b) 3 : 2 (c) 1 : 1
(d) 4 : 3 (e) 3 : 4
33. In school B, 10% students failed in science, 20% failed
in commerce and 30% failed in arts. What is the percentage of
failures in school B?
(a) 15 (b) 25 (c) 12
(d) 12
1
2
(e) 17
34. The total number of arts students expressed as a
percentage of total number of commerce students is
(approximately):
(a) 32 (b) 36 (c) 35
(d) 30 (e) 38
35. How many girls are there in science stream of school
D?
(a) 264 (b) 704 (c) 88
(d) 264 (e) Data inadequate
36. A square is inscribed inside a circle of radius 4 cm.
What is the area of the shaded region in the diagram?
(a) 10 cm2 (b) 8 cm2 (c)
20
3
cm2
(d) 16 cm2 (e) 8 cm2
37. A square is drawn inside a right-angled triangle with
the two perpendicular sides as 12 cm and 8 cm. What is the
side of the largest possible square that can be drawn?
(a) 4 cm (b) 4.8 cm (c) 4.5 cm
(d) 4.4 cm (e) 5 cm
38. Three terrorists are employed to shoot a reknowned
person Mr X. Only one bullet is sufficient to kill him if it
strikes in the head. The probabilities of the terrorists striking
Mr X’s head by their bullets are 0.2, 0.3 and 0.4. What is the
probability that Mr X is shot dead?
(a) 0.336 (b) 0.9 (c) 0.1
(d) 0.760 (e) 0.664
39. Which of the following numbers is a perfect square?
(a) 10201 (b) 12222 (c) 11112
(d) 55555 (e) 10101
40. The following two figures have the same perimeter.
Which of the following statements is true?
(a) The square and the circle have equal areas.
(b) The area of square is greater than that of the circle.
(c) The area of the circle is greater.
(d) Area of circle is π times that of the square.
(e) None of these.
41. Prasoon’s bike needs a fresh paint. He wants 2 shades
on his bike. The painter shows all the available 5 shades. In
how many ways can Prasoon paint his bike?
(a) 7 (b) 10 (c) 25
(d) 20 (e) 52
School Science : Boys :
Commerce : Girls
Arts
A 1 : 4 : 1 5 : 4
B 4 : 1 : 1 5 : 1
C 2 : 5 : 2 2 : 7
D 3 : 5 : 1 1 : 8
Students’ Ratio
Square Circle
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1142 JULY 2002 THE COMPETITION MASTER
42. A bag contains 4 white and 3 black balls. 2 balls are
drawn out one at a time, randomly in succession. What is the
probability that both the balls drawn out are white in colour,
if the first ball is replaced before the second draw is made.
(a)
9
49
(b)
4
12
(c)
12
49
(d)
4
7
(e)
16
49
43. What is the maximum value of the function
f(x) = x2 + 5x + 16
(a)
16
5
(b)
4
3
(c) 16
(d)
39
4
(e)
25
16
44. On giving a reduction of 20% on clothes, a clothmerchant’s
sale increased by 30% to become Rs 26,000 in the
month of June 2002. What was the previous sale?
(a) Rs 25,000 (b) Rs 30,000
(c) Rs 13,000 (d) Rs 24,500
(e) Rs 20,000
45. The difference between the product of two numbers
and their sum is 24. What is the difference between the two
numbers?
(a) 24
(b) 25
(c) Data inadequate
(d) Several solutions
(e) None of these
46. Mr Sanjay Sharma, Mr Pravesh Khare and Mr Mitra
are partners in a business. Mr Mitra started the business with
Rs 40,000. After 3 months, Mr Sanjay and Mr Pravesh joined
him with Rs 60,000 each. If the total profits at the end of the
year amount to Rs 31,200, what would be Mr Khare’s share in
it?
(a) Rs 15600 (b) Rs 9600
(c) Rs 10800 (d) Rs 21600
(e) None of these
47. Mr Rajiv and Mr Jogesh can complete a piece of work
in 6 days and 8 days respectively. They started together but
Rajiv left the work after 2 days. In how many days will Mr
Jogesh complete the remaining work now?
(a) 4
3
4
days
(b) 2
2
3
days
(c) 4
1
2
days
(d) 4 days
(e) 3
1
3
days
Directions (Q. 48-50): Study the given series (A) carefully
and answer the incomplete series (B), in the following questions.
Assume that series B follows the same rule as series A.
48. Series A: 12 24 48 96 192
Series B: 0.5 x y z t
What should come in place of t?
(a) Both (d) and (e) (b) 4 (c) 84.5
(d) 8 (e) 180.5
49. Series A: 2 5 17 71 ...
Series B: 1 a b c
What will replace the symbol b?
(a) 47 (b) 11 (c) 3
(d) 17 (e) 12
50. Series A: 2 11 47 128 ...
Series B: 5 m n o ...
The value of n is ...
(a) 27 (b) 50 (c) 25
(d) 21 (e) 62
Answers (with Hints and Solutions)
1. (b) (0.04)−1 =
100
4
= ±FH G IK J
10
2
2. (e) If D = 0 → the roots are real and equal
3. (c) We can write: 2x + 3y < 24
The highest value of second can take place when the
first is having lowest value.
Lowest value of 1st = 1
(Q smallest natural number = 1)
Thus, 2(1) + 3y < 24
3y < 22
y <
22
3
~ 7
4. (a) Let SP per kg be Rs x
→ Total SP = 80 x and loss = 16 x
Now, loss = CP – SP
→ 16x = 3840 – 80x
i.e. x = 40
5. (a) By cross-multiplication (or otherwise), compare the 4
fractions. Thus,
7
9
>
5
7
>
9
14
>
8
13
6. (e) Scholarship holders =
25(3x)
100
20
100
+ (7x)
=
3x
4
1
5
+ .7x =
43x
20
Total = 3x + 7x = 10 x
% of scholarship holders =
43x
20
10x
F
H
GGG
I
K
JJJ
× 100 = 21.5
∴ % of those without scholarships = 78.5
7. (c) V1 = V2 →π × 100 × 100 × l1 = π × 110 × 110 × l2
SPECIAL SUPPLEMENT
1143 JULY 2002 THE COMPETITION MASTER
→
l
l
2
1
=
100
121
∴ % decrease =
21
121
× 100 ≅ 17
8. (c) R = S2
S increases by 10% and becomes S +
S
10
i.e.
11
10
S
Since R = S2 → R becomes (
11
10
S)2 =
121
100
S2
→ % increase in R = 21%
9. (e) February has 28 days while July and August have 31
days. Find average using number of days.
10. (d) Given expression = log10 (
16 5
8
×
) = log10 10 = 1
11. (c) Pipes A and B fill
1
3
tank in 12 and 8 minutes
→ They fill the whole tank in 36 and 24 minutes
1 minute’s work =
1
36
1
24
5
72
+ =
→ Time taken =
72
5
minutes = 14.4 minutes
= 14 minutes, 24 seconds
12. (e) Refer special P.O. Exam issues of THE COMPETITION
MASTER (May and June 2002)
13. (e) 7C3 = 35 (Refer May and June issues of CM)
14. (d) 6P6 or ∠6 = 720 (Refer May and June issues of CM)
15. (c)
16. (b) LCM of 4, 6, 3 is 12
Thus, 4 9 = 9 b g1 4 = 9 b g3 12 = 93 1 12 e j = 12 729
6 26 = 26 b g1 6 = 26 b g2 12 = 262 1 12 e j = 12 676
and 3 5 = 5 b g1 3 = 5 b g4 12 = 54 1 12 e j = 12 625
∴ 12 729 > 12 676 > 12 625
17. (b) Factorise by splitting the middle term
Thus, we have: (x 2)(x 5)
(x 5) (x 3)
+ +
+ −
= (x 2)
(x 3)
+
−
18. (a) Solve as simple linear equations.
19. (a) Quantity of salt = 90 ×
10
100
= 9 kg
If ‘x’ kg water is evaporated,
9
90 − x
FH G
IK J
× 100 = 25
Thus, x = 54
20. (b) Let the son be x years old and father be 7x years old
Their ages after 10 years are (x + 10) and (7x + 10)
Thus, according to the question, we have
(7x + 10) = 3(x + 10)
x = 5 and father’s age = 7 × 5 = 35 years
21. (e) Shyamu and Company—It first registered a growth
and then a fall in the sales.
22. (c) Sum of sales figures in both the years are the same i.e.
328 crore each
23. (c) The total sales of Ramu and Company is Rs 200 crore
which is closest to Rs 166 crore of Shyamu and
Company.
24. (a) For Dabbu Bhai and Sons, the rate of change is only
46 40
46
−
i.e.
6
46
25. (a) For Dabbu Bhai and Sons, the rate of change is
40 30
40
1
4
− =
For others it is less than this.
26. (b) Refer June issue for direct formula on TRAIN problems
10 =
18
5
x
90
FH G
IK J
x = 250 → length = 250 – 160 = 90 m
27. (a) Let the first number be x
Then, x (x 2) (x 4) (x 6)
4
+ + + + + +
= 9
→ 4x = 36 – 12
→ x =
24
4
= 6
28. (d) Average speed =
2 30 40
30 40
2400
70
× ×
+
= = 34.33 km/hr
29. (b) Sum of 1st n natural numbers is
n
2
[a + l]
Where n = number of terms, a= 1st term, l = last term
Thus, average =
n
2
a
n
[ + l]
=
a + l
2
=
1 50
2
+
= 25.5
30. (c) S.I. = Amount – Principal
→ S.I. = 2P – P = P
Now, S.I. =
PTR
100
→ P =
P 10 R
100
× ×
→ R = 10%
31. (d) First put all the data in a relevant form
From the pie-chart, we get,
A =
20
100
3600
3600
5
× = = 720
B =
25
100
3600
3600
4
× = = 900
C =
33
100
× 3600 = 1188
D =
22
100
× 3600 = 792
SPECIAL SUPPLEMENT
1144 JULY 2002 THE COMPETITION MASTER
Now, dividing the students as per the table, we have:
32. (c) 12.5% increase in A = 720 + 90 = 810
10% decrease in B = 900 – 90 = 810
Ratio =
810
810
= 1 : 1 = 1
33. (a) Number of failures
=
10
100
600
20
100
150
30
100
× + × + × 150
= 60 + 30 + 45 = 135
% failures =
135
900
× 100 = 15
34. (b) Required percentage
=
120 150 264 88
480 150 660 440
100
+ + +
+ + +
×
=
622
1730
× 100 = 36 (approx.)
35. (e) The given data only provides the number of students
in the 3 streams and the ratio of boys to girls in a
particular school.
36. (b) Area =
1
2
× r2 =
1
2
× 42 = 8 cm2
(Solve using Pythagoras’ theorem)
37. (b)
The triangles AXY and YZC are similar
Thus, we have:
x
8 − x =
12 x
x
i.e., x2 = 96 – 20x + x2
i.e., x = 4.8 cm
38. (e) (Refer May issue of CM for a more systematic approach
to this topic)
Required Probability = 1– q1 q2 q3 (Where q = proba-
= 1– 0.8×0.7×0.6 bility of fail-
= 1– 0.336 ing to strike)
= 0.664
39. (a) (101)2 = 10201
40. (c) Let side of square be ‘x’ cm, i.e. perimeter = 4x cm
Since perimeters are same → 4x = 2π r i.e. r =
2x
π
Now, Area of square = x2 and area of circle
48. (a) We can have two possibilities
= π
π π
2 4 2
2 x FH G
IK J
=FH G
IK J
x
Since
4
π > 1 → Area of circle > Area of square
41. (d) (Refer April 2002 issue of CM for more details on the
topic of permutations and combinations).
Using arrangements or permutations, we have
5
2 P =
5!
3!
=
5 × 4 × 3
3
!
!
= 20
42. (e) (Refer May 2002 issue of CM for more details on
probability).
Required probability = p (1st white) × p (2nd white)
=
4
4 + 3 ×
4
4 + 3 =
16
49
43. (d) Maximum value = c
b
4a
2
= 16
5
4
2
− ( )
= 16
25
4
− =
39
4
(of ax2 + bx + c)
44. (a) Required figure = 26,000 ×
100
80
100
130
× = Rs 25,000
45. (c) Let the two numbers be x and y
It is only given that xy – (x + y) = 24
We need at least two equations to solve for the two
unknown variables
46. (c) Money will be divided according to the time and
investment ratios
Thus, required ratio
= 40,000 × 12 : 60,000 × 9 : 60,000 × 9
= 48 : 54 : 54 = 8 : 9 : 9
Mr Khare’s share =
9
8 9 9
31200
+ +
× = Rs 10800
47. (e) Their 1-day work =
1
6
1
8
7
24
+ =
∴ They do 2
7
24
FH G
IK J
work in 2 days
Remaining work = 1 –
14
24
=
5
12
Days taken by Jogesh =
5
12
8
10
3
× =
× 2, × 2, × 2 ...
+ 12, + 24, + 48 ...
Thus, both (d) and (e)
49. (b) The series A is: ×2 + 1, × 3 + 2, × 4 + 3 ...
Thus, 1 × 2 + 1 = 3, 3 × 3 + 2
= 11, 11 × 4 + 3 = 47 etc
50. (b) The given series A is: +32, +62, +92, etc
Thus, series B is: m = 5 + 32 = 14,
n = 14 + 62 = 50, etc
A
B C
X
8-x
Y
Z
8 x
x
x
x
12-x
12
School Science : Comm. : Arts Boys : Girls
A(720) 120 / 480 / 120 400 / 320
B(900) 600 / 150 / 150 750 / 150
C(1188) 264 / 660 / 264 264 / 924
D(792) 264 / 440 / 88 88 / 704