CAT 1991 QA questions and solutions

Previous year paper questions for CAT 1991 QA

1. CAT 1991 QA | Arithmetic - Simple & Compound Interest

The initial amount in rupees, deposited by Ghosh Babu was:

A.
25,000
B.
75,000
C.
50,000
D.
None of these

2. CAT 1991 QA | Arithmetic - Simple & Compound Interest

The year, at the end of which, Ghosh Babu withdrew the smallest amount was:

A.
First
B.
Second
C.
Third
D.
Fourth

3. CAT 1991 QA | Arithmetic - Simple & Compound Interest

The year, at the end of which, Ghosh Babu collected the maximum interest was:

A.
First
B.
Second
C.
Third
D.
Fourth

4. CAT 1991 QA | Arithmetic - Simple & Compound Interest

The year, at the end of which, Ghosh Babu withdrew the maximum amount was:

A.
First
B.
Second
C.
Third
D.
Fourth

5. CAT 1991 QA | Arithmetic - Simple & Compound Interest

The total interest, in rupees, collected by Ghosh Babu was:

A.
12,000
B.
20,000
C.
4,000
D.
11,000

6. CAT 1991 QA | Arithmetic - Percentage

Prakash should not test if the number of bad widgets in the lot is:

A.
less than 100
B.
more than 200
C.
between 120 & 190
D.
Cannot be found out.

Answer: Option A

Video Explanation

Explanation :

Let the number of defective widgets be ‘x’.

Cost to Prakash if he does not use any test = 50x

Cost to Prakash if he uses test 1 = 2 × 1000 + $\frac{4 x}{5}$ × 25 + $\frac{x}{5}$ × 50 = 2000 + 30x

Cost to Prakash if he uses test 2 = 3000 + 25x

Prakash should not test when 50x ≤ (2000 + 30x) and (3000 + 25x)

50x ≤ (2000 + 30x) and 50x ≤ (3000 + 25x)

20x ≤ 2000 and 25x ≤ 3000

x ≤ 100 and x ≤ 120

x ≤ 100

Prakash should use test 1 when test 1 is cheaper than test 2

3000 + 25x ≥ 2000 + 30x

5x ≤ 1000

x ≤ 200

For x ≥ 200 he can use test 2.

Below 100, no test would be cheaper.

Hence, option (a).

7. CAT 1991 QA | Arithmetic - Percentage

If there are 120 defective widgets in the lot, Prakash:

A.
should either use Test I or not test.
B.
should either use Test II or not test
C.
should use Test I or Test II.
D.
should use Test I only.

8. CAT 1991 QA | Arithmetic - Percentage

If the number of defective widgets in the lot is between 200 and 400, Prakash:

A.
may use Test I or Test II
B.
should use Test I only
C.
should use Test II only
D.
cannot decide

9. CAT 1991 QA | Arithmetic - Percentage

If Prakash is told that the lot has 160 defective widgets, he should:

A.
use Test I only
B.
use Test II only
C.
do no testing
D.
either use Test I or do not test

10. CAT 1991 QA | Arithmetic - Percentage

If there are 200 defective widgets in the lot, Prakash:

A.
may use either Test I or Test II
B.
should use Test I or not use any test
C.
should use Test II or not use any test
D.
cannot decide

11. CAT 1991 QA | Algebra - Functions & Graphs

A function can sometimes reflect on itself, i.e. if y = f(x), then x = f(y). Both of them retain the same structure and form. Which of the following functions has this property?

A.
y = $\frac{2 x + 3}{3 x + 4}$
B.
y = $\frac{2 x + 3}{3 x - 2}$
C.
y = $\frac{3 x + 4}{4 x - 5}$
D.
None of the above

Answer: Option B

Video Explanation

Explanation :

For each of the given expressions, you may have to simplify and express x in terms of y and hence verify for which one does the form & structure remain the same.

In general, any function of the form y = $\frac{a x + b}{b x - a}$ reflects on to itself as we can rearrange it as

x = $\frac{a y + b}{b y - a}$

Let us check option (b).

y = f(x)

⇒ y = $\frac{2 x + 3}{3 x - 2}$

⇒ 3xy - 2y = 2x + 3

⇒ 3xy - 2x = 2y + 3

⇒ x(3y - 2) = (2y + 3)

⇒ x = $\frac{2 y + 3}{3 y - 2}$

∴ x = f(y)

Hence, the answer is (b).

12. CAT 1991 QA | Algebra - Simple Equations

What is the value of k for which the following system of equations has no solution:

2x – 8y = 3 and kx +4y = 10

A.
-2
B.
1
C.
-1
D.
2

Answer: Option C

Video Explanation

Explanation :

For two lines a₁x + b₁y = c₁ and a₂x + b₂y = c₂ to be parallel, the condition is

$\frac{a_{1}}{a_{2}}$ = $\frac{b_{1}}{b_{2}}$ ≠ $\frac{c_{1}}{c_{2}}$

For no solution, lines must be parallel and not over lapping

⇒ $\frac{2}{k} = \frac{- 8}{4} \neq \frac{3}{10}$

∴ k = − 1.

Hence, option (c).

13. CAT 1991 QA | Modern Math - Permutation & Combination

How many 3-digit even numbers can you form such that if one of the digits is 5 then the following digit must be 7?

A.
5
B.
405
C.
365
D.
495

14. CAT 1991 QA | Arithmetic - Time & Work

Alord got an order from a garment manufacturer for 480 Denim Shirts. He brought 12 sewing machines and appointed some expert tailors to do the job. However, many didn’t report to duty. As a result, each of those who did, had to stitch 32 more shirts than originally planned by Alord, with equal distribution of work. How many tailors had been appointed earlier and how many had not reported for work?

A.
12, 4
B.
10, 3
C.
10, 4
D.
None of these

15. CAT 1991 QA | Algebra - Simple Equations

Iqbal dealt some cards to Mushtaq and himself from a full pack of playing cards and laid the rest aside. Iqbal then said to Mushtaq. “If you give me a certain number of your cards, I will have four times as many cards as you will have. If I give you the same number of cards, I will have thrice as many cards as you will have “. Of the given choices, which could represent the number of cards with Iqbal?

A.
9
B.
31
C.
12
D.
35

16. CAT 1991 QA | Venn Diagram

Fifty college teachers are surveyed as to their possession of colour TV, VCR and tape recorder. Of them, 22 own colour TV, 15 own VCR and 14 own tape recorders. Nine of these college teachers own exactly two items out of colour TV, VCR and tape recorder; and, one college teacher owns all three. How many of the 50 teachers own none of the three, colour TV, VCR or tape recorder?

A.
4
B.
9
C.
10
D.
11

17. CAT 1991 QA | Algebra - Simple Equations

Three times the first of three consecutive odd integers is 3 more than twice the third. What is the third integer?

A.
15
B.
9
C.
11
D.
5

18. CAT 1991 QA | Modern Math - Permutation & Combination

What is the total number of ways to reach A to B in the network given?

A.
12
B.
16
C.
20
D.
22

19. CAT 1991 QA | Geometry - Quadrilaterals & Polygons

Let the consecutive vertices of a square S be A, B, C & D. Let E, F & G be the mid-points of the sides AB, BC & AD respectively of the square. Then the ratio of the area of the quadrilateral EFDG to that of the square S is nearest to

A.
1/2
B.
1/3
C.
1/4
D.
1/8

20. CAT 1991 QA | Algebra - Surds & Indices

2⁷³ – 2⁷² – 2⁷¹ is the same as

A.
2⁶⁹
B.
2⁷⁰
C.
2⁷¹
D.
2⁷²

21. CAT 1991 QA | Algebra - Number Theory

The number of integers n satisfying –n + 2 ≥ 0 and 2n ≥ 4 is

A.
0
B.
1
C.
2
D.
3

22. CAT 1991 QA | Algebra - Number Theory

The sum of two integers is 10 and the sum of their reciprocals is 5/12. Then the larger of these integers is

A.
2
B.
4
C.
6
D.
8

23. CAT 1991 QA | Geometry - Circles

A circle is inscribed in a given square and another circle is circumscribed about the square. What is the ratio of the area of the inscribed circle to that of the circumscribed circle?

A.
2 : 3
B.
3 : 4
C.
1 : 4
D.
1 : 2

24. CAT 1991 QA | Algebra - Functions & Graphs

If y = f(x) and f(x) = $\frac{(1 - x)}{(1 + x)}$ , which of the following is true?

A.
f(2x) = f(x) - 1
B.
x = f(2y) - 1
C.
f(1/x) = f(x)
D.
x = f(y)

Answer: Option D

Video Explanation

Explanation :

y = f(x) = $\frac{1 - x}{1 + x}$

⇒ y(1 + x) = 1 – x

⇒ y + xy = 1 – x

⇒ x + xy = 1 – y

⇒ x(1 + y) = (1 – y)

⇒ x = $\frac{1 - y}{1 + y}$ = f(y)

Since here, y = f(x) and x = f(y), the given function refleccts on itself.

Hence, option (d).

25. CAT 1991 QA | Venn Diagram

How many schools had none of the three viz., laboratory, library or play – ground?

A.
20
B.
5
C.
30
D.
35

26. CAT 1991 QA | Venn Diagram

What was the ratio of schools having laboratory to those having library?

A.
1 : 2
B.
5 : 3
C.
2 : 1
D.
2 : 3

27. CAT 1991 QA | Modern Math - Probability

A player rolls a die and receives the same number of rupees as the number of dots on the face that turns up. What should the player pay for each roll if he wants to make a profit of one rupee per throw of the die in the long run?

A.
Rs. 2.50
B.
Rs. 2
C.
Rs. 3.50
D.
Rs. 4

28. CAT 1991 QA | Arithmetic - Time & Work

Three machines, A, B and C can be used to produce a product. Machine A will take 60 hours to produce a million units. Machine B is twice as fast as Machine A. Machine C will take the same amount of time to produce a million units as A and B running together. How much time will be required to produce a million units if all the three machines are used simultaneously? (in hous)

A.
12
B.
10
C.
8
D.
6

Answer: Option B

Video Explanation

Explanation :

B being twice as fast as A, will take half of time of A to produce a million units i.e. = 30 hrs.

Since, Machine C takes the same amount of time as A & B running together,

∴ $\frac{1}{C}$ = $\frac{1}{A}$ + $\frac{1}{B}$ or

$\frac{1}{A}$ + $\frac{1}{B}$ + $\frac{1}{C}$ = $\frac{2}{A}$ + $\frac{2}{B}$

⇒ $\frac{1}{A}$ + $\frac{1}{B}$ + $\frac{1}{C}$ = $2 (\frac{1}{A} + \frac{1}{B})$ = $2 (\frac{1}{60} + \frac{1}{30})$ = $\frac{1}{10}$

Hence, it will take 10 hours for all three of them to produce 1 million units.

Hence, option (b).

29. CAT 1991 QA | Algebra - Functions & Graphs

Let Y = minimum of {(x + 2), (3 – x)}. What is the maximum value of Y for 0 ≤ x ≤ 1?

A.
1.0
B.
1.5
C.
3.1
D.
2.5

30. CAT 1991 QA | Venn Diagram

There are 3 clubs A, B & C in a town with 40, 50 & 60 members respectively. While 10 people are members of all 3 clubs, 70 are members in only one club. How many belong to exactly two clubs?

A.
20
B.
25
C.
50
D.
70

31. CAT 1991 QA | Geometry - Mensuration

A square piece of cardboard of sides ten inches is taken and four equal squares pieces are removed at the corners, such that the side of this square piece is also an integer value. The sides are then turned up to form an open box. Then the maximum volume such a box can have is

A.
72 cubic inches.
B.
24.074 cubic inches.
C.
2000/27 cubic inches.
D.
64 cubic inches.

32. CAT 1991 QA | Algebra - Number Theory

x, y and z are three positive integers such that x > y > z. Which of the following is closest to the product xyz?

A.
(x - 1)yz
B.
x(y - 1)z
C.
xy(z - 1)
D.
x(y + 1)z

33. CAT 1991 QA | Algebra - Number Theory

What is the greatest power of 5 which can divide 80! exactly.

A.
16
B.
20
C.
19
D.
None of these

Answer: Option C

Video Explanation

Explanation :

Number of powers of 5 in 80! = $(\frac{80}{5} = 16)$ + $(\frac{16}{5} = 3)$ = 19.

Hence, option (c).

34. CAT 1991 QA | Algebra - Number Theory

A third standard teacher gave a simple multiplication exercise to the kids. But one kid reversed the digits of both the numbers and carried out the multiplication and found that the product was exactly the same as the one expected by the teacher. Only one of the following pairs of numbers will fit in the description of the exercise. Which one is that?

A.
14, 22
B.
13, 62
C.
19, 33
D.
42, 28

35. CAT 1991 QA | Algebra - Number Theory

Find the minimum integral value of n such that the division 55n/124 leaves no remainder.

A.
124
B.
123
C.
31
D.
62

36. CAT 1991 QA | Algebra - Number Theory

Let k be a positive integer such that k + 4 is divisible by 7. Then the smallest positive integer n, greater than 2, such that k + 2n is divisible by 7 equals

A.
9
B.
7
C.
5
D.
3

37. CAT 1991 QA | Miscellaneous

A calculator has two memory buttons, A and B. Value 1 is initially stored in both memory locations.

The following sequence of steps is carried out five times:

add 1 to B
multiply A to B
store the result in A

What is the value stored in memory location A after this procedure?

A.
120
B.
450
C.
720
D.
250

	A	B	Step I (B + 1)	Step II (A × B)	Step III A	Step III B
Beginning	1	1
1^st time	1	1	B = 2	(1 × 2) = 2	2	2
2^nd time	2	2	B = 3	(2 × 3) = 6	6	3
3^rd time	6	3	B = 4	(6 × 4) = 24	24	4
4^th time	24	4	B = 5	(24 × 5) = 120	120	5
5^th time	120	5	B = 6	(120 × 6) = 720	720	6

38. CAT 1991 QA | Geometry - Circles

A one rupee coin is placed on a table. The maximum number of similar one rupee coins which can be placed on the table, around it, with each one of them touching it and only two others is

A.
8
B.
6
C.
10
D.
4

39. CAT 1991 QA | Algebra - Simple Equations | Arithmetic - Percentage

Gopal went to a fruit market with certain amount of money. With this money he can buy either 50 oranges or 40 mangoes. He retains 10% of the money for taxi fare. If he buys 20 mangoes, then the number of oranges he can buy is

A.
25
B.
20
C.
18
D.
6

40. CAT 1991 QA | Arithmetic - Time, Speed & Distance

Every day Neera’s husband meets her at the city railway station at 6.00 p.m. and drives her to their residence. One day she left early from the office and reached the railway station at 5.00 p.m. She started walking towards her home, met her husband coming from their residence on the way and they reached home 10 minutes earlier than the usual time. For how long did she walk?

A.
1 hour
B.
50 minutes
C.
1/2 hour
D.
55 minutes

41. CAT 1991 QA | Algebra - Number Theory

In Sivakasi, each boy’s quota of match sticks to fill into boxes is not more than 200 per session. If he reduces the number of sticks per box by 25, he can fill 3 more boxes with the total number of sticks assigned to him. Which of the following is the possible number of sticks assigned to each boy?

A.
200
B.
150
C.
125
D.
175

Answer: Option B

Video Explanation

Explanation :

Let x be the total number of sticks assigned to each boy and y be the number of boxes in which he has to fill them. If he reduces number of sticks per box by 25, he would fill (x/y - 25) in each box and hence he would now fill (y + 3) boxes.

∴ x = (x/y - 25)(y + 3) = x + 3x/y - 25y - 75.

⇒ 3x = (25y + 75)y

⇒ x = $\frac{(25 y^{2} + 75 y)}{3} = \frac{25 y (y + 3)}{3}$

For x to have an integer value, y(y + 3) has to be a multiple of 3. This is possible only when y is a multiple of 3.

Subsequently, (y + 3) will be a multiple of 3.

Therefore, 25y(y + 3)/3 = x will be a multiple of 3.

Hence, option (b).

42. CAT 1991 QA | Arithmetic - Simple & Compound Interest

A sum of money compounded annually becomes Rs. 625 in two years and Rs. 675 in three years. The rate of interest per annum is

A.
7%
B.
8%
C.
6%
D.
5%

43. CAT 1991 QA | Miscellaneous

In a six-node network, two nodes are connected to all the other nodes. Of the remaining four, each is connected to four nodes. What is the total number of links in the network?

A.
13
B.
15
C.
7
D.
26

44. CAT 1991 QA | Algebra - Number Theory

If x is a positive integer such that 2x + 12 is perfectly divisible by x, then the number of possible values of x is

A.
2
B.
5
C.
6
D.
12

45. CAT 1991 QA | Arithmetic - Percentage

An outgoing batch of students wants to gift PA system worth Rs. 4200 to their school. If the teachers offer to pay 50% more than the students, and an external benefactor gives three times teachers’ contribution, how much should the teachers donate?

A.
600
B.
840
C.
900
D.
1200

46. CAT 1991 QA | Algebra - Number Theory

A positive integer is said to be a prime number if it is not divisible by any positive integer other than itself and 1. Let p be a prime number greater than 5. Then (p² – 1) is

A.
never divisible by 6
B.
always divisible by 6, and may or may not be divisible by 12
C.
always divisible by 12, and may or may not be divisible by 24
D.
always divisible by 24

47. CAT 1991 QA | Algebra - Number Theory

To decide whether a n digit number is divisible by 7, we can define a process by which its magnitude is reduced as follows: (i₁, i₂, i₃, … , are the digits of the number, starting from the most significant digit).

i₁i₂ ……. i_n ⇒ i₁. 3^n-1 + i₂ . 3^n-2 + ……… + i_n . 3⁰.

e.g. 259 ⇒ 2.3² + 5.3¹ + 9.3⁰ = 18 + 15 + 9 = 42

Ultimately the resulting number will be seven after repeating the above process a certain number of times. After how many such stages, does the number 203 reduce to 7?

A.
2
B.
3
C.
4
D.
1

48. CAT 1991 QA | Geometry - Coordinate Geometry

What is the distance between the points A(3, 8) and B(–2, –7)?

A.
5√2
B.
5
C.
5√10
D.
10√2

Answer: Option A

Video Explanation

Explanation :

The distance between two points (x₁, y₁) and (x₂, y₂) is given as $\sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}$

Hence, required distance = $\sqrt{{(- 2 - 3)}^{2} + {(- 7 - 8)}^{2}}$ = 5√10.

Hence, option (c).

49. CAT 1991 QA | Data Sufficiency

Each of these items has a question followed by two statements. As the answer,

Type 1, If the question can be answered with the help of statement I alone,
Type 2, If the question can be answered with the help of statement II alone,
Type 3, If both the statement I and statement II are needed to answer the question, and
Type 4, If the question cannot be answered even with the help of both the statements.

Is it more profitable for Company M to produce Q?

Product R is sold at a price four times that of Q.
One unit of Q requires 2 units of labour, while one unit of R requires 5 units of labour. There is no other constraint on production.

A.
1
B.
2
C.
3
D.
4

50. CAT 1991 QA | Arithmetic - Time, Speed & Distance | Data Sufficiency

Each of these items has a question followed by two statements. As the answer,

Type 1, If the question can be answered with the help of statement I alone,
Type 2, If the question can be answered with the help of statement II alone,
Type 3, If both the statement I and statement II are needed to answer the question, and
Type 4, If the question cannot be answered even with the help of both the statements.

A train started from Station A, developed engine trouble and reached Station B, 40 minutes late. What is the distance between Stations A and B?

The engine trouble developed after travelling 40 km from Station A and the speed reduced to 1/4^th of the original speed.
The engine trouble developed after travelling 40 km from station A in two hours and the speed reduced to ¼th of the original speed.

A.
1
B.
2
C.
3
D.
4

51. CAT 1991 QA | Data Sufficiency

Each of these items has a question followed by two statements. As the answer,

Type 1, If the question can be answered with the help of statement I alone,
Type 2, If the question can be answered with the help of statement II alone,
Type 3, If both the statement I and statement II are needed to answer the question, and
Type 4, If the question cannot be answered even with the help of both the statements.

What is the value of prime number x?

x² + x is a two digit number greater than 50.
x³ is a three digit number.

A.
1
B.
2
C.
3
D.
4

52. CAT 1991 QA | Data Sufficiency

Each of these items has a question followed by two statements. As the answer,

Type 1, If the question can be answered with the help of statement I alone,
Type 2, If the question can be answered with the help of statement II alone,
Type 3, If both the statement I and statement II are needed to answer the question, and
Type 4, If the question cannot be answered even with the help of both the statements.

The average of three unequal quotations for a particular share is Rs.110. If all are quoted in integral values of rupee, does the highest quotation exceed Rs.129?

The lowest quotation is Rs.100.
One of the quotations is Rs.115.

A.
1
B.
2
C.
3
D.
4

53. CAT 1991 QA | Data Sufficiency

Each of these items has a question followed by two statements. As the answer,

Type 1, If the question can be answered with the help of statement I alone,
Type 2, If the question can be answered with the help of statement II alone,
Type 3, If both the statement I and statement II are needed to answer the question, and
Type 4, If the question cannot be answered even with the help of both the statements.

How many people (from the group surveyed) read both Indian Express and Times of India?

Out of total of 200 readers, 100 read Indian Express, 120 read Times of India and 50 read Hindu.
Out of a total of 200 readers, 100 read Indian Express, 120 read Times of India and 50 read neither

A.
1
B.
2
C.
3
D.
4

54. CAT 1991 QA | Data Sufficiency

Each of these items has a question followed by two statements. As the answer,

Type 1, If the question can be answered with the help of statement I alone,
Type 2, If the question can be answered with the help of statement II alone,
Type 3, If both the statement I and statement II are needed to answer the question, and
Type 4, If the question cannot be answered even with the help of both the statements.

X says to Y, “I am 3 times as old as you were 3 years ago”. How old is X?

Y’s age 17 years from now will be same as X’s present age.
X’s age nine years from now is 3 times Y’s present age.

A.
1
B.
2
C.
3
D.
4

55. CAT 1991 QA | Data Sufficiency

Each of these items has a question followed by two statements. As the answer,

Type 1, If the question can be answered with the help of statement I alone,
Type 2, If the question can be answered with the help of statement II alone,
Type 3, If both the statement I and statement II are needed to answer the question, and
Type 4, If the question cannot be answered even with the help of both the statements.

What is the area under the line GHI – JKL in the given quadrilateral OPQR, knowing that all the small spaces are squares of the same area?

Length ABCDEQ is greater than or equal to 60.
Area OPQR is less than or equal to 1512.

A.
1
B.
2
C.
3
D.
4

56. CAT 1991 QA | Data Sufficiency

Each of these items has a question followed by two statements. As the answer,

Type 1, If the question can be answered with the help of statement I alone,
Type 2, If the question can be answered with the help of statement II alone,
Type 3, If both the statement I and statement II are needed to answer the question, and
Type 4, If the question cannot be answered even with the help of both the statements.

What is the radius of the circle?

Ratio of its area to circumference is > 7.
Diameter of the circle is ≤ 32.

A.
1
B.
2
C.
3
D.
4

57. CAT 1991 QA | Data Sufficiency

Each of these items has a question followed by two statements. As the answer,

Type 1, If the question can be answered with the help of statement I alone,
Type 2, If the question can be answered with the help of statement II alone,
Type 3, If both the statement I and statement II are needed to answer the question, and
Type 4, If the question cannot be answered even with the help of both the statements.

What is the time difference between New York and London?

The departure time at New York is exactly 9:00 a.m. local time and the arrival time at London is at 10:00 a.m. local time.
The flight time is 5 hours.

A.
1
B.
2
C.
3
D.
4

58. CAT 1991 QA | Data Sufficiency

Each of these items has a question followed by two statements. As the answer,

Type 1, If the question can be answered with the help of statement I alone,
Type 2, If the question can be answered with the help of statement II alone,
Type 3, If both the statement I and statement II are needed to answer the question, and
Type 4, If the question cannot be answered even with the help of both the statements.

Mr. Murthy takes the morning train to his office from station A to station B, and his colleague Mr.Rahman joins him on the way. There are three stations C, D and E on the way not necessarily in that sequence. What is the sequence of stations?

Mr. Rahman boards the train at D.
Mr. Thomas, who travels between C & D has two segments of journey in common with Mr. Murthy but none with Mr. Rahman.

A.
1
B.
2
C.
3
D.
4

CAT 1991 QA | Previous Year CAT Paper

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