CAT 1993 QA questions and solutions

Previous year paper questions for CAT 1993 QA

1. CAT 1993 QA | Data Sufficiency

Each of these questions is followed by two statements. As the answer,
Mark (a), If the question can be answered with the help of statement I alone,
Mark (b), If the question can be answered with the help of statement II, alone,
Mark (c), If both, statement I and statement II are needed to answer the question, and
Mark (d), If the question cannot be answered even with the help of both the statements.

Given that X and Y are non-negative. What is the value of X?
I. 2X + 2Y ≤ 40
II. X − 2Y ≥ 20

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2. CAT 1993 QA | Data Sufficiency

Each of these questions is followed by two statements. As the answer,
Mark (a), If the question can be answered with the help of statement I alone,
Mark (b), If the question can be answered with the help of statement II, alone,
Mark (c), If both, statement I and statement II are needed to answer the question, and
Mark (d), If the question cannot be answered even with the help of both the statements.

What are the values of 3 integers a, b, and c?
I. ab = 8
II. bc = 9

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3. CAT 1993 QA | Data Sufficiency

Each of these questions is followed by two statements. As the answer,
Mark (a), If the question can be answered with the help of statement I alone,
Mark (b), If the question can be answered with the help of statement II, alone,
Mark (c), If both, statement I and statement II are needed to answer the question, and
Mark (d), If the question cannot be answered even with the help of both the statements.

Is the average of the largest and the smallest of four given numbers greater than the average of the four numbers?
I. The difference between the largest and the second largest numbers is greater than the difference
between the second smallest and the smallest numbers.
II. The difference between the largest and the second largest numbers is less than the difference
between the second largest and the second smallest numbers.

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Answer: 1

Explanation :

If the numbers are a, b, c and d such that a < b < c < d, then from statement I, we get (d – c) > (b – a).
So we can say, (d + a) > (b + c) or (d + a) + (d + a) > (b + c) + (d + a). Dividing both the sides by 4, we get

$\frac{(d + a)}{2} > \frac{(a + b + c + d)}{4}$ .

This shows that the average of the largest and the smallest of four numbers is indeed greater than the average of all the 4 numbers. Hence, we can answer the question using first statement only.

4. CAT 1993 QA | Data Sufficiency

Each of these questions is followed by two statements. As the answer,
Mark (a), If the question can be answered with the help of statement I alone,
Mark (b), If the question can be answered with the help of statement II, alone,
Mark (c), If both, statement I and statement II are needed to answer the question, and
Mark (d), If the question cannot be answered even with the help of both the statements.

What are the ages of the three brothers?
I. The product of their ages is 21.
II. The sum of their ages is not divisible by 3.

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5. CAT 1993 QA | Data Sufficiency

Each of these questions is followed by two statements. As the answer,
Mark (a), If the question can be answered with the help of statement I alone,
Mark (b), If the question can be answered with the help of statement II, alone,
Mark (c), If both, statement I and statement II are needed to answer the question, and
Mark (d), If the question cannot be answered even with the help of both the statements.

Two types of widgets, namely type A and type B, are produced on a machine. The number of machine hours available per week is 80. How many widgets of type A must be produced?
I. One unit of type A widget requires 2 machine hours and one unit of type B widget requires 4 machine hours.
II. The widget dealer wants supply of at least 10 units of type A widget per week and he would not accept less than 15 units of type B widget.

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6. CAT 1993 QA | Data Sufficiency

Each of these questions is followed by two statements. As the answer,
Mark (a), If the question can be answered with the help of statement I alone,
Mark (b), If the question can be answered with the help of statement II, alone,
Mark (c), If both, statement I and statement II are needed to answer the question, and
Mark (d), If the question cannot be answered even with the help of both the statements.

What is the area of a regular hexagon?
I. The length of the boundary line of the hexagon is 36 cm.
II. The area of the hexagon is 6 times the area of an equilateral triangle formed on one of the sides.

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7. CAT 1993 QA | Data Sufficiency

Each of these questions is followed by two statements. As the answer,
Mark (a), If the question can be answered with the help of statement I alone,
Mark (b), If the question can be answered with the help of statement II, alone,
Mark (c), If both, statement I and statement II are needed to answer the question, and
Mark (d), If the question cannot be answered even with the help of both the statements.

What is the price of mangoes per kg?
I. Ten kg of mangoes and two dozens of oranges cost Rs.252.
II. Two kg of mangoes could be bought in exchange for one dozen oranges.

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8. CAT 1993 QA | Algebra - Simple Equations

Two oranges, three bananas and four apples cost Rs.15. Three oranges, two bananas and one apple cost Rs 10. I bought 3 oranges, 3 bananas and 3 apples. How much did I pay?

A.
Rs. 10
B.
Rs. 8
C.
Rs. 15
D.
Cannot be determined

9. CAT 1993 QA | Arithmetic - Percentage

The rate of increase of the price of sugar is observed to be two percent more than the inflation rate expressed in percentage. The price of sugar, on January 1, 1994, is Rs. 20 per kg. The inflation rate for the years 1994 and 1995 are expected to be 8% each. The expected price of sugar on January 1, 1996 would be

A.
Rs. 23.60
B.
Rs. 24.00
C.
Rs. 24.20
D.
Rs. 24.60

Answer: Option C

Video Explanation

Explanation :

Since the inflation rate is 8% in both the years 1994 and 1995, therefore, the rate of increase of the price of sugar is 10%.

∴ Price of sugar on January 1, 1996 = Price of sugar on January 1, 1994 × ${(1 + \frac{10}{100})}^{2}$ = 20 × 1.21 = Rs. 24.20 per kg.

Hence, option (c).

10. CAT 1993 QA | Modern Math - Permutation & Combination

An intelligence agency decides on a code of 2 digits selected from 0, 1, 2, …. , 9. But the slip on which the code is hand–written allows confusion between top and bottom, because these are indistinguishable. Thus, for example, the code 91 could be confused with 16. How many codes are there such that there is no possibility of any confusion?

A.
25
B.
75
C.
80
D.
None of these

11. CAT 1993 QA | Algebra - Number Theory

Suppose one wishes to find distinct positive integers x, y such that (x + y)/xy is also a positive integer. Identify the correct alternative.

A.
This is never possible.
B.
This is possible and the pair (x,y) satisfying the stated condition is unique.
C.
This is possible and there exist more than one but a finite number of ways of choosing the pair (x,y).
D.
This is possible and the pair (x,y) can be chosen in infinite ways.

Answer: Option A

Video Explanation

Explanation :

It can be very easy to figure out that (x + y) will always be greater than xy, only if one of them is 1. For eg. If x = 1 and y =2, then (x + y) = 3 and xy = 2.

Hence, (x + y) > xy.

Other than this, for all other values of x & y, (x + y) will always be less than xy, and hence, the ratio of $\frac{(x + y)}{x y} < 1,$ and hence, cannot be an integer. Also, even if one of the values is 1, $\frac{(x + y)}{x y}$ will never be an integer.

Hence, option (a).

12. CAT 1993 QA | Algebra - Number Theory

Given odd positive integers x, y and z, which of the following is not necessarily true?

A.
x²y² z²is odd
B.
3(x² + y³)z² is even.
C.
5x + y + z⁴ is odd
D.
z² (x⁴ + y⁴)/2 is even

Answer: Option D

Video Explanation

Explanation :

You can solve this by assuming values of x, y and z.

Let the three numbers be 1, 3 & 5.

Option (a) is 1² 3² 5² = 225, which is odd.

Option (b) is 3(1² + 3³)5² = 2100, which is even.

Option (c) is 5 + 3 + 5⁴ = 633, which is odd.

Option (d) is $5^{2} \frac{(14 + 34)}{2} =$ 1025, which is not even and hence, the answer.

Hence, option (d).

13. CAT 1993 QA | Miscellaneous

139 persons have signed up for an elimination tournament. All players are to be paired up for the first round, but because 139 is an odd number one player gets a bye, which promotes him to the second round, without actually playing in the first round. The pairing continues on the next round, with a bye to any player left over. If the schedule is planned so that a minimum number of matches is required to determine the champion, the number of matches which must be played is

A.
136
B.
137
C.
138
D.
139

14. CAT 1993 QA | Miscellaneous

There are ten 50 paise coins placed on a table. Six of these show tails, four show heads. A coin is chosen at random and flipped over (not tossed). This operation is performed seven times. One of the coins is then covered. Of the remaining nine coins, five show tails and four show heads. The covered coin shows

A.
a head
B.
a tail
C.
more likely a head
D.
more likely a tail

15. CAT 1993 QA | Arithmetic - Ratio, Proportion & Variation

From each of the two given numbers, half the smaller number is subtracted. Of the resulting numbers the larger one is three times as large as the smaller. What is the ratio of the two numbers?

A.
2 : 1
B.
3 : 1
C.
3 : 2
D.
None

Answer: Option A

Video Explanation

Explanation :

Let the two given numbers be x and y such that x > y.

According to the question,

x - $\frac{y}{2} = 3 (y - \frac{y}{2})$

⇒ 2x - y = 6y - 3y

⇒ 2x = 4y

$\Rightarrow \frac{x}{y} = \frac{2}{1} .$

Hence, option (a).

16. CAT 1993 QA | Geometry - Circles

Three identical cones with base radius r are placed on their bases so that each is touching the other two. The radius of the circle drawn through their vertices is

A.
smaller than r
B.
equal to r
C.
larger than r
D.
depends on the height of the cones.

Answer: Option C

Video Explanation

Explanation :

It can be seen that, if we place the 3 cones in such a way that they touch each other, it will be similar to placing 3 circles touching, with vertices of the cone corresponding to the centers of the circles. The centers of the circle form an equilateral triangle with each side being 2r. A circle that passes through the centers will be the circumcircle to such a triangle. The radius of the circumcircle of an equilateral triangle is $(\frac{1}{\sqrt{3}})$ times its side.

Hence, in our case it would be be $(\frac{2 r}{\sqrt{3}})$ and $(\frac{2 r}{\sqrt{3}})$ > r, since $\sqrt{3}$ = 1.73 (approx).

Hence, option (c).

17. CAT 1993 QA | Geometry - Circles

The line AB is 6 metres in length and is tangent to the inner one of the two concentric circles at point C. It is known that the radii of the two circles are integers. The radius of the outer circle is

A.
5 metres
B.
4 metres
C.
6 metres
D.
3 metres

18. CAT 1993 QA | Modern Math - Permutation & Combination

Four cities are connected by a road network as shown in the figure. In how many ways can you start from any city and come back to it without travelling on the same road more than once?

A.
8
B.
12
C.
16
D.
20

19. CAT 1993 QA | Algebra - Simple Equations

How many children did not try any of the rides?

A.
5
B.
10
C.
15
D.
20

20. CAT 1993 QA | Algebra - Simple Equations

How many children took exactly one ride?

A.
5
B.
10
C.
15
D.
20

21. CAT 1993 QA | Algebra - Simple Equations

John bought five mangoes and ten oranges together for forty rupees. Subsequently, he returned one mango and got two oranges in exchange. The price of an orange would be

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

22. CAT 1993 QA | Algebra - Number Theory

The number of positive integers not greater than 100, which are not divisible by 2, 3 or 5 is

A.
26
B.
18
C.
31
D.
None

23. CAT 1993 QA | Algebra - Progressions

Let U_n+1 = 2U_n + 1 (n = 0, 1, 2, ...) and u₀ = 0. Then u₁₀ is nearest to

A.
1023
B.
2047
C.
4095
D.
8195

24. CAT 1993 QA | Algebra - Functions & Graphs

The function given by f(x) = |x|³ is

A.
even
B.
odd
C.
neither
D.
both

25. CAT 1993 QA | Algebra - Functions & Graphs

The sum of two odd functions

A.
is always an even functions
B.
is always an odd function
C.
is sometimes odd and sometimes even
D.
may be neither odd nor even

26. CAT 1993 QA | Modern Math - Permutation & Combination

A five digit number is formed using digits 1, 3, 5, 7 and 9 without repeating any one of them. What is the sum of all such possible numbers?

A.
6666600
B.
6666660
C.
6666666
D.
None

27. CAT 1993 QA | Modern Math - Probability

A box contains 6 red balls, 7 green balls and 5 blue balls. Each ball is of a different size. The probability that the red ball selected is the smallest red ball, is

A.
1/18
B.
1/3
C.
1/6
D.
2/3

Answer: Option C

Video Explanation

Explanation :

Since there are 6 red balls and all six of them are of different sizes, probability of choosing the smallest among them is $\frac{1}{6}$ .

Hence, option (c).

28. CAT 1993 QA | Geometry - Triangles

Then D is

A.
3 km east and 1 km north of A
B.
3 km east and $\sqrt{3}$ km north of A
C.
$\sqrt{3}$ km east and 1 km south of A
D.
$\sqrt{3}$ km west and 3 km north of A

Answer: Option B

Video Explanation

Explanation :

Since Δ ABC is an equilateral triangle with length of the side 2 km, so its altitude will be $\sqrt{3}$ km.

The horizontal distance of D from A = half of AB + CD = 1 + 2 = 3 kms

The vertical distance of D from A = height of the triangle ABC = √3 kms

∴ D is 3 km east and $\sqrt{3}$ km north of A.

Hence, option (b).

29. CAT 1993 QA | Geometry - Triangles

The total distance walked by the person is

A.
3 km
B.
4 km
C.
2 $\sqrt{3}$ km
D.
6 km

30. CAT 1993 QA | Geometry - Mensuration

A slab of ice 8 inches in length, 11 inches in breadth, and 2 inches thick was melted and resolidified into the form of a rod of 8 inches diameter. The length of such a rod, in inches, is nearest to

A.
3
B.
3.5
C.
4
D.
4.5

Answer: Option B

Video Explanation

Explanation :

Let l be the length of the rod, then

Volume of slab = Volume of rod

⇒ 8 × 11 × 2 = π × ${(\frac{8}{2})}^{2}$ × l

⇒ l = 11/π = 3.5 inches.

Hence, option (b).

31. CAT 1993 QA | Algebra - Progressions

Let x < 0.50, 0 < y < 1, z > 1. Given a set of numbers, the middle number, when they are arranged in ascending order, is called the median. So the median of the numbers x, y, and z would be

A.
less than one
B.
between 0 and 1
C.
greater than 1
D.
cannot say

32. CAT 1993 QA | Algebra - Functions & Graphs

The maximum possible value of y = min (1/2 – 3x²/4, 5x²/4) for the range 0 < x < 1 is

A.
1/3
B.
1/2
C.
5/27
D.
5/16

Answer: Option D

Video Explanation

Explanation :

So maximum possible value will be at the point of intersection of the two graphs.

$∴ \frac{1}{2} - \frac{3 x^{2}}{4} = \frac{5 x^{2}}{4} \Rightarrow x^{2} = \frac{1}{4}$

Hence, required maximum value = $\frac{5 x^{2}}{4} = \frac{5}{4} \times \frac{1}{4} = \frac{5}{16} .$

Hence, option (d).

33. CAT 1993 QA | Arithmetic - Time & Work

A group of workers was put on a job. From the second day onwards, one worker was withdrawn each day. The job was finished when the last worker was withdrawn. Had no worker been withdrawn at any stage, the group would have finished the job in two-third the time. How many workers were there in the group?

A.
2
B.
3
C.
5
D.
11

Answer: Option B

Video Explanation

Explanation :

Let the work done by a worker be x units, w be the total work and n be the number of workers in the group.

Then, w = Work done on the nth day i.e. last day + Work done on the second last day + … + Work done on the first day

⇒ w = x + 2x + ... + nx = $\frac{n (n + 1) x}{2}$ ...(i)

When none of the workers is removed, then

w = nx × $\frac{2 n}{3} = \frac{2 n^{2} x}{3}$ ...(ii)

From equation (i) and (ii), we get

$\frac{n (n + 1) x}{2} = \frac{2 n^{2} x}{3}$

⇒ n = 3.

Hence, option (b).

34. CAT 1993 QA | Geometry - Quadrilaterals & Polygons

Consider the five points comprising of the vertices of a square and the intersection point of its diagonals. How many triangles can be formed using these points?

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

35. CAT 1993 QA | Venn Diagram

Out of 100 families in the neighbourhood, 45 own radios, 75 have TVs, 25 have VCRs. Only 10 families have all three and each VCR owner also has a TV. If 25 families have radio only, how many have only TV?

A.
30
B.
35
C.
40
D.
45

36. CAT 1993 QA | Miscellaneous

If A = 2 and B = 4, the value of @(/(*(A, B), B), A) would be

A.
2
B.
4
C.
6
D.
16

37. CAT 1993 QA | Miscellaneous

The sum of A and B is given by

A.
*(@(A, B), 2)
B.
/(@(A, B), 2)
C.
@(*(A, B), 2)
D.
@(/(A, B), 2)

38. CAT 1993 QA | Miscellaneous

The sum of A, B, and C is given by

A.
*(@(*(@(B, A), 2), C), 3)
B.
/(@(*(@(B, A), 3), C), 2)
C.
/(*(@(*(B, A), 2), C), 3)
D.
None of these

39. CAT 1993 QA | Arithmetic - Percentage

A report consists of 20 sheets each of 55 lines and each such line consist of 65 characters. This report is retyped into sheets each of 65 lines such that each line consists of 70 characters. The percentage reduction in number of sheets is closest to

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

Answer: Option A

Video Explanation

Explanation :

The total number of the characters = (20 × 55 × 65).

Let the number of pages in the new format be n.
Thus, the total number of the characters = (65 × 70 × n).

Since the total number of the characters remains same,
⇒ 20 × 55 × 65 = 65 × 70 × n
⇒ n ≈ 16.

Hence, the required percentage $= \frac{20 - 16}{20} \times$ 100 = 20%.

Hence, option (a).

40. CAT 1993 QA | Algebra - Number Theory

Let x < 0, 0 < y < 1, z > 1. Which of the following may be false?

A.
(x² – z²) has to be positive.
B.
yz can be less than one.
C.
xy can never be zero.
D.
(y² – z²) is always negative.

Answer: Option A

Video Explanation

Explanation :

Let us evaluate each option.

Option (a): As x < 0 and z > 1, let x = – 1 and z = 3, then (x² –z²) = – 8. Hence, this option is not true.

Option (b): As 0 < y < 1 and z > 1, let y = $\frac{1}{4}$ and z = 2, then yz = $\frac{1}{4} \times 2 = \frac{1}{2} .$
Therefore, yz can be less than 1.

Option (c): Since, none of the x and y is equal to zero, therefore xy can never be zero.

Option (d): 0 < y < 1 and z > 1, therefore (y² – z²) is always negative.

Hence, option (a).

41. CAT 1993 QA | Modern Math - Permutation & Combination

A young girl counted in the following way on the fingers of her left hand. She started calling the thumb 1, the index finger 2, middle finger 3, ring finger 4, little finger 5, then reversed direction, calling the ring finger 6, middle finger 7, index finger 8, thumb 9, then back to the index finger for 10, middle finger for 11, and so on. She counted up to 1994. She ended on her.

A.
thumb
B.
index finger
C.
middle finger
D.
ring finger

42. CAT 1993 QA | Arithmetic - Time, Speed & Distance

The point where P and Q would meet is

A.
Closer to A
B.
Exactly between A and B
C.
Closer to B
D.
P and Q will not meet at all

43. CAT 1993 QA | Arithmetic - Time, Speed & Distance

How many hours would P take to reach B?

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

Answer: Option D

Video Explanation

Explanation :

Let the speed of P be x and the distance between A and B be d, so the speed of Q will be 2x.

According to the question, (1 + t)v = 2vt = $\frac{d}{6}$ (Let t be the travel time of Q)

⇒ t = 1, and d = 12v

Hence, the time taken by P to reach to B = $\frac{d}{v}$ = 12 hours.

Hence, option (d).

44. CAT 1993 QA | Arithmetic - Time, Speed & Distance

How many more hours would P (compared to Q) take to complete his journey?

A.
4
B.
5
C.
6
D.
7

Answer: Option C

Video Explanation

Explanation :

As P takes 12 hours to complete his journey, so Q moving with twice the speed of P will take $\frac{12}{2} = 6$ hours to complete his journey.

Hence P will take 6 hours more than Q to complete the journey.

Hence, option (c).

45. CAT 1993 QA | Algebra - Number Theory

The smallest number which when divided by 4, 6 or 7 leaves a remainder of 2, is

A.
44
B.
62
C.
80
D.
86

46. CAT 1993 QA | Geometry - Mensuration

The diameter of a hollow cone is equal to the diameter of a spherical ball. If the ball is placed at the base of the cone, what portion of the ball will be outside the cone?

A.
50%
B.
less than 50%
C.
more than 50%
D.
100%

47. CAT 1993 QA | Arithmetic - Time, Speed & Distance

A ship leave on a long voyage. When it is 18 miles from the shore, a seaplane, whose speed is ten times that of the ship, is sent to deliver mail. How far from the shore does the seaplane catch up with the ship?

A.
24 miles
B.
25 miles
C.
22 miles
D.
20 miles

Answer: Option D

Video Explanation

Explanation :

The separation between the ship and the seaplane is 18 miles.

Since the two are travelling in the same direction, the relative speed would be 9 times the speed of the ship (If speed of ship is x miles/hour, speed of the seaplane would be 10x and 10x – x = 9x).

Hence, to catch up with the ship, the seaplane would take $\frac{18}{9 x} = \frac{2}{x}$ hours.

Now, the ship covers x miles in an hour, so in $\frac{2}{x}$ hours it would cover 2 miles.

So when the seaplane catches up with the ship, it would be 18 + 2 = 20 miles from the shore.

Hence, option (d).

48. CAT 1993 QA | Algebra - Number Theory

The product of all integers from 1 to 100 will have the following numbers of zeros at the end.

A.
20
B.
24
C.
19
D.
22

Answer: Option B

Video Explanation

Explanation :

The number of zeros at the end will be same as the highest power of 10 in 100!.

Number of 2 in the product of all integers from 1 to 100 $= \frac{100}{2} + \frac{100}{4} + \frac{100}{8} + \frac{100}{16} + \frac{100}{32} + \frac{100}{64}$

= 50 + 25 +12 + 6 + 3 + 1= 97 and number of 5 in the product of all integers from 1 to 100 = $\frac{100}{5} + \frac{100}{25}$ = 20 + 4 = 24.

∴ 100! = 2⁹⁷ × 5²⁴ × ...
⇒ 100! = 2⁷³ × (2 × 5²⁴) × ...
⇒ 100! = 2⁷³ × (10²⁴) × ...

∴ Highest power of 10 in 100! is 24.

Hence, option (b).

Concept:

Highest power of a prime number p in N!

49. CAT 1993 QA | Algebra - Number Theory

Let x, y and z be distinct positive integers satisfying x < y < z and x + y + z = k. What is the smallest value of K that does not determine x, y, z uniquely?

A.
9
B.
6
C.
7
D.
8

50. CAT 1993 QA | Venn Diagram

Amar, Akbar, and Anthony came from the same public school in the Himalayas. Every boy in that school either fishes for trout or plays frisbee. All fishermen like snow while no frisbee player likes rain. Amar dislikes whatever Akbar likes and likes whatever Akbar dislikes. Akbar likes rain and snow. Anthony likes whatever the other two like. Who is a fisherman but not a frisbee player?

A.
Amar
B.
Akbar
C.
Anthony
D.
None

CAT 1993 QA | Previous Year CAT Paper

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