[Download PDF] CAT 1998 Question Paper Paper with Solutions

Instructions

For the following questions answer them individually

Question 41

A company has a job to prepare certain number cans and there are three machines A, B and C for this job. A can complete the job in 3 days, B can complete the job in 4 days, and C can complete the job in 6 days. How many days will the company take to complete the job if all the machines are used simultaneously?

Video Solution

Solution

If they work together, then total work done in single day = $$\frac{1}{3} + \frac{1}{4} + \frac{1}{6} = \frac{9}{12}$$
So $$\frac{9}{12}$$ work is done in 1 day
Hence unit work will be done in $$\frac{12}{9}$$ or $$\frac{4}{3}$$ days.

Question 42

$$n^3$$ is odd. Which of the following statement(s) is/are true?
I. $$n$$ is odd.
II.$$n^2$$ is odd.
III.$$n^2$$ is even.

Video Solution

Solution

if $$n^3$$ is odd then $$n$$ will be odd. let's say it is $$2k+1$$
then $$n^2$$ will be = $$(4k^2 + 4k + 1)$$ which will be odd
Hence answer will be C.

Instructions

Answer the questions based on the following information. Production pattern for number of units (in cubic feet) per day.

The finished goods are to be transported to the market by a truck having a capacity of 2000 cubic feet. Any finished goods (ready at the end of the day) retained overnight at the factory will incur a storage cost of Rs 5 per cubic foot for each night of storage. The hiring cost for the truck is Rs 1000 per day.

Question 43

If all the units should be sent to the market, then on which days should the trucks be hired to minimize the cost?

Video Solution

Solution

The truck should be sent on a day on which maximum number of units can be sent. It happens on 2nd, 4th, 5th and 7th day. Hence the answer is b.

Question 44

If the storage cost is reduced to Re 0.80 per cubic feet per day, then on which day(s), should the truck be hired?

Video Solution

Solution

If the truck is hired only for the 7th day, the cost will be (150*6 + 180*5 + 120*4 + 250*3 + 160*2 + 120)*0.8 = 2776
So, total cost incurred = 2776 + 1000 = 3776

If a truck is hired on 4th day and 7th day, the cost incurred will be (150*3 + 180*2 + 120)*0.8 + 1000 + (160*2 + 120)*0.8 + 1000 = 3096

So, a truck should be hired on the 4th and 7th day.

Instructions

For the following questions answer them individually

Question 45

One bacterium splits into eight bacteria of the next generation. But due to environmental condition only 50% survives and remaining 50% dies. If the seventh generation number is 4,096 million, what is the number in first generation?

Video Solution

Solution

let's say x is the initial number of bacterias :
So in 2nd generation no. of bacterias = $$\frac{8x}{2} = 4x$$
In 3rd generation, it will be = 16x
4th gen. = 64x
5th gen. = 256x
6th gen. = 1024x
7th gen. = 4096x
Hence x = 1 million

Question 46

Three circles, each of radius 20, have centres at P, Q and R. Further, AB = 5, CD = 10 and EF = 12. What is the perimeter of ΔPQR ?

Video Solution

Solution

As radius of circle is 20, lengths of PR, QR and PQ will be (15+5+15) , (10+10+10) , (8+12+8) respectively.
So perimeter will be = 28 + 30 + 35 = 93

Instructions

Answer the questions based on the following information. The following operations are defined for real numbers.

a # b=a + b, if a and b both are positive else a # b=1

a $$\triangledown$$ b=$$(a \times b)^{a+b}$$ if $$a\times\ b$$ is positive else a $$\triangledown$$b=1.

Question 47

$$\frac{(2\#1)}{(1 \triangledown 2)}$$ =

Video Solution

Solution

$$2\#1 = 2+1 = 3$$

$$1 \triangledown 2 = (1 \times 2)^(1+2) = 2^3$$

So answer will be $$\frac{3}{8}$$

Question 48

$$\frac{((1\#1) \# 2) - (10^{1.3} \triangledown \log_{10}{0.1})}{1 \triangledown 2}$$

Video Solution

Solution

$$((1\#1) \# 2) = 2 \# 2 = 4$$

$$\log_{10} 0.1$$ = $$\log_{10} 10^{-1}$$ = -1

As $$(10^{1.3} \times -1)$$ is not positive, $$(10^{1.3} \triangledown \log_{10}{0.1})$$ = 1

$$1 \triangledown 2$$ = $$(1x2)^{(1+2)}$$ = $$2^3$$ = 8

Hence, the fraction is (4-1)/8 = 3/8

Instructions

For the following questions answer them individually

Question 49

$$(BE)^2 = MPB$$, where B, E, M and P are distinct integers. Then M =

Video Solution

Solution

A digit number when squared produces a 3 digit number. This means that the number ranges from [10, 31].
First digit of $$BE^2$$ should be unit digit of $$E^2$$. But unit digit of $$E^2$$ is B. Look at the numbers and the unit digit of their square.

0-0, 1-1, 2-4, 3-9, 4-6, 5-5, 6-6, 7-9, 8-4, 9-1. Only 2-4, 3-9, 4-6, 7-9, 8-4 and 9-1 are kind of pairs we are looking after.But all the pairs except 9-1 produce a number greater than 31. Now, the number we can form from 9-1 is 19 whose square is 361 which satisfies all the condition we are looking for. This is the only such number.

Question 50

Five-digit numbers are formed using only 0, 1, 2, 3, 4 exactly once. What is the difference between the maximum and minimum number that can be formed?

Video Solution

Solution

Maximum five digit number which can be formed by using numbers is 43210
And minimum five digit number = 10234
Difference = 43210 - 10234 = 32976

CAT Content

CAT 1998 Question Paper

$$n^3$$ is odd. Which of the following statement(s) is/are true?
I. $$n$$ is odd.
II.$$n^2$$ is odd.
III.$$n^2$$ is even.

If all the units should be sent to the market, then on which days should the trucks be hired to minimize the cost?

If the storage cost is reduced to Re 0.80 per cubic feet per day, then on which day(s), should the truck be hired?

One bacterium splits into eight bacteria of the next generation. But due to environmental condition only 50% survives and remaining 50% dies. If the seventh generation number is 4,096 million, what is the number in first generation?

Three circles, each of radius 20, have centres at P, Q and R. Further, AB = 5, CD = 10 and EF = 12. What is the perimeter of ΔPQR ?

$$\frac{(2\#1)}{(1 \triangledown 2)}$$ =

$$\frac{((1\#1) \# 2) - (10^{1.3} \triangledown \log_{10}{0.1})}{1 \triangledown 2}$$

$$(BE)^2 = MPB$$, where B, E, M and P are distinct integers. Then M =

Five-digit numbers are formed using only 0, 1, 2, 3, 4 exactly once. What is the difference between the maximum and minimum number that can be formed?

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CAT Content

CAT 1998 Question Paper

$$n^3$$ is odd. Which of the following statement(s) is/are true? I. $$n$$ is odd. II.$$n^2$$ is odd. III.$$n^2$$ is even.

If all the units should be sent to the market, then on which days should the trucks be hired to minimize the cost?

If the storage cost is reduced to Re 0.80 per cubic feet per day, then on which day(s), should the truck be hired?

One bacterium splits into eight bacteria of the next generation. But due to environmental condition only 50% survives and remaining 50% dies. If the seventh generation number is 4,096 million, what is the number in first generation?

Three circles, each of radius 20, have centres at P, Q and R. Further, AB = 5, CD = 10 and EF = 12. What is the perimeter of ΔPQR ?

$$\frac{(2\#1)}{(1 \triangledown 2)}$$ =

$$\frac{((1\#1) \# 2) - (10^{1.3} \triangledown \log_{10}{0.1})}{1 \triangledown 2}$$

$$(BE)^2 = MPB$$, where B, E, M and P are distinct integers. Then M =

Five-digit numbers are formed using only 0, 1, 2, 3, 4 exactly once. What is the difference between the maximum and minimum number that can be formed?

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$$n^3$$ is odd. Which of the following statement(s) is/are true?
I. $$n$$ is odd.
II.$$n^2$$ is odd.
III.$$n^2$$ is even.