Instructions

For the following questions answer them individually

Question 11

If a man cycles at 10 km/hr, then he arrives at a certain place at 1 p.m. If he cycles at 15 km/hr, he will arrive at the same place at 11 a.m. At what sped must he cycle to get there at noon?

Video Solution

Solution

Let the distance to be travelled be D.

In the first case, D/10 = t

In the second case, D/15 = t-2

=> D/15 = D/10 - 2

=> 2D = 3D - 60

=> D = 60 km and T = 6 hours

Therefore, to get there at noon, he has to travel at 60/5 = 12 km/hr

Question 12

On January 1, 2004 two new societies S1 and S2 are formed, each n numbers. On the first day of each subsequent month, S1 adds b members while S2 multiples its current numbers by a constant factor r. Both the societies have the same number of members on July 2, 2004. If b = 10.5n, what is the value of r?

Video Solution

Solution

According to given condition we have ,

n+6b =$$nr^6$$ and b=10.5n ,

63n+n = $$nr^6$$

$$r^6 = 64$$

r = 2

Question 13

If $$f(x)=x^3-4x+p$$ , and f(0) and f(1) are of opposite signs, then which of the following is necessarily true

[CAT 2004]

Video Solution

Solution

f(1) = 1-4+p = p-3
f(0) = p
Since they are of opposite signs, p(p-3) < 0
=> 0 < p < 3

Question 14

Suppose n is an integer such that the sum of digits on n is 2, and $$10^{10} < n < 10^{11}$$. The number of different values of n is

Video Solution

Solution

The sum of digits should be 2. The possibilities are 1000000001,1000000010,10000000100,..these 10 cases . Also additional 1 case where 20000000000. Hence option A .

Question 15

A milkman mixes 20 litres of water with 80 litres of milk. After selling one-fourth of this mixture, he adds water to replenish the quantity that he had sold. What is the current proportion of water to milk?

[CAT 2004]

Video Solution

Solution

After selling 1/4th of the mixture, the remaining quantity of water is 15 liters and milk is 60 liters. So the milkman would add 25 liters of water to the mixture. The total amount of water now is 40 liters and milk is 60 liters. Therefore, the required ratio is 2:3.

Question 16

Let $$ y = \frac{1}{2+\frac{1}{3+\frac{1}{2+\frac{1}{3+…}}}}$$. Then y equals?

Video Solution

Solution

$$ y = \frac{1}{2+\frac{1}{3+\frac{1}{2+\frac{1}{3+…}}}}$$

which is equal to $$ y = \frac{1}{2+\frac{1}{3+y}} $$ solving we get

$$ y = \frac{3+y}{7+2y} $$ we get

$$2y^2+6y-3=0$$ .

Solution of this equation is $$\frac{\sqrt{15}-3}{2}$$.

Question 17

N persons stand on the circumference of a circle at distinct points. Each possible pair of persons, not standing next to each other, sings a two-minute song one pair after the other. If the total time taken for singing is 28 minutes, what is N?

[CAT 2004]

Video Solution

Solution

Total number of pairs is $$^{N}C_2$$. Number of pairs standing next to each other = N. Therefore, number of pairs in question = $$^{N}C_2$$ - N

= 28/2 = 14.

If N = 7,

$$^{7}C_2$$ - 7 = 21 - 7 = 14.

N = 7.

Question 18

A father and his son are waiting at a bus stop in the evening. There is a lamp post behind them. The lamp post, the father and his son stand on the same straight line. The father observes that the shadows of his head and his son's head are incident at the same point on the ground. If the heights of the lamp post, the father and his son are 6 metres, 1.8 metres and 0.9 metres respectively, and the father is standing 2.1 metres away from the post then how far (in metres) is son standing form his father?

Video Solution

Solution

The three triangles are similar.

Let the distance of the tip of the shadow from the child be y. Let the child be standing at distance x from the father.

So, 6/(2.1+x+y) = 1.8/(x+y) = 0.9/y

=> 2y = x+y => x = y

=> 6/(2.1 + 2x) = 0.9/x

=> 6x = 0.9*(2.1+2x)

=> 6x = 1.89 + 1.8x

=> 4.2x = 1.89

=> x = 1.89/4.2 = 0.45

Question 19

Let $$f(x) = ax^2 - b|x|$$ , where a and b are constants. Then at x = 0, f(x) is

[CAT 2004]

Video Solution

Solution

$$f(x) = ax^2 - b|x|$$. When $$x=0, f(x) = 0$$
When $$a > 0$$ and $$b < 0$$,
For x > 0, $$f(x) = ax^2 - bx$$, will be greater than 0 as $$ax^2 > 0$$ and $$bx<0$$ as $$b$$ is negative and $$x$$ is positive.
For x < 0, $$f(x) = ax^2 + bx$$ will again be greater than 0 as $$ax^2 >0$$ and $$bx>0$$ as both $$b$$ and $$x$$ are negative.

Therefore, the function $$f(x)$$ is positive when $$x<0$$ and when $$x>0$$ but becomes 0 when $$x=0$$.

Therefore, for $$a > 0$$ and $$b < 0$$, f(x) will attain its minimum value at $$x = 0$$.

Question 20

Each family in a locality has at most two adults, and no family has fewer than 3 children.

Considering all the families together, there are more adults than boys, more boys than girls, and more girls than families.

Then the minimum possible number of families in the locality is

Video Solution

Solution

Consider 3 family. Let 1st one have 2 A , 1B and 2 G , 2nd one have 2 A , 2B and 1 G, 3rd one have 2 A , 2B and 1 G . SO total A-6 , B - 5 , g - 4 , F - 3 . Hence minimum is 3 .

CAT Content

CAT 2004 Question Paper

If a man cycles at 10 km/hr, then he arrives at a certain place at 1 p.m. If he cycles at 15 km/hr, he will arrive at the same place at 11 a.m. At what sped must he cycle to get there at noon?

If $$f(x)=x^3-4x+p$$ , and f(0) and f(1) are of opposite signs, then which of the following is necessarily true

[CAT 2004]

Suppose n is an integer such that the sum of digits on n is 2, and $$10^{10} < n < 10^{11}$$. The number of different values of n is

A milkman mixes 20 litres of water with 80 litres of milk. After selling one-fourth of this mixture, he adds water to replenish the quantity that he had sold. What is the current proportion of water to milk?

[CAT 2004]

Let $$ y = \frac{1}{2+\frac{1}{3+\frac{1}{2+\frac{1}{3+…}}}}$$. Then y equals?

N persons stand on the circumference of a circle at distinct points. Each possible pair of persons, not standing next to each other, sings a two-minute song one pair after the other. If the total time taken for singing is 28 minutes, what is N?

[CAT 2004]

Let $$f(x) = ax^2 - b|x|$$ , where a and b are constants. Then at x = 0, f(x) is

[CAT 2004]

Each family in a locality has at most two adults, and no family has fewer than 3 children.

Considering all the families together, there are more adults than boys, more boys than girls, and more girls than families.

Then the minimum possible number of families in the locality is

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CAT Content

CAT 2004 Question Paper

If a man cycles at 10 km/hr, then he arrives at a certain place at 1 p.m. If he cycles at 15 km/hr, he will arrive at the same place at 11 a.m. At what sped must he cycle to get there at noon?

If $$f(x)=x^3-4x+p$$ , and f(0) and f(1) are of opposite signs, then which of the following is necessarily true [CAT 2004]

Suppose n is an integer such that the sum of digits on n is 2, and $$10^{10} < n < 10^{11}$$. The number of different values of n is

A milkman mixes 20 litres of water with 80 litres of milk. After selling one-fourth of this mixture, he adds water to replenish the quantity that he had sold. What is the current proportion of water to milk? [CAT 2004]

Let $$ y = \frac{1}{2+\frac{1}{3+\frac{1}{2+\frac{1}{3+…}}}}$$. Then y equals?

N persons stand on the circumference of a circle at distinct points. Each possible pair of persons, not standing next to each other, sings a two-minute song one pair after the other. If the total time taken for singing is 28 minutes, what is N? [CAT 2004]

Let $$f(x) = ax^2 - b|x|$$ , where a and b are constants. Then at x = 0, f(x) is [CAT 2004]

Each family in a locality has at most two adults, and no family has fewer than 3 children. Considering all the families together, there are more adults than boys, more boys than girls, and more girls than families. Then the minimum possible number of families in the locality is

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Related Formulas With Tests

CAT Quant Questions | CAT Quantitative Ability

CAT DILR Questions | LRDI Questions For CAT

CAT Verbal Ability Questions | VARC Questions For CAT

Also Read

Get Started with 5000+ Free CAT Questions & Mock Tests with Detailed Solutions!

If $$f(x)=x^3-4x+p$$ , and f(0) and f(1) are of opposite signs, then which of the following is necessarily true

[CAT 2004]

A milkman mixes 20 litres of water with 80 litres of milk. After selling one-fourth of this mixture, he adds water to replenish the quantity that he had sold. What is the current proportion of water to milk?

[CAT 2004]

N persons stand on the circumference of a circle at distinct points. Each possible pair of persons, not standing next to each other, sings a two-minute song one pair after the other. If the total time taken for singing is 28 minutes, what is N?

[CAT 2004]

Let $$f(x) = ax^2 - b|x|$$ , where a and b are constants. Then at x = 0, f(x) is

[CAT 2004]

Each family in a locality has at most two adults, and no family has fewer than 3 children.

Considering all the families together, there are more adults than boys, more boys than girls, and more girls than families.

Then the minimum possible number of families in the locality is