[Download PDF] CAT 2005 Question Paper Paper with Solutions

Instructions

Ram and Shyam run a race between points A and B, 5 km apart. Ram starts at 9 a.m. from A at a speed of 5 km/hr, reaches B, and returns to A at the same speed. Shyam starts at 9:45 a.m. from A at a speed of 10 km/hr, reaches B and comes back to A at the same speed.

Question 1

At what time do Ram and Shyam first meet each other?

Video Solution

Solution

Let the time at which they meet be t minutes past 10.

So, distance run by Ram + distance run by Shyam = 10 km

=> (60+t)5/60 [t+60 because he would have traveled for 9 am to 10 am and t minutes more before meeting Shyam]+ (15+t)*10/60 [15+t because he would have traveled from 9:45 to 10:00 and t minutes more]= 10

=> 300+5t+150+10t = 600 => t = 10

So, they meet at 10.10 am

Question 2

At what time does Shyam overtake Ram?

Video Solution

Solution

Let the time at which Shyam overtakes Ram be t minutes past 10.
So, distance run by both of them is the same till that moment.
(60+t)5 = (15+t)10 => 300 + 5t = 150 + 10t => 5t = 150 => t = 30 min.
So, at 10.30 am, Shyam overtakes Ram

Instructions

For the following questions answer them individually

Question 3

If R = $$(30^{65}-29^{65})/(30^{64}+29^{64})$$ ,then

Video Solution

Solution

$$\frac{(30^{65}-29^{65})}{(30^{64}+29^{64})} = ((30-29)*\frac{(30^{64}+30^{63}*29+....+29^{64})}{(30^{64}+29^{64})}$$ , which is greater than 1 . Hence option D.

Question 4

What is the distance in cm between two parallel chords of lengths 32 cm and 24 cm in a circle of radius 20 cm?

Video Solution

Solution

The distances of the chords from the center are 12 cm and 16 cm respectively.

If the chords lie on the same side of the center, the distance between the chords is 4 cm, if they lie on opposite sides of the center, the distance between them is 28 cm.

Question 5

For which value of k does the following pair of equations yield a unique solution for x such that the solution is positive?

$$x^2 - y^2 = 0$$

$$(x-k)^2 + y^2 = 1$$

Video Solution

Solution

From 1st equation we know that $$(x)^2 = y^2 $$

Substituting this in 2nd equation. we get , $$2*x^2 - 2*x*k + k^2-1 =0 $$ and for unique solution $$b^2-4ac=0$$ must satisfy.

This is possible only when k = $$\sqrt{2}$$

Question 6

If x = $$(16^3 + 17^3+ 18^3+ 19^3 )$$, then x divided by 70 leaves a remainder of

Video Solution

Solution

We know that x = $$16^3 + 17^3 + 18^3 + 19^3 = (16^3 + 19^3) + (17^3 + 18^3)$$

= $$(16 + 19)(16^2 - 16 * 19 + 19^2) + (17 + 18)(17^2 - 17 * 18 + 18^2)$$ = 35 × odd + 35 × odd = 35 × even = 35 × (2k)

=> x = 70k

=> Remainder when divided by 70 is 0.

Question 7

A chemical plant has four tanks (A, B, C, and D), each containing 1000 litres of a chemical. The chemical is being pumped from one tank to another as follows:

From A to B @ 20 litres/minute

From C to A @ 90 litres/minute

From A to D @ 10 litres/minute

From C to D @ 50 litres/minute

From B to C @ 100 litres/minute

From D to B @ 110 litres/minute

Which tank gets emptied first, and how long does it take (in minutes) to get empty after pumping starts?

Video Solution

Solution

After 1 min the cans will contain following amount of chemicals : A - 1060 B - 1030 C - 970 D - 950

So, we can see that the can D loses 50 ltrs in 1 min which is highest. So the can D will lose 1000 ltrs in 20*1 = 20 mins.

Question 8

Two identical circles intersect so that their centres, and the points at which they intersect, form a square of side 1 cm. The area in sq. cm of the portion that is common to the two circles is

Video Solution

Solution

We know that quad ambn is a square of side 1.

Area of the sector a-mqn is $$\frac{90}{360}* \pi *1*1$$ = $$\frac{\pi }{4}$$.

Area of square = 1*1 = 1

Area of common portion = 2 * Area of sector - Area of square

= 2 * $$\frac{\pi }{4}$$ - 1 = $$\frac{\pi }{2}$$ - 1

Question 9

A jogging park has two identical circular tracks touching each other, and a rectangular track enclosing the two circles. The edges of the rectangles are tangential to the circles. Two friends, A and B, start jogging simultaneously from the point where one of the circular tracks touches the smaller side of the rectangular track. A jogs along the rectangular track, while B jogs along the two circular tracks in a figure of eight. Approximately, how much faster than A does B have to run, so that they take the same time to return to their starting point?

Video Solution

Solution

Let the radius of 2 circles be r . Speed of A would be 12r/t and Speed of B would be 4*Pi*r/t . To find percentage faster B have to run than A we have : (4*Pi - 12)*100/ (4*Pi) = 4.46% approx. Hence, option D.

Question 10

In a chess competition involving some boys and girls of a school, every student had to play exactly one game with every other student. It was found that in 45 games both the players were girls, and in 190 games both were boys. The number of games in which one player was a boy and the other was a girl is

Video Solution

Solution

Number of games in which both the players are girls = $$^GC_2$$ where G is the number of girls
$$^GC_2 = 45$$
$$^{10}C_2 = 45$$
So, G = 10
Similarly, number of games in which both the players are boys = $$^BC_2$$, where B is the number of boys
$$^BC_2 = 190$$
$$^{20}C_2 = 190$$
So, B = 20
So, number of games in which one player is a boy and the other player is a girl is 20*10 = 200

CAT Content

CAT 2005 Question Paper

At what time do Ram and Shyam first meet each other?

At what time does Shyam overtake Ram?

If R = $$(30^{65}-29^{65})/(30^{64}+29^{64})$$ ,then

What is the distance in cm between two parallel chords of lengths 32 cm and 24 cm in a circle of radius 20 cm?

For which value of k does the following pair of equations yield a unique solution for x such that the solution is positive?

$$x^2 - y^2 = 0$$

$$(x-k)^2 + y^2 = 1$$

If x = $$(16^3 + 17^3+ 18^3+ 19^3 )$$, then x divided by 70 leaves a remainder of

Two identical circles intersect so that their centres, and the points at which they intersect, form a square of side 1 cm. The area in sq. cm of the portion that is common to the two circles is

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CAT Content

CAT 2005 Question Paper

At what time do Ram and Shyam first meet each other?

At what time does Shyam overtake Ram?

If R = $$(30^{65}-29^{65})/(30^{64}+29^{64})$$ ,then

What is the distance in cm between two parallel chords of lengths 32 cm and 24 cm in a circle of radius 20 cm?

For which value of k does the following pair of equations yield a unique solution for x such that the solution is positive? $$x^2 - y^2 = 0$$ $$(x-k)^2 + y^2 = 1$$

If x = $$(16^3 + 17^3+ 18^3+ 19^3 )$$, then x divided by 70 leaves a remainder of

Two identical circles intersect so that their centres, and the points at which they intersect, form a square of side 1 cm. The area in sq. cm of the portion that is common to the two circles 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

For which value of k does the following pair of equations yield a unique solution for x such that the solution is positive?

$$x^2 - y^2 = 0$$

$$(x-k)^2 + y^2 = 1$$