CAT 2002 Question Paper

The given equations are  x + y + z = 5 -- (1) , xy + yz + zx = 3 -- (2)
xy + yz + zx = 3
x(y + z) + yz = 3
=> x ( 5 -x ) +y ( 5 - x - y) = 3
=> $$ - y^2 - y (5 -x) - x ^2 + 5x = 3$$
=> $$ y^2 + y (x-5) + ( x ^2 - 5x +3)  = 0 $$
The above equation should have real roots for y, => Determinant >= 0

=>$$b^2-4ac>0$$

=> $$ ( x - 5)^2 - 4(x ^2 - 5x +3 ) \geq 0$$
=>  $$ 3x^2 -10x - 13 \leq 0$$
=>  $$ -1 \leq x \leq \frac{13}{3}$$
Hence maximum value x can take is $$\frac{13}{3}$$, and the corresponding values for y,z are $$\frac{1}{3},\frac{1}{3}$$

To mow half of lawn is to mow 400 squares of 1m width each. Neeraj mows 40+18+40+18 = 116 squares of 1m width each in 1st round, in 2nd round he mows 38+16+38+16 = 108 squares and in 3rd round, he mows 36+14+36+14 = 100 squares. So, in total, he mows 324 squares; but he needs 400-324 = 76 more.

If he covers the 4th round completely, he mows 92 squares, to cover only 76, he needs to cover 76/92 = 0.826 round.

So in total 3.826 $$\approx$$ 3.83 rounds are required.

Suppose the thief stole 'x' diamonds. \

After giving the share to the first watchman, the thief has (x/2)-2 diamonds.

After giving to the second watchman, the thief has (x/4)-3 diamonds.

After giving to the third watchman, the thief has (x/8)-(7/2) diamonds.

This is equal to 1. So, (x/8) - 7/2 = 1

Solving this equation, we get x = 36

Let the amount paid by Mayank be x. So, amount paid by the other three = 2x

=> Total bill = x+2x = 3x = 60 => x = 20. So, Mayank paid 20

Similarly, amount paid by Mirza + 3*Amount paid by Mirza = 60

=> Amount paid by Mirza = 15

Amount paid by Little + 4*Amount paid by Little = 60

=> Amount paid by Little = 12

So, amount paid by Jaspal = 60 - (20+15+12) = 60 - 47 = $13

Let x = y+z such that z > y.

We know that $$48*(z - y)^2 = z^2 - y^2$$ 

Solving the above equation, we get z + y = 48

So, option d) is the correct answer.

Let the number of total days=N
They played tennis for=N-14  days
They did yoga for =N-24 days

And the question says that total days when they did yoga or played tennis are 22

which means

N-14 + N-24 = 22

2N - 38 = 22

2N = 60

N = 30

Hence total days they stayed together were 30

Let $$S = 2+5x+9x^2+....$$
$$S*x = 2x+5x^2+9x^3+...$$
$$S(1-x) = 2+3x+4x^2+...$$
$$S(1-x)*x = 2x+3x^2+4x^3+...$$
$$S(1-x)(1-x) = 2+x+x^2+x^3+... = 2+x/(1-x)$$
So, $$S = [2(1-x) + x]/(1-x)^3 => S = (2-x)/(1-x)^3$$

The equation is not satisfied for only x = 2y.

Using statements B and C, i.e., x = 2z and 2x = z, we see that the equation is not satisfied.

Using statements A and B, i.e., x = 2y and x = 2z, i.e., z = y = x/2, the equation is satisfied.

Option c) is the correct answer.

Let the actual average be n. So, the new average is n + 1.8
Actual total = 10n
New total = 10n + 18

Let the number which was miswritten = ab(a is the tenth's digit and b is the units digit) = 10a+b

and reversed number ba = 10b+a

So, 10b + a - (10a + b) = 18
=> 9(b-a) = 18
=> b-a = 2

Suppose Akil drove the car for less than 5 hrs. In this case, by distance basis, Rs 360 should be charged. This is not the case.

So he dove for more than 5 hrs. Cost comes more using time basis; which is Rs 300, i.e. he used the car for 6 hours.