# CAT Model Paper 2 Questions and Answers with Explanation Part 5

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Q: 24. What are the number of positive integral solutions for ?

(A) 1

(B) 2

(C) 3

(D) 4

Ans:

Sol:

It is clear that 5y is always a multiple of y. so is should also be a multiple of must be a multiple of can be or 15. The corresponding values of y will be . There are 3 solutions.

The correct choice is (C)

Q: 25. If are the roots of this quadratic equation, which of the following statements is true?

(A) are always real irrespective of the values of ‘b’ and ‘c’

(B) and are always complex irrespective of the values of ‘b’ and ‘c’

(C) and are always real/complex depending on the values of ‘b’ and ‘c’

(D) One root of the quadratic equation is real and the other root is complex.

Ans:

Sol:

For a quadratic equation of the form, the roots are real if the discriminant .

Consider the given equation

The discriminant is

Which can also be re-written as (Since)

But sum of the roots, and product of the roots,

Substituting the above values in the discriminant,

The above value is always positive and therefore the discriminant is also always positive and hence the roots are always real.

The correct choice is (A)

Q: 26. The weight of the empty bottle is of the weight of the bottle when filled with a liquid. Some of the liquid from the completely filled bottle is removed, and then the weight of the empty bottle is found to be of the total weight. What percentage of liquid is removed?

(A)

(B)

(C)

(D)

Ans: A

Sol:

Let the initial weight be 100gms.

Weight of bottle is

Liquid weight is .

After some liquid is removed, the weight of .

Out of , weight of bottle is , so weight of liquid is .

of liquid is removed.

Percentage of liquid removed

Q: 27. If is divisible by find the value of ?

(A) 89

(B) 121

(C)

(D)

Ans: A

Sol:

Let

f(x) is divisible by which can be rewritten as

Hence, is divisible by both and . (In other words the remainder when f(x) is divided by or is equal to zero).

As per remainder theorem, when a polynomial function is divided by the remainder is f(a).

Solving the above two equations,

Hence,

The correct choice is (A)

Q: 28. If , then for all real x which among the following hold good?

(A)

(B)

(C)

(D)

Ans: B

Sol;

We have

Hence the maximum value of A occurs when cos2x is maximum i.e. whence and the minimum value of A occurs when , when .

The correct choice is (B)

Q: 29. Consider the equation. How many ordered integer triplets satisfy the given equation?

(A) Only one

(B) Only two

(C) Infinitely many

(D) None

Ans: D

Sol:

If possible, let there be integers x, y, and z such that, we have,

or,

or,

or

Clearly, the RHS is a multiple of 8 and hence even. The LHS also must be even. However if one among

is even, then all of them are. This means that the LHS is at least a multiple of 16 whereas the RHS is a multiple of 8 which is a contradiction. Hence, no solution in integers exists.

The correct choice is (D)

Q: 30. Find the distance between the pair of straight lines given by the equation

(A) 4

(B)

(C)

(D)

Ans:

Sol:

We have,

Or;

Putting we get, the given quadratic takes the form,

Whence

Hence the equations of the lines are

The lines are parallel and applying the distance formula, we get the distance between the lines is given by.

The correct choice is (C)