# CAT Questions

A circle is inscribed in a given square and another circle is circumscribed about the square. What is the ratio of the area of the inscribed circle to that of the circumscribed circle?

1. 2 : 3
2. 3 : 4
3. 1 : 2
4. 1 : 4

A semicircle is drawn with AB as its diameter. From C, a point on AB, a line perpendicular to AB is drawn meeting the circumference of the semi-circle at D. Given that AC = 2 cm and CD = 6 cm, the area of the semicircle (in sq. cm) will be:

1. 32 π
2. 40.5 π
3. 50 π
4. 81 π

Consider obtuse angled triangles with sides 8 cm, 15 cm and x cm. If x is an integer then how many such triangles exist?

1. 5
2. 10
3. 15
4. 21

If in the figure below, angle XYZ = 90° and the length of the arc XZ = 10π, then the area of the sector XYZ is 1. 10π
2. 25π
3. 100π
4. None of the above

AB is a chord of a circle. The length of AB is 24 cm. P is the midpoint of AB. Perpendiculars from P on either side of the chord meets the circle at M and N respectively. If PM < PN and PM = 8 cm. then what will be the length of PN?

1. 17 cm
2. 18 cm
3. 19 cm
4. 20 cm
5. 21 cm

AB, CD and EF are three parallel lines, in that order. Let d1 and d2 be the distances from CD to AB and EF respectively. d1 and d2 are integers, where d1 : d2 = 2 : 1. P is a point on AB, Q and S are points on CD and R is a point on EF. If the area of the quadrilateral PQRS is 30 square units, what is the value of QR when value of SR is the least?

1. slightly less than 10 units
2. 10 units
3. slightly greater than 10 units
4. slightly less than 20 units
5. slightly greater than 20 units

The points (2, 5) and (6, 3) are two end points of a diagonal of a rectangle. If the other diagonal has the equation y = 3x + c, then c is

1. -5
2. -6
3. -7
4. -8

If n is a positive integer, then (√3+1)2n - (√3−1)2n is

1. An even positive integer
2. A rational number other than positive integers
3. An odd positive integer
4. An irrational number

The last digit of the number 32015 is

1. 1
2. 3
3. 5
4. 7

When we multiply a certain two-digit number by the sum of its digits, 405 is achieved. If you multiply the number written in reverse order of the same digits, we get 486. Find the number?

1. 81
2. 45
3. 36
4. 54

A positive integer is said to be a prime number if it is not divisible by any positive integer other than itself and 1. Let p be a prime number greater than 5. Then (p2 - 1) is

1. always divisible by 6, and may or may not be divisible by 12
2. always divisible by 24
3. never divisible by 6
4. always divisible by 12, and may or may not be divisible by 24

Consider four-digit numbers for which the first two digits are equal and the last two digits are also equal. How many such numbers are perfect squares?

1. 1
2. 2
3. 3
4. 4

The number of integers n satisfying -n+2 ≥ 0 and 2n ≥ 4 is

1. 0
2. 1
3. 2
4. 3

What is the right most non-zero digit of the number 302720?

1. 1
2. 3
3. 7
4. 9

The product of all integers from 1 to 100 will have the following numbers of zeros at the end?

1. 19
2. 20
3. 22
4. 24

The sum of four consecutive two-digit odd numbers, when divided by 10, becomes a perfect square. Which of the following can possibly be one of these four numbers?

1. 21
2. 25
3. 41
4. 67

The number of positive integer valued pairs (x, y) satisfying 4x - 17y = 1 and x ≤ 1000 is

1. 55
2. 57
3. 58
4. 59

What are the last two digits of 72008?

1. 01
2. 21
3. 41
4. 61

What is the digit in the unit’s place of 251?

1. 1
2. 2
3. 4
4. 8

When you reverse the digits of the number 13, the number increases by 18. How many other two-digit numbers increase by 18 when their digits are reversed?

1. 5
2. 6
3. 7
4. 8

How many times will the digit 5 come in counting from 1 to 99 excluding those which are divisible by 3?

1. 16
2. 15
3. 14
4. 13

What is the number of all possible positive integer values of n for which n2 + 96 is a perfect square?

1. 2
2. 4
3. 5
4. Infinite

If N = (11p + 7)(7q – 2)(5r + 1)(3s) is a perfect cube, where p, q, r and s are positive integers, then the smallest value of p + q + r + s is:

1. 5
2. 6
3. 7
4. 8
5. 9

If the product of three consecutive positive integers is 15600 then the sum of the squares of these integers is

1. 1777
2. 1785
3. 1875
4. 1877

Let f(x) = x2 and g(x) = 2x, for all real x. Then the value of f(f(g(x)) + g(f(x))) at x = 1 is

1. 16
2. 18
3. 36
4. 40

Largest value of min(2 + x2, 6 - 3x), when x > 0, is

1. 1
2. 2
3. 3
4. 4

A quadratic function f(x) attains a maximum of 3 at x = 1. The value of the function at x = 0 is 1. What is the value f(x) at x = 10?

1. -180
2. -159
3. -110
4. -119

If both a and b belong to the set {1, 2, 3, 4}, then the number of equations of the form ax2+ bx + 1 = 0 having real roots is

1. 6
2. 7
3. 10
4. 12

Let p and q be the roots of the quadratic equation x2 - (α - 2)x - α - 1 = 0. What is the minimum possible value of p2 + q2?

1. 0
2. 3
3. 4
4. 5

The number of roots common between the two equations x3 + 3x2 + 4x + 5 = 0 and x3 + 2x2 + 7x + 3 = 0 is

1. 0
2. 1
3. 2
4. 3

One root of x2 + kx - 8 = 0 is square of the other. Then the value of k is

1. 2
2. -2
3. 8
4. -8

What is the value of m which satisfies 3m2 - 21m + 30 < 0?

1. m < 2 or m > 5; 2 < m < 5
2. 2 < m < 5
3. m < 2 or m > 5
4. m > 2

Which of the following values of x do not satisfy the inequality x2 - 3x + 2 > 0 at all?

1. 0 ≥ x ≥ –2
2. 0 ≤ x ≤ 2
3. –1 ≥ x ≥ –2
4. 1 ≤ x ≤ 2

The minimum possible value of the sum of the squares of the roots of the equation x2 + (a + 3)x - (a + 5) = 0 is

1. 1
2. 2
3. 3
4. 4

What is the sum of the following series?

- 64, - 66, - 68, …… , - 100

1. - 1458
2. - 1558
3. - 1568
4. - 1664
5. None of the above

If three positive real numbers x, y and z satisfy y - x = z - y  and xyz = 4, then what is the minimum possible value of y?

1. 2(1/4)
2. 2(2/3)
3. 2(1/3)
4. 2(3/4)

If log3 2, log3 (2x - 5), log3 (2x - 7/2) are in arithmetic progression, then the value of x is equal to

1. 2
2. 3
3. 4
4. 5

Consider a triangle drawn on the X-Y plane with its three vertices of (41, 0), (0, 41) and (0, 0), each vertex being represented by its (X, Y) coordinates. The number of points with integer coordinates inside the triangle (excluding all the points on the boundary) is

1. 741
2. 780
3. 800
4. 820

Consider the set S = {1, 2, 3, ..., 1000}. How many arithmetic progressions can be formed from the elements of S that start with 1 and end with 1000 and have at least 3 elements?

1. 3
2. 4
3. 6
4. 7

For a Fibonacci sequence, from the third term onwards, each term in the sequence is the sum of the previous two terms in that sequence. If the difference in squares of 7th and 6th terms of this sequence is 517, what is the 10th term of this sequence?

1. 76
2. 108
3. 123
4. 147

Fourth term of an arithmetic progression is 8. What is the sum of the first 7 terms of the arithmetic progression?

1. 7
2. 35
3. 56
4. 64

If the harmonic mean between two positive numbers is to their geometric mean as 12 : 13; then the numbers could be in the ratio

1. 12 : 13
2. 4 : 9
3. 2 : 3
4. 1/12 : 1/13

The number of common terms in the two sequences 17, 21, 25, ..., 417 and 16, 21, 26, ..., 466 is

1. 19
2. 20
3. 77
4. 78

The integers 1, 2, ..., 40 are written on a blackboard. The following operation is then repeated 39 times: In each repetition, any two numbers, say a and b, currently on the blackboard are erased and a new number a + b - 1 is written. What will be the number left on the board at the end?

1. 821
2. 820
3. 819
4. 781

If p, q and r are three unequal numbers such that p, q and r are in A.P., and p, r-q and q-p are in G.P., then p : q : r is equal to

1. 1 : 2 : 3
2. 2 : 3 : 4
3. 3 : 2 : 1
4. 1 : 3 : 4

Seema has joined a new Company after the completion of her B.Tech from a reputed engineering college in Chennai. She saves 10% of her income in each of the first three months of her service and for every subsequent month, her savings are Rs. 50 more than the savings of the immediate previous month. If her joining income was Rs. 3000, her total savings from the start of the service will be Rs. 11400 in

1. 6 months
2. 12 months
3. 18 months
4. 24 months

The sum of series, (–100) + (–95) + (–90) + …………+ 110 + 115 + 120, is:

1. 0
2. 220
3. 340
4. 450
5. None of the above

An infinite geometric progression a1, a2, a3, ... has the property that an = 3(an+1 + an+2 +…) for every n ≥ 1. If the sum a1 + a2 + a3 + … = 32, then a5 is

1. 1/32
2. 2/32
3. 3/32
4. 4/32

The Maximum Retail Price (MRP) of a product is 55% above its manufacturing cost. The product is sold through a retailer, who earns 23% profit on his purchase price. What is the profit percentage (expressed in nearest integer) for the manufacturer who sells his product to the retailer? The retailer gives 10% discount on MRP.

1. 31%
2. 22%
3. 15%
4. 13%
5. 11%

A dealer offers a cash discount of 20% and still makes a profit of 20%, when he further allows 16 articles to a dozen to a particularly sticky bargainer. How much percent above the cost price were his wares listed?

1. 75%
2. 66 2/3%
3. 100%
4. 80%