CAT Questions

In a triangle ABC, the lengths of the sides AB and AC equal 17.5 cm and 9 cm respectively

• Mensuration

In a triangle ABC, the lengths of the sides AB and AC equal 17.5 cm and 9 cm respectively. Let D be a point on the line segment BC such that AD is perpendicular to BC. If AD = 3 cm, then what is the radius (in cm) of the circle circumscribing the triangle ABC?

1. 17.05
2. 22.45
3. 26.25
4. 27.85

If a right circular cylinder of height 14 is inscribed in a sphere of radius 8

• Mensuration

If a right circular cylinder of height 14 is inscribed in a sphere of radius 8, then the volume of the cylinder is

1. 110
2. 220
3. 440
4. 660

Let ABCDEF be a regular hexagon with each side of length 1 cm

• Mensuration

Let ABCDEF be a regular hexagon with each side of length 1 cm. The area (in sq cm) of a square with AC as one side is

1. 3√2
2. 3
3. 4
4. √3

The base of a vertical pillar with uniform cross section is a trapezium

• Mensuration

The base of a vertical pillar with uniform cross section is a trapezium whose parallel sides are of lengths 10 cm and 20 cm while the other two sides are of equal length. The perpendicular distance between the parallel sides of the trapezium is 12 cm. If the height of the pillar is 20 cm, then the total area, in sq cm, of all six surfaces of the pillar is

1. 1300
2. 1340
3. 1480
4. 1520

A circle is inscribed in a given square and another circle is circumscribed about the square

• Geometry

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

• Geometry

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

• Geometry

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

• Geometry

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

• Geometry

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

• Geometry

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

• Geometry

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

• Numbers

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 3^2015 is

• Numbers

The last digit of the number 3²⁰¹⁵ is

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

When we multiply a certain two-digit number

• Numbers

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

Let p be a prime number greater than 5. Then (p^2 - 1) is

• Numbers
• Divisibility

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

• Numbers

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

• Numbers

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 30^2720

• Numbers

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

• Numbers

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

• Numbers

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

• Numbers
• Linear Equations

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 7^2008

• Numbers

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 2^51

• Numbers

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

• Numbers

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

If N = (11^(p+7)) (7^(q-2)) (5^(r+1)) (3^s) is a perfect cube

• Numbers

If N = (11^{p + 7})(7^{q – 2})(5^{r + 1})(3^s) 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

• Numbers

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) = x^2 and g(x) = 2x, for all real x

• Numbers

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 + x^2, 6 - 3x), when x > 0, is

• Quant Quadratic

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

• Quant Quadratic

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

• Quant Quadratic

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 x^2 - (α-2)x - α - 1 = 0

• Quant Quadratic

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 x^3 + 3x^2 + 4x + 5 = 0

• Quant Quadratic

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 x^2 + kx - 8 = 0 is square of the other. Then the value of k is

• Quant Quadratic

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 3m^2 - 21m + 30

• Quant Quadratic

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 x^2 - 3x + 2 > 0 at all

• Quant Quadratic

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

• Quant Quadratic

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

• Progressions

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

• Progressions

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 (2^x - 5), log3 (2^x - 7/2) are in arithmetic progression

• Progressions
• Logarithms

If log₃ 2, log₃ (2^x - 5), log₃ (2^x - 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)

• Progressions
• Linear Equations

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

• Progressions

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

• Progressions

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

• Progressions

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

• Progressions

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, ...,

• Progressions

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

• Progressions

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 AP

• Progressions

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

• Progressions

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) + …

• Progressions

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

• Progressions

An infinite geometric progression a₁, a₂, a₃, ... has the property that a_n = 3(a_n+1 + a_n+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