Let the sum of two positive integers be 24. If the probability, that their product is not less than 3/4 times their greatest positive product, is m/n, where gcd(m, n) = 1, then n – m equals :

The values of x satisfying x^{log₅ x} > 5 lie in the interval

E₁ and E₂ are two independent events of a random experiment with P(E₁) = 1/2 and P(E₂) = 1/3.

1. Probability that event E₁ occurs given event E₂ has already occurred is 1/2

2. Probability that event E₂ occurs given event E₁ has already occurred is 1/2

Which of the following statement(s) is/are correct?

If the term independent of x in the expansion of  is 105, then a² is equal to :

Find the order and degree of  respectively.

Let a be the sum of all coefficients in the expansion of (1 – 2x + 2x² )²⁰²³ (3 – 4x² +2x³ )²⁰²⁴ and  If the equations cx² + dx + e = 0 and 2bx² + ax + 4 = 0 have a common root, where c, d, e ∈ R, then d : c : e equals

Consider the following statements

Which of the above statement(s) is/are correct?

If A, B and C are angles of a triangle, then the determinant  is equal to

The area (in sq. units) of the region described by A = {(x,y)|y ≥ x² − 5x + 4,x + y ≥ 1, y ≤ 0} is:

Choose the correct option for the given statements

Statement 2 : f (x) and g (x) are discontinuous for all x ∈ R.

log₃ tan 1° + log₃ tan 2° + ..... + log₃ tan 88° + log₃ tan 89° =

The distance of the point (1, 1, 9) from the point of intersection of the line (x – 3)/1 = (y – 4)/2 = (z – 5)/2 and the plane x + y + z = 17 is:

The number of integers, between 100 and 1000 having the sum of their digits equals to 15, is

The number of complex numbers z satisfying |z – 3 – i| = |z – 9 – i| and |z – 3 + 3i| = 3 are

The number of roots of the equation,

in the interval [0, π] is equal to:

Let R₁ and R₂ be two relations defined as follows:

R₁ = {(a, b) ∈ R² : a² + b² ∈ Q}

and R₂ = {(a, b) ∈ R² : a² + b² ∉ Q}. Then,

A shopkeeper wants to check the average number of cars sold per call. Past record of sales is shown below: 

The expected number of cars sold is :

Let A be a 3 × 3 invertible matrix. If |adj(24 A)| = |adj(3 adj (2A))|, then |A|² is equal to :

Consider the following relations:

R={(x, y) : x, y are real numbers and x = wy for some rational number w}

Then,

There are n points in a plane, in which m points are collinear. Let f(n, m) denotes the number of triangles which can be formed with the given points as vertices. If f(n + 1, m + 1) - f(n, m) = 11, a possible value of (m, n) is:

The sum and product of matrices A and B exist. Which of the following implications are necessarily true?

I. A and B are square matrices of same order.

II. A and B are non-singular matrices.

Select the correct answer using the code give below.

If the correlation coefficient between X and Y is 0.7, then the correlation coefficient between U and V, where U = X - 5 and V = Y + 2, is:

Consider the function  where [x] denotes the greatest integer less than or equal to x. If S denotes the set of all ordered paris (a, b) such that f(x) is continuous at x = 3, then the number of elements in S is:

What is the number of words formed using the letters of the word COMPUTER, so that all the vowels are always together?

Let a, b∈R. If the mirror image of the point P(a, 6, 9) with respect to the line (x – 3)/7 = (y – 2)/5 = (z – 1)/(-9) is (20, b, -a -9), then |a + b| is equal to:

What is the period of the function f(x) = 2sin x cos x?

If g is the inverse of a function f and  then g′(x) is equal to

If g(x) = x² + x – 1 and (gof)(x) = 4x² – 10x + 5, then  is equal to

Let S = {1, 2, … , 499, 500}. The number of ways of choosing 4 dictinct integers from S such that they form an increasing GP and its common ratio is a positive integer number is

Let f(x) and ϕ(x) be two continuous functions on R satisfying  and another continuous function g(x) satisfying g(x + α) + g(x) = 0, ∀ x ∈ R, α > 0 and  is independent of b, then

I. If f(x) is an even function, then ϕ(x) is also even.

II. If f(x) is an even function, then ϕ(x) is an odd function.

III. f(x) and ϕ(x) are independent.

Which of the above statement(s) is/are correct?

Let  where O is the origin. If S is the parallelogram with adjacent sides OA and OC, then  is equal to ___

Let R be the focus of the parabola y² = 20x and the line y = mx + c intersect the parabola at two points P and Q.

Let the point G(10, 10) be the centroid of the triangle PQR. If c – m = 6, then (PQ)² is

In a class of 160 students, each of them opt at least one language from among English, Hindi and Sanskrit. It is found that 130 students opt English, 120 students Hindi and 110 Sanskrit. If the students opt either only one language or all three languages, then what is the number of students who study all three languages?

If a, b, c are positive real numbers, then 

Let the equations of two ellipses be  then the length of the minor axis of ellipse E₂ is:

Let h (x) = f₁(x) + f₂(x) + f₃(x) + ........ + f_n(x) where f₁(x), f₂(x), f₃(x) ,........ , f_n(x) are real valued functions of x.

Statement 1 : f(x) = |cos |x|| + cos^–1 (sgn x) + |ln x| is not differentiable at 3 points in (0, 2π)

Statement 2 : Exactly one function f_i(x), i = 1, 2, ........, n not differentiable and the rest of the function differentiable at x = a makes h(x) not differentiable at x = a.

Let the set S = {2, 4, 8, 16, ....., 512} be partitioned into 3 sets A, B, C with equal number of elements such that A ∪ B ∪ C = S and A ∩ B = B ∩ C = A ∩ C = ϕ. The maximum number of such possible partitions of S is equal to :

Let a computer program generate only the digits 0 and 1 to form a string of binary numbers with probability of occurrence of 0 at even places be 1/2 and probability of occurrence of 0 at the odd place be 1/3. Then the probability that '10' is followed by ‘01' is equal to :

The set (A ∩ B′)′ ∪ (B ∩ C) is equal to

Let 2^nd, 8^th and 44^th, terms of a non-constant A.P. be respectively the 1^st, 2^nd and 3^rd terms of G.P. If the first term of A.P. is 1 then the sum of first 20 terms is equal to-

A random experiment is conducted 5 times. If the number of success of the experiment follows binomial distribution such that difference of mean and variance of the success is 5/9, then :

1. Mean of the binomial distribution is 5/3.

2. Probability of getting at most 2 success is 64/81.

Which of the following statement(s) is/are correct?

PMF (Probability mass Function) of X is given by P(X = x) = (x² + 1)/15; x = -2, -1, 0, 1, 2.

Find the conditional probability of 

If P is a point (x, y) on the line y = 2x such that P and the point (4, 2) are on the opposite sides of the line 2x + 5y = 8 then –

Let a, b, c be in arithmetic progression. Let the centroid of the triangle with vertices (a, c), (2, b) and (a, b) be (10/3, 7/3). If α, β are the roots of the equation ax² + bx + 1 = 0, then the value of α² + β² – αβ is:

The equation of the plane passing through the point (1, 2, –3) and perpendicular to the planes 3x + y – 2z = 5 and 2x – 5y – z = 7, is:

Consider the following statements

I. The product of two non-zero matrices can never be identity matrix.

II. The product of two non-zero matrices can never be zero matrix.

Which of the above statement(s) is/are correct ?

If the domain of the function f(x) = log_e  is (α, β], then the value of 5β - 4α is equal to

If f is a function such that f(0) = 2, f(1) = 3 and f(x + 2) = 2f(x) – f(x + 1) for every real x then f(5) is

If 27 is the p^th, 8 is the q^th and 12 is the s^th term of a G.P., then

Let y = y(x) be the solution of the differential equation 

If ω is a complex cube root of unity, then 

Let ƒ, g : R → R be defined as : ƒ(x) = |x – 1| and  Then the function f(g(x)) is 

If three real numbers a, b, c are in harmonic progression, then which of the following is true

Let α be a root of the equation (a - c) x² + (b - a)x (c - b) = 0 where a, b, c are distinct real numbers such that the matrix  is singular. 

Then, the value of 

The coefficient of x⁵ in the expansion of 

A point is chosen at random inside a circle.

I: Probability that the point is closer to the center of the circle than to its boundary is 1/4.

II: Probability that the point is closer to the circle’s boundary than its center is 1/4.

If α, β, γ, δ are the roots of the equation x⁴ + x³ + x² + x + 1 = 0, then α²⁰²¹ + β²⁰²¹ + γ²⁰²¹ + δ²⁰²¹ is equal to

Let a unit vector which makes an angle of 60° with 2î + 2ĵ - k̂ and an angle of 45° with 

If the system of linear equations

x + y + z = 3

x + by + b²z = 7

and ax + a²y + a³z = 1

has infinitely many solutions, then the ordered pair (a, b) is

Consider the following statements:

2. If a square matrix A is orthogonal as well as symmetric, then A is an idempotent matrix.

Which of the above statements is/are correct?

The differential equation of the family of curves y = Ae^3x + Be^5x, where A and B are arbitrary constants, is

Find the non-negative remainder when 41⁶⁵ and 2⁵⁰ are divided by 7

If 0 ≤ x < 2π, then the number of real values of x, which satisfy the equation cos x + cos2x + cos3x + cos4x = 0:

The general solution of the equation sin¹⁰⁰ x – cos¹⁰⁰x = 1, is

Let A and B be two matrices such that AB = A and BA = B. which of the following statements are correct ?

1. A² = A

2. B² = B

3. (AB)² = AB

Select the correct answer using the code given below:

Suppose l, m, n respectively represent the coefficient of x¹⁰, the constant term and the coefficient of x^-10 in the expansion of  then a² ∶ b² =

If ax² - by² + 2hxy + 2gx + 2fy + c = 0 is the locus of a point, which moves such that it is always equidistant from the lines 2x - y + 3 = 0 and x + 2y - 5 = 0, then the value of a - b + g - c + h - f equals

Bag A contains 3 white, 7 red balls and bag B contains 3 white, 2 red balls. One bag is selected at random and a ball is drawn from it. The probability of drawing the ball from the bag A, if the ball drawn in white, is:

Let C be the point (6, 5) and let A and B be two points on the line 3x - 4y = 1 such that PQR is an equilateral triangle. Then the area of ΔPQR is :

The slope of the tangent to the curve x = 2t² - 4t + 3 and y = t³ - 4t2 + 6 at t = 2 is:

The coordinates of the middle point of the chord 2x – 5y + 18 = 0 cut off by the circle x² + y² – 6x + 2y – 54 = 0 are

Consider the following statements:

2. If A and B are symmetric matrices, then AB - BA is a symmetric matrix.

3. If A is a square matrix such that A² = I, then (A - I)³ + (A + I)³ - 7A = A.

Which of the above statements is/are correct?

If α, β ≠ 0 and f(n) = αⁿ + βⁿ and  = K(1 - α)² (1 - β)² (α - β)² then K is equal to:

A circle is passing through the points (3, -5), (-1, 7), and (1, 3).

What are the coordinates of the center of the circle?

A circle is passing through the points (3, -5), (-1, 7), and (1, 3).

If r is the radius of the circle, then which one of the following is correct?

Consider the following data for the next three (03) items that follow:

The function f(x) = (x + 1)³(x - 3)³  and g(x) = 3e^2x + 2e^3x

Find the interval where function f(x) is strictly increasing.

Consider the following data for the next three (03) items that follow:

The function f(x) = (x + 1)³(x - 3)³  and g(x) = 3e^2x + 2e^3x

Consider the following data for the next three (03) items that follow:

The function f(x) = (x + 1)³(x - 3)³  and g(x) = 3e^2x + 2e^3x

Find The interval where function g(x) is strictly increasing.

Consider the following for the items that follow:

Consider the following for the items that follow:

Consider the following for the items that follow :

What is the value of β ?

Consider the following for the items that follow :

What is the value of α ?

Directions: For the next three (03) items that follow:

Consider the function 

Directions: For the next three (03) items that follow:

Consider the function

Consider the following statements:

1. The function is discontinuous at x = 3

2. The function is not differentiable at x = 0

Which of the above statements is/are correct?

Directions: For the next three (03) items that follow:

Consider the function

Consider the following for the items that follow:

A, B, and C are three events such that P(A) = 0.6 , P(B) = 0.4 , P(C) = 0.5 , P(A ∪ B) = 0.8 , P(A ∩ C) = 0.3 , P(A ∩ B ∩ C) = 0.2 , and P(A ∪ B ∪ C) ≥ 0.85

What is the minimum value of P(A ∩ B) ?

Consider the following for the items that follow:

A, B, and C are three events such that P(A) = 0.6 , P(B) = 0.4 , P(C) = 0.5 , P(A ∪ B) = 0.8 , P(A ∩ C) = 0.3 , P(A ∩ B ∩ C) = 0.2 , and P(A ∪ B ∪ C) ≥ 0.85

What is the maximum value of P(B ∩ C) ?

What is the value of fof(-7/4) - gog(-1)?

What is the value of fog(-2/3) - gof(-2/3)?

For the next two (2) items that follow:

The median of the following distribution is 14.4 and the total frequency is 20:

What is the relation between x and y?

For the next two (2) items that follow:

The median of the following distribution is 14.4 and the total frequency is 20:

What is x equal to

What is the coefficient of variance?

What is the standard deviation ?

What is the value of f₂ ?

How much excess money (in crore) is allocated to Miscellaneous over Education ?

How much money (in crore) is allocated to both Agriculture and Employment?

Direction: Based on the following information, answer the questions

Direction: Based on the following information, answer the questions

What is x -y is equal to?

Direction: Based on the following information, answer the questions

What is x equal to ?

Consider the following data for the next two (02) items that follow:

Given that, acos³θ + 3acosθsin²θ = x and asin³θ + 3acos²θsinθ = y

Consider the following data for the next two (02) items that follow:

Given that, acos³θ + 3acosθsin²θ = x and asin³θ + 3acos²θsinθ = y

Direction: Based on the following information, answer the questions.

Matrix A has x rows and x + 5 columns. Matrix B has y rows and 11 - y columns. Both AB and BA exist.

The order of AB is ?

Direction: Based on the following information, answer the questions.

Matrix A has x rows and x + 5 columns. Matrix B has y rows and 11 - y columns. Both AB and BA exist.

Find the value of y