If $Δ_1 = \begin{vmatrix} 1 & p & q \\ 1 & q & r \\ 1 & r & p\end{vmatrix} \ \text{and} \ Δ_2 = \begin{vmatrix} 1 & 1 & 1 \\ q & r & p \\ r & p & q \end{vmatrix}$ where p ≠ q ≠ r, then Δ₁ + Δ₂ is

If (a - b) (b - c) (c - a) = 2 and abc = 6, then what is the value of $\begin{vmatrix}a & b & c \\ a^2 & b^2 & c^2 \\ a^3 & b^3 & c^3 \end{vmatrix}?$

Under which of the following conditions does the determinant $\begin{vmatrix}a & b & c \\ b & c & a \\ c & a & b\end{vmatrix}$ vanish?

1. a + b + c = 0

2. a³ + b³ + c³ = 3abc

3. a² + b² + c² - ab - bc - ca = 0

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Consider the following in respect of the matrices:

A = [m n], B = [-n -m] & $ C = \begin{bmatrix} m \\ -m\end{bmatrix} $

 

1. CA = CB

2. AC = BC

3. C(A + B) = CA + CB

Which of the above statements is/are correct?

If $A = \begin{bmatrix} 2 \sin \theta & \cos \theta & 0 \\ -2\cos \theta & \sin \theta & 0 \\ -1 & 1 & 1 \end{bmatrix},$ then what is A(adj A) equal to?

(where) I is the identity matrix.

For what value of k is the matrix $\begin{bmatrix} 2\cos 2\theta & 2\cos 2\theta & 6 \\ 1 -2 \sin^2\theta & 2 \cos^2\theta -1 & 3 \\ k & 2k & 1 \end{bmatrix}$ singular?

Let A be a non-singular matrix and B = adj A. Which of the following statements is/are correct?

1. AB = BA

2. AB is a scalar matrix

3. AB can be a null matrix

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Consider the following statements in respect of square matrices A and B of same order :

1. If AB is a null matrix, then at least one of A and B is a null matrix.

2. If AB is an identity matrix, then BA = AB.

Which of the above statements is/are correct?

If A is the identity matrix of order 3 and B is its transpose, then what is the value of the determinant of the matrix C = A + B?

Let A and B be non-singular matrices of the same order such that AB = A and BA = B. Which of the following statements is/are correct ?

1. A² = A

2. AB² = A²B

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How many terms are there in the expansion of $\left(1 + \frac{2}{x}\right)^9\left(1-\frac{2}{x}\right)^9 ? $

Consider the following statements in respect of the expansion of (x + y)¹⁰

1. Among all the coefficients of the terms, the coefficient of the 6th term has the highest value

2. The coefficient of the 3rd term is equal to coefficient of the 9th term

Which of the above statements is /are correct ?

If C(3n, 2n) = C(3n, 2n - 7), then what is the value of C(n, n - 5)?

What is the value of

C(51, 21) - C(51, 22) + C(51, 23) - C(51, 24) + C(51, 25) - C(51, 26) + C(51, 27) - C(51, 28) + C(51, 29) - C(51, 30) ?

How many odd numbers between 300 and 400 are there in which none of the digits is repeated ?

How many permutations are there of the letters of the word 'TIGER' in which the vowels should not occupy the even positions ?

Let α and β be the roots of the equation x² + px + q = 0. If α³ and β³ are the roots of the equation x² + mx + n = 0, then what is the value of m + n ?

Let α and β be the roots of the equation x² - ax - bx + ab - c = 0. What is the quadratic equation whose roots are a and b?

If the roots of the equation x² - ax - bx - cx + bc + ca = 0 are equal, then which one of the following is correct?

Let α and β (α > β) be the roots of the equation x² - 8x + q = 0. If α² - β² = 16, then what is the value of q?

What is the maximum value of n such that 5ⁿ divides (30! + 35!), where n is a natural number?

What is the value of 2(2 × 1) + 3(3 × 2 × 1) + 4(4 × 3 × 2 × 1) + 5(5 × 4 × 3 × 2 × 1) + .................. + 9(9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1) + 2 ?

If A = {{1, 2, 3}}, then how many elements are there in the power set of A?

If a, b, c are in GP where a > 0, b > 0, c > 0, then which of the following are correct?

1. a², b², c² are in GP

2. $\frac{1}{a}, \frac{1}{b}, \frac{1}{c}$ are in GP

3. $\sqrt {a}, \sqrt{b}, \sqrt{c} $ are in GP

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If $\frac{a+b}{2}, b, \frac{b+c}{2}$ are in HP, then which one of the following is correct?

What is value of cot² 15° + tan² 15° ?

In a triangle ABC, sin A - cos B - cos C = 0. What is angle B equal to?

If $α + β = \frac{\pi}{4}$ and 2tan α = 1, then what is tan 2β equal to?

If tan (45° + θ) = 1 + sin 2θ, where $-\frac{\pi}{4}< θ <\frac{\pi}{4},$ then what is the value of cos 2θ?

Let sin 2θ = cos 3θ, where θ is acute angle. What is the value of 1 + 4 sin θ? (given that $\sin 18^\circ =\frac{\sqrt 5 - 1}{4}$)

If $\tan θ = - \frac{5}{12},$ then what can be the value of sin θ?

What is the value of $\cos^4\frac{7\pi}{8}+\cos^4\frac{5\pi}{8}?$

What is $\sin^2\left(\frac{\pi}{4}+\theta\right)-\sin^2\left(\frac{\pi}{4}-\theta\right)$ equal to?

A vertical tower stands on a horizontal plane and is surmounted by a vertical flagstaff of height h. At a point on the plane the angles of elevation of the bottom and top of the flagstaff are θ and 2θ respectively. What is the height of the tower ?

The shadow of a tower becomes x metre longer, when the angle of elevation of sun changes from 60° to θ. If the height of the tower is $\sqrt{3}x$ metre, then which one of the following is correct ?

If $\tan^{-1} \left(\frac{1}{2}\right)+\tan^{-1} \left(\frac{x}{3}\right)=\frac{\pi}{4},$ where 0 < x < 6, then what is x equal to?

If 3 sin^-1x + cos^-1x = π, then what is x equal to?

If tan α + tan β = 1 - tan α.tan β, where tan α tan β ≠ 1, then which of the following is one of the values of (α + β) ?

If (1 + tan θ) (1 + tan 9θ) = 2, then what is the value of tan(10θ)?

What is the value of sin 0° + sin 10° + sin 20° + sin 30° + ⋯ + sin 360°?

Consider all the subsets of the set A = {1, 2, 3, 4}. How many of them are supersets of the set {4}?

Consider the following statements in respect of two non-empty sets A and B :

1. x ∉ (A ∪ B) ⇒ x ∉ A or x ∉ B

2. x ∉ (A ∩ B) ⇒ x ∉ A and x ∉ B

Which of the above statements is/are correct?

Consider the following statements in respect of two non-empty sets A and B :

1. A ∪ B = A ∩ B if A = B

2. A Δ B = ϕ  if A = B

Which of the above statements is/are correct ?

Consider the following statements in respect of the relation R in the set IN of natural numbers defined by xRy if x² - 5xy + 4y² = 0 :

1. R is reflexive

2. R is symmetric

3. R is transitive

Which of the above statements is /are correct ?

Consider the following statements in respect of any relation R on a set A :

1. If R is reflexive, then R^-1 is also reflexive

2. If R is symmetric, then R^-1 is also symmetric

3. If R is transitive, then R^-1 is also transitive

Which of the above statements are correct?

What is the principal argument of $\frac{1}{1 + i}$ where $i = \sqrt{-1}?$

What is the modulus of $\left(\frac{\sqrt{-3}}{2}-\frac{1}{2}\right)^{200}?$

Consider the following statements:

1. $\frac{n!}{3!}$ is divisible by 6, where n > 3

2. $\frac{n!}{3!}+3 $ is divisible by 7, where n > 3

Which of the above statements is/are correct?

In how many ways can a team of 5 players be selected out of 9 players so as to exclude two particular players ?

In the expansion of $\left(x + \frac{1}{x}\right)^{2n} $, what is the (n + 1)th term from the end (when arranged in descending powers of x) ?

If the sum of the first 9 terms of an AP is equal to sum of the first 11 terms, then what is the sum of the first 20 terms ?

If the 5^th term of an AP is $\frac{1}{10}$ and its 10^th term is $\frac{1}{5},$ then what is the sum of first 50 terms ?

What is (1110011)₂ ÷ (10111)₂ equal to ?

If x³ + y³ = (100010111)₂ and x + y = (11111)₂, then what is (x - y)² + xy equal to ?

Consider the inequations 5x - 4y + 12 < 0, x + y < 2, x < 0 and y > 0. Which one of the following points lies in the common region?

Consider the following statements in respect of the function y = [x], x ∈ (-1, 1) where [.] is the greatest integer function:

1. Its derivative is 0 at x = 0.5

2. It is continuous at x = 0

Which of the above statements is/are correct?

What is the degree of the differential equation? $1+\left(\frac{dy}{dx}\right)^2 =\left(\frac{d^2y}{dx^2}\right)^{\frac{4}{3}}?$

A radioactive substance decays at a rate proportional to the amount of substance present. If half of the substance decays in 100 years, then what is the decay constant (proportionality constant) ?

What is the domain of the function $f(x) =\sqrt{1 - (x-1)^2} \ ?$

The area of the region bounded by the parabola y² = 4kx, where k > 0 and its latus rectum is 24 square units. What is the value of k ?

What is $\int\limits^{\frac{\pi}{4}}_0 \frac{dx}{(\sin x + \cos x)^2}$ equal to?

What is $\int (\sin x)^{-1/2} (\cos x)^{-3/2}dx$ equal to?

If $I_1 =\int\frac{e^x dx}{e^x + e^{-x}}$ and $I_2 =\int\frac{dx}{e^{2x} + 1},$then what is I₁ + I₂ equal to?

What is $\int\limits^{-1}_{-2}\frac{x}{|x|}dx$ equal to?

How many extreme values does sin4x + 2x, where $0 < x < \frac{\pi}{2} $ have ?

What is the maximum value of the functions $f(x) = \frac{1}{\tan x+\cot x},$ where $0 < x < \frac{\pi}{2}?$

If $4f(x) - f \left(\frac{1}{x}\right)=\left(2x+\frac{1}{x}\right)\left(2x-\frac{1}{x}\right),$ then what is f(2) equal to?

If f(x) = 4x + 3, then what is f o f o f(-1) equal to?

If x^yy^x = 1, then what is $\frac{dy}{dx}$ at (1, 1) equal to?

If y = (x^x)^x, then what is the value of $\frac{dy}{dx}$ at x = 1?

Let y = [x + 1], -4 < x < -3 where [.] is the greatest integer function. What is the derivative of y with respect to x at x = -3.5?

If $\frac{dy}{dx} = (\ln 5)y $  with y(0) = ln 5, then what is y(1) equal to?

Consider the following in respect of the function f(x) = 10^x :

1. Its domain is (-∞, ∞)

2. It is a continuous function

3. It is differentiable at x = 0

Which of the above statements are correct?

What is $\lim\limits_{x \to 0}x^3(\text{cosec} \ x)^2$ equal to ?

What is $\lim\limits_{x \to 1}\frac{x^3 - 1}{\sqrt x - 1}$ equal to?

In which one of the following intervals is the function $f(x) = \frac{x^3}{3}-\frac{7x^2}{2} + 6x + 5$ decreasing?

If the derivative of the function$f(x) =\frac{m}{x} +2nx + 1$ vanishes at x = 2, then what is the value of m + 8n ?

What is the area included in the first quadrant between the curves y = x and y = x³ ?

If xy = 4225 where x, y are natural numbers, then what is the minimum value of x + y ?

What does the equation $x \frac{dy}{dx}-2y= 0$ represent ?

If the points with coordinates (-5, 0), (5p², 10p) and (5q², 10q) are collinear, then what is the value of pq where p ≠ q ?

What is the equation of the straight line which passes through the point (1, -2) and cuts off equal intercepts from the axes ?

What is the equation of the circle which touches both the axes in the first quadrant and the line y - 2 = 0?

What is the equation of the parabola with focus (-3, 0) and directrix x - 3 = 0 ?

What is the distance between the foci of the ellipse x² + 2y² = 1 ?

Let a, b, c be the lengths of sides BC, CA, AB respectively of a triangle ABC. If p is the perimeter and q is the area of the triangle, then what is $p(p-2a) \tan \left(\frac{A}{2}\right)$ equal to ?

A straight line passes through the point of intersection of x + 2y + 2 = 0 and 2x - 3y - 3 = 0. It cuts equal intercepts in the fourth quadrant. What is the sum of the absolute values of the intercepts?

Under which one of the following conditions are the lines ax + by + c = 0 and bx + ay + c = 0 parallel (a ≠ 0, b ≠ 0)?

What is the equation of the locus of the mid-point of the line segment obtained by cutting the line x + y = p, (where p is a real number) by the coordinate axes ?

If the point (x, y) is equidistant from the points (2a, 0) and (0, 3a) where a > 0, then which one of the following is correct ?

What is the value of k?

If p is the perpendicular distance from the centre of the sphere to the plane, then which one of the following is correct ?

What is the equation of the line through the origin and the centre of the sphere ?

What are the direction ratios of the normal to the plane ?

If p, q and r are the intercepts made by the plane on the coordinate axes respectively, then what is (p + q + r) equal to ?

If $4\hat i + \hat j - 3\hat k$ and $p\hat i + q\hat j - 2\hat k$ are collinear vectors, then what are the possible values of p and q respectively?

If $\vec a, \vec b, \vec c$, are the position vectors of the vertices A, B, C respectively of a triangle ABC and G is the centroid of the triangle, then what is $\overrightarrow{AG}$ equal to ? .

Consider the following statements :

1. Dot product over vector addition is distributive

2. Cross product over vector addition is distributive

3. Cross product of vectors is associative

Which of the above statements is/are correct ?

Let  $\vec{a}, \vec{b}, \vec{c}$ be three non-zero vectors such that  $\vec{a}\times \vec{b} = \vec{c} $. Consider the following statements:

1. $\vec a$ is unique if $\vec b$ and $\vec c$ are given

2. $\vec c$ is unique if $\vec a$ and $\vec b$ are given

Which of the above statements is/are correct?

Let $\vec a$ and $\vec b$ be two unit vectors such that $|\vec a - \vec b|<2.$ If 2θ is the angle between $\vec a$ and $\vec b,$ then which one of the following is correct ?

Two digits out of 1, 2, 3, 4, 5 are chosen at random and multiplied together. What is the probability that the last digit in the product appears as 0?

The frequency curve (assuming unimodal) corresponding to the data obtained in an experiment is skewed to the left. What conclusion can be drawn from the curve ?

The variance of five positive observations is 3.6. If four of the observations are 2, 2, 4, 5 then what is the remaining observation ?

What is the arithmetic mean of 50 terms of an AP with first term 4 and common difference 4 ?

What is the coefficient of mean deviation of 21, 34, 23, 39, 26, 37, 40, 20, 33, 27 (taken from mean)?

What is the mean of the values?

What is the algebraic sum of the deviations of the same set of values measured from 99?

If the algebraic sum of the deviations of the same set of values measured from y is 180, then what is the value of y ?

What is the mean of the marks ?

What is the median of the marks?

What is the sum of the deviations measured from the median?

What is $P (G \cap \overline T)$ equal to?

What is $P(G | \overline T) $ equal to?

What is $P(\overline T | \overline G)$ equal to?

What is the probability that exactly 3 out of 6 workers suffer from a disease?

What is the probability that no one out of 6 workers suffers from a disease?

What is the probability that at least one out of 6 workers suffer from a disease ?

What is the value of p ?

What is the value of q ?

If the frequency of each class is doubled, then what would be the mean?