If $H$ is the harmonic mean between $P$ and $Q$, then $\dfrac{H}{P} + \dfrac{H}{Q}$ is

The number of values of $k$ for which the system of equations $(k+1)x + 8y = 4k$ and $kx + (k+3)y = 3k-1$ has infinitely many solutions, is

The sum $^{20}C_8 + ^{20}C_9 + ^{21}C_{10} + ^{22}C_{11} - ^{23}C_{11}$

The value of $\cot^{-1}(21) + \cot^{-1}(13) + \cot^{-1}(-8)$ is:

Normal to the curve $y = x^3 - 3x + 2$ at the point $(2,4)$ is:

The value of $\displaystyle \lim_{n\to\infty} \frac{\pi}{n}\left[\sin\frac{\pi}{n}+\sin\frac{2\pi}{n}+\cdots+\sin\frac{(n-1)\pi}{n}\right]$ is:

The point on the curve $y = 6x - x^2$ where the tangent is parallel to the x-axis is:

If $ I_1 = \displaystyle \int_{0}^{1} 2^{x^2},dx,\quad I_2 = \displaystyle \int_{0}^{1} 2^{x^3},dx,\quad I_3 = \displaystyle \int_{1}^{2} 2^{x^2},dx,\quad I_4 = \displaystyle \int_{1}^{2} 2^{x^3},dx,$ then

The value of integral $\displaystyle \int_{0}^{\pi/2} \log \tan x dx$ is

A determinant is chosen at random from the set of all determinants of matrices of order 2 with elements 0 and 1 only. The probability that the determinant chosen is non-zero is:

If $\sin^2 x = 1 - \sin x$, then $\cos^4 x + \cos^2 x$ is equal to:

The equation of the plane passing through the point $(1,2,3)$ and having the normal vector $N = 3\mathbf{i} - \mathbf{j} + 2\mathbf{k}$ is:

The value of $\displaystyle \int_{0}^{\sin^2 x} \sin^{-1}\sqrt{t} dt + \int_{0}^{\cos^2 x} \cos^{-1}\sqrt{t} dt$ is:

Coefficients a, b, c of $ax^2 + bx + c = 0$ are chosen by tossing 3 fair coins. Head means 1, Tail means 2. Find the probability that the roots are imaginary

In a class of 100 students, 55 passed in Mathematics and 67 passed in Physics.The number of students who passed in Physics only is:

If $(4,-3)$ and $(-9,7)$ are two vertices of a triangle and $(1,4)$ is its centroid, find the area of the triangle.

The equation of ellipse with major axis along the x–axis and passes through the point $(4,3)$ and $(-1,4)$.

If the circles $ x^2 + y^2 + 2x + 2ky + 6 = 0$ and $x^2 + y^2 + 2ky + k = 0$ intersect orthogonally, then $k$ is:

Focus of the parabola $x^2 + y^2 - 2xy - 4(x + y - 1) = 0$ is:

If $\mathbf{a},\; \mathbf{b},\; \mathbf{c}$ are unit vectors such that $\mathbf{a} + \mathbf{b} + \mathbf{c} = 0$, then the value of $\mathbf{a}\cdot \mathbf{b} + \mathbf{b}\cdot \mathbf{c} + \mathbf{c}\cdot \mathbf{a}$ is:

The number of words that can be formed by using the letters of the word 'MATHEMATICS' that start as well as end with T is

Let $P(E)$ denote the probability of event $E$. Given $P(A) = 1$, $P(B) = \frac{1}{2}$, the values of $P(A \mid B)$ and $P(B \mid A)$ respectively are

The number of words that can be formed by using the letters of the word MATHEMATICS that start as well as end with T is

If $A-B=\frac{\pi}{4}$, then (1 + tan A)(1 – tan B) is equal to

Let P(E) denote the probability of event E. Given P(A) = 1, P(B) =$\frac{1}{2}$ the value of P(A|B) and P(B|A) respectively are

The number of different license plates that can be formed in the format 3 English letters (A….Z) followed by 4 digits (0, 1, …9) with repetitions allowed in letters and digits is equal to