List of top Mathematics Questions

Let $\vec{a} = \hat{i} + \hat{j} + \sqrt{2} \hat{k} , \vec{b} = b_1 \hat{i} + b_2 \hat{j} + \sqrt{2} \hat{k}$ and $\vec{c} = 5 \hat{i} + \hat{j} + \sqrt{2} \hat{k}$ be three vectors such that the projection vector of $\vec{b}$ on $\vec{a}$ , If $\vec{a} + \vec{b}$ is perpendicular to $\vec{c}$, then $| \vec{b}|$ is equal to :

Assume that each born child is equally likely to be a boy or a girl. If two families have two children each, then the conditional probability that all children are girls given that at least two are girls is :

The third term of a G.P. is 9. The product of its first five terms is

A circle touching the x-axis at $(3, 0)$ and making an intercept of length $8$ on the y-axis passes through the point :

If a circle of radius $R$ passes through the origin $O$ and intersects the coordinate axes at $A$ and $B$, then the locus of the foot of perpendicular from $O$ on $AB$ is :

If the angle of intersection at a point where the two circles with radii 5 cm and 12 cm intersect is $90^{\circ}$, then the length (in cm) of their common chord is :

If a circle $C$ passing through the point $(4,0)$ touches the circle $x^2 + y^2 + 4x - 6y = 12$ externally at the point $(1, -1)$, then the radius of $C$ is :

If one end of a focal chord of the parabola, $y^2 = 16x$ is at $(1, 4),$ then the length of this focal chord is

If cos x = |sin x| then, the general solution is

A man speaks truth 2 out of 3 times. He picks one of the natural numbers in the set S = {1, 2, 3, 4, 5, 6, 7} and reports that it is even. The probability that it is actually even is

If $A = \{ x | x \in N, x \le 5 \}, B = \{ x | x \in Z, x^2 - 5x + 6 = 0\}$, then the number of onto functions from A to B is

$\sqrt{3}\, cosec \,20^{\circ} - \sec\, 20^{\circ}$ =

If $| 3x -5| \le 2$, then

The interval in which the function $f(x) = x^3 - 6x^2 + 9x + 10$ is increasing in

If $a + \frac{\pi}{2} < 2 \tan^{-1} x + 3 \cot^{-1} x < b $ then 'a' and 'b' are respectively.

The domain of the function f : R $\to$ R defined by $f(x) = \sqrt{x^2 - 7x + 12} $ is

If $P$ and $Q$ are symmetric matrices of the same order then $PQ - QP$ is

On the set of positive rationals, a binary operation * is defined by a * b = $\frac{2ab}{5}$ . If 2 * x = $3^{-1}$ then x =

The eccentricity of the ellipse $9x^2 + 25y^2 = 225$ is

Two circles with equal radii are intersecting at the points $(0, 1)$ and $(0, -1)$. The tangent at the point $(0, 1)$ to one of the circles passes through the centre of the other circle. Then the distance between the centres of these circles is :

Let $D=R-\{0,1\}$ and $f: D \rightarrow D, g: D \rightarrow D$ and $h: D \rightarrow D$ be three functions defined by $f(x)=\frac{1}{x} ; g(x)=1-x$ and $h(x)=\frac{1}{1-x} .$ If $j: D \rightarrow D$ is such that $(gojof)$ $(x)=f(x)$ for all $x \in D$, then which one of the following is $j(x) ?$

The distance of the point (1, 2, 1) from the line $\frac{x-1}{2} = \frac{ y-2}{1}=\frac{ z-3}{2} $ is

If $y = f(x)$ is twice differentiable function such that at a point $P , \frac{dy}{dx} = 4 , \frac{d^2 y}{dx^2} = - 3$ , then $\left( \frac{d^2 x}{dy^2} \right)_P = $

Let a, b, c be real numbers such that $a + b + c < 0$ and the quadratic equation $ax^2 + bx + c = 0$ has imaginary roots. Then