List of top Mathematics Questions

If $\hat{a}$ is a unit vector such that $(\vec{x} - \hat{a}) \cdot (\vec{x} + \hat{a}) = 8$, then $|\vec{x}| =$

The area of the triangle ABC is $10\sqrt{3}\ \text{cm}^2$, angle B is $60^\circ$ and its perimeter is $20\ \text{cm}$, then $\ell(\text{AC}) =$

If the slopes of the lines given by the equation $ax^2 + 2hxy + by^2 = 0$ are in the ratio $5 : 3$, then the ratio $h^2 : ab =$

For all real $x$, the minimum value of the function $f(x) = \frac{1 - x + x^2}{1 + x + x^2}$ is

If the function defined by $f(x) = K(x - x^2)$ if $0 < x < 1$ and $f(x) = 0$ otherwise, is the probability density function (p.d.f.) of a random variable X, then the value of $P\left(X < \frac{1}{2}\right)$ is

Evaluate the indefinite integral $$\int \frac{dx}{\cos x \sqrt{\cos 2x}}$$

The equation of the tangent to the curve $y = \sqrt{2} \sin\left(2x + \frac{\pi}{4}\right)$ at $x = \frac{\pi}{4}$ is

The general solution of the differential equation $\cos(x + y) \frac{dy}{dx} = 1$ is

The domain of the function $\log_{10}(x^2 - 5x + 6)$ is

With usual notations in $\triangle \text{ABC}$, if $\frac{\sin \text{A}}{\sin \text{C}} = \frac{\sin(\text{A} - \text{B})}{\sin(\text{B} - \text{C})}$, then $a^2, b^2, c^2$ are in

If $\vec{a} = 2\hat{\text{i}} - \hat{\text{j}} + \hat{\text{k}}$, $\vec{b} = \hat{\text{i}} + 2\hat{\text{j}} - 3\hat{\text{k}}$ and $\vec{c} = 3\hat{\text{i}} + \lambda\hat{\text{j}} + 5\hat{\text{k}}$ are coplanar, then $\lambda$ is the root of the equation

The derivative of the function $\cot^{-1}\left[(\cos 2x)^{1/3}\right]$ at $x = \frac{\pi}{6}$ is

If $\int \frac{\sqrt{x}}{x(x+1)} \, dx = k \tan^{-1} m + c$ (where $c$ is a constant of integration), then

The order and degree of the differential equation $$\frac{d^2y}{dx^2} = \sqrt{\frac{dy}{dx}}$$ are respectively

Evaluate the definite integral $$\int_{1}^{3} \left[ \tan^{-1}\left(\frac{x}{x^2-1}\right) + \tan^{-1}\left(\frac{x^2-1}{x}\right) \right] dx$$

Let two cards be drawn at random from a pack of 52 playing cards. Let X be the number of aces obtained. Then the value of E(X) is

If the amplitude of $(z - 2 - 3i)$ is $\frac{3\pi}{4}$, then the locus of $z$ is (where $z = x + iy$)

A fair coin is tossed 100 times. The probability of getting a head for an even number of times is

If $\sin(y + z - x)$, $\sin(z + x - y)$ and $\sin(x + y - z)$ are in AP, then

Let $\vec{a} = 2\hat{\text{i}} + \hat{\text{j}} - 2\hat{\text{k}}$ and $\vec{b} = \hat{\text{i}} + \hat{\text{j}}$. If $\vec{c}$ is a vector such that $\vec{a} \cdot \vec{c} = |\vec{c}|$, $|\vec{c} - \vec{a}| = 2\sqrt{2}$ and the angle between $\vec{a} \times \vec{b}$ and $\vec{c}$ is $60^\circ$, then $|(\vec{a} \times \vec{b}) \times \vec{c}| =$

The projection of $\vec{a} = \hat{\text{i}} - 2\hat{\text{j}} + \hat{\text{k}}$ on $\vec{b} = 2\hat{\text{i}} - \hat{\text{j}} + \hat{\text{k}}$ is

If $\tan^{-1}(2x) + \tan^{-1}(3x) = \frac{\pi}{4}$, where $x > 0$, then $x =$

If the function given by
$f(x) = -2 \sin x$ for $-\pi \le x < -\frac{\pi}{2}$
$f(x) = a \sin x + b$ for $-\frac{\pi}{2} < x < \frac{\pi}{2}$
$f(x) = \cos x$ for $\frac{\pi}{2} \le x \le \pi$
is continuous in $[-\pi, \pi]$, then the value of $(3a + 2b)^3$ is

The negation of the inverse of $\sim p \rightarrow q$ is

If $x = a(\theta + \sin \theta)$ and $y = a(1 - \cos \theta)$, then $\left(\frac{d^2y}{dx^2}\right)_{\text{at } \theta = \pi / 2} =$