List of practice Questions

The length of the straight line $x - 3y = 1$ intercepted by the hyperbola $x^2 - 4y^2 = 1$ is

If the rank of the matrix $\begin{bmatrix}-1 &2&5\\ 2&-4&a-4\\ 1&-2&a+1\end{bmatrix}$ is 1, then the value of $a$ is

If $ P\left(A\right)=\frac{1}{12}, P\left(B\right)=\frac{5}{12},$ and $P\left(B|A\right) =\frac{1}{15}$ then $p\left(A\cup B\right)$ is equal to

If $\sin^{-1} x + \sin^{-1} y = \frac{\pi}{2}$ , then $\cos^{-1} x + \cos^{-1} y$ is equal to

The function $f(x) = x^2 \; e^{-2} x, x > 0$. Then the maximum value of $f(x)$ is

If b$^2 \ge 4 ac$ for the equation $ax^4 + bx^2 + c = 0$, then all the roots of the equation will be real if

Let $A = \{1,2,3,....., n\}$ and $B = \{a,b,c\}$, then the number of functions from $A$ to $B$ that are onto is

The differential equation of the system of all circles of radius $r$ in the $xy$ plane is

The product of all values of $(\cos \alpha + i \sin \alpha)^{3/5}$ is equal to

The sum of the series $ \frac{1}{2}+\frac{3}{4}+\frac{7}{8}+\frac{15}{16}+... $ upto $ n $ term is

What is the stoichiometric coefficient of $Ca$ in the reaction? $Ca + Al^{3+} \to Ca^{2+} + Al$

The solution of the differential equation $\frac{d y}{d x}+\frac{2 y x}{1+x^{2}}=\frac{1}{1+x^{22}}$ is

If one root of the quadratic equation $a x^{2}+b x+c=0$ is equal to $n^{\text {th }}$ power of the other root, then the value of: $a^{\frac{n}{n-1}} C^{\frac{1}{n-1}}+c^{\frac{n}{n-1}} a^{\frac{1}{n-1}}$ is equal to

If $(\cos \theta+i \sin \theta)(\cos 2 \theta+i \sin 2 \theta) \ldots . .(\cos n \theta+i \sin n \theta)=1$, then the value of $\theta$ is, $m \in N$

In how many ways can $5$ boys and $5$ girls sit in a circle so that no two boys sit together?

The length of the common chord of the ellipse $\frac{(x+1)^{2}}{9}+\frac{(y-2)^{2}}{4}=1$ and the circle $x-1^{2}+y-2^{2}=1$ is

Area of the region satisfying $x \leq 2, y \leq|x|, x-$ axis and $x \geq 0$ is:

If average velocity becomes $4$ times then t-hat will be the effect on rms velocity at that Temperature?

Two rings of radius $R$ and $n R$ made of same material have the ratio of moment of inertia about an axis passing through centre in $1: 8$. The value of $n$ is

There are two planets. The ratio of radius of the two planets is $k$ but ratio of acceleration due to gravity of both planets is g. What will be the ratio of their escape velocity?

The extension in a string obeying Hooke's law $v$ is $x$. The speed of sound in the stretched string is $v$. If the extension in the string is increased to $1.5\, x$, the speed of sound will be

The equation of progressive wave is $y=0.2 \sin 2 \pi\left[\frac{t}{0.01}-\frac{x}{0.3}\right]$, where $x$ and $y$ are in metre and t is second. The velocity of propagation of the wave is

A ball whose density is $0.4 \times 10^3 \, kg/m^3$ falls into water from a height of $9\,cm$. To what depth does the ball sink?

The relative humidity on a day when partial pressure of water vapour is $0.012 \times 10^{5} Pa$ at $12^{\circ} C$ is (Take vapour pressure of water at this temperature as $0.016 \times 10^{5}Pa)$

In the absence of intermolecular forces of attraction, the observed pressure $P$ will be