List of practice Questions

Let $X$ be a random variable with its expectation $ E(X)=3 $ and its variance $ V(X)=2 $ . If $V$ is another random variable defined by $ Y=10X, $ then the ordered pair $ (E(Y),\,V(Y)) $ is equal to

A unit cell with edge length $ a\ne b\ne c $ and axial angles $ \alpha =\beta =\gamma =90^{\circ} $ is called

The following tripeptide can be synthesized from the following ammo acid

If the density of methanol is $0.8\, kg\, L ^{-1}$, what is its volume needed for making $4\, L$ of its $0.25\, M$ solution?

If the forward bias voltage in a $p-n$ junction diode is decreased, the length of depletion region will

An observer standing near the sea-coast counts $48$ waves per min. If the wavelength of the wave is $ 10\,m, $ the velocity of the waves will be

The ratio of mass defect of the nucleus to its mass number is maximum for

An ideal gas is heated at constant volume until its pressure doubles. Which one of the following statements is correct?

A body is travelling towards East with a speed of $9\, m / s$ and with an acceleration of $2\, m / s ^{2}$ acting along West on it. The displacement of the body during the $5^{\text {th }}$ second of its motion is

A $30\,V-90\,W$ lamp is operated on a $120\,V$ DC line. A resistor is connected in series with the lamp in order to glow it properly. The value of resistance is

The integrating factor of the differential equation $\frac{d y}{d x}+\left(3 x^{2} \tan ^{-1} y-x^{3}\right)\left(1+y^{2}\right)=0$is

A large number of particles are placed around the origin, each at a distance $R$ from the origin. The distance of the center of mass of the system from the origin is

The rms speed of oxygen is $v$ at a particular temperature. If the temperature is doubled and oxygen molecules dissociate into oxygen atoms, the rms speed becomes

Given that $x$ is a real number satisfying $\frac{5x^{2}-26x+5}{3x^{2}-10x+3} < 0 ,$ then

Let $x_{n}=\left(1-\frac{1}{3}\right)^{2}\left(1-\frac{1}{6}\right)^{2}\left(1-\frac{1}{10}\right)^{2} ........ \left(1-\frac{1}{\frac{n\left(n+1\right)}{2}}\right)^2, n \ge 2.$ Then the value of $\displaystyle \lim_{n \to \infty} x_n$ is

The least value of $2x^2 + y^2 + 2xy + 2x - 3y + 8$ for real numbers $x$ and y is

The variance of first $20$ natural numbers is

A radiation of energy 'E ' falls normally on a perfectly reflecting surface. The momentum transferred to the surface is (C = Velocity of light)

If $\omega$ is an imaginary cube root of unity, then the value of the determinant $\begin{vmatrix}1+\omega&\omega^{2}&-\omega\\ 1+\omega^{2}&\omega&-\omega^{2}\\ \omega+\omega^2&\omega&-\omega^{2}\end{vmatrix}$

The value of $2 \cot ^{-1} \frac{1}{2}-\cot ^{-1} \frac{4}{3}$ is

A gas can be liquefied at temperature $T$ and pressure $P$ provided

Cold ferrous sulphate solution on absorption of $NO$ develops brown colour due to the formation of

In a certain town, $60\%$ of the families own a car, $30\%$own a house and $20\%$ own both a car and a house. If a family is randomly chosen, what is the probability that this family owns a car or a house but not both ?

The length of a metal wire is $L_1$ when the tension is $T_1$ and $L_2$ when the tension is $T_2$. The unstretched length of the wire is