List of practice Questions

Beryllium shows diagonal relationship with aluminium. Which of the following similarity is incorrect ?
Chloroform, $CHCl _{3}$, boils at $61.7^{\circ} C$. If the $K_{b}$ for chloroform is $3.63^{\circ} C / molal$, what is the boiling point of a solution of $15.0\, kg$ of $CHCl _{3}$ and $0.616\, kg$ of acenaphthalene, $C _{12} H _{10}$ ?
If $V$ is the volume of one molecule of gas under given conditions, the van der Waal?s constant $b$ is
The pH of $0.1\, M$ solution of the following salts increases in the order :
Which of the following is strongest nucleophile
The $25 \,mL$ of a $0.15 \, M$ solution of lead nitrate, $Pb \left( NO _{3}\right)_{2}$ reacts with all of the aluminium sulphate, $Al _{2}\left( SO _{4}\right)_{3}$, present in $20 \,mL$ of a solution. What is the molar concentration of the $Al _{2}\left( SO _{4}\right)_{3}$ ? $3 Pb \left( NO _{3}\right)_{2}( aq )+ Al _{2}\left( SO _{4}\right)_{3}( aq ) \longrightarrow 3 PbSO _{4}( s )+2 Al \left( NO _{3}\right)_{3}( aq )$
$100\, mL \,O _{2}$ and $H _{2}$ kept at same temperature and pressure. What is true about their number of molecules
Students of an MBA school have got together to constitute a fund where each student has contributed Rs. 10000. This group of 20 students laid down certain rules for investment where it was agreed that 50 percent of the fund would be invested in riskless government securities while the rest 50 percent would be invested in equities. At the end of the year the group realizes that the crash in the stock market has left them with zero return on equities but 1 percent dividend gain on the shares bought. The return on the riskless government security has been 7.8 percent in the last year. If the gains are to be divided equally, the gain per student is
$10$ forks are arranged in increasing order of frequency in such a way that any two nearest tuning forks produce $4$ beats/sec. The highest frequency is twice of the lowest. Possible highest and the lowest frequencies (in $Hz$ ) are
Two long parallel wires carry equal current $i$ flowing in the same direction are at a distance $2d$ apart. The magnetic field $B$ at a point lying on the perpendicular line joining the wires and at a distance $x$ from the midpoint is -
The variation of magnetic susceptibility $(x)$ with temperature for a diamagnetic substance is best represented by
A body of mass $M$ hits normally a rigid wall with velocity $V$ and bounces back with the same velocity. The impulse experienced by the body is
A particle of mass $m$ executes simple harmonic motion with amplitude a and frequency $n$. The average kinetic energy during its motion from the position of equilibrium to the end is.
The dipole moment of the given charge distribution is:
The acceleration due to gravity on the surface of the moon is $1/6$ that on the surface of earth and the diameter of the moon is one-fourth that of earth. The ratio of escape velocities on earth and moon will be
Give $\vec{P} = 2 \hat{i} - 3 \hat{j} + 4 \hat {k}$ and $\vec{Q} = \hat{j} - 2 \hat {k} $ The magnitude of their resultant is.

Let \(\{a_n\}_{n\ge1}\) and \(\{b_n\}_{n\ge1}\) be two convergent sequences of real numbers. For \( n \geq 1 \), define \( u_n = \max\{a_n, b_n\} \) and \( v_n = \min\{a_n, b_n\} \). Then
 

Let \( M = \begin{bmatrix} \tfrac{1}{4} & \tfrac{3}{4} \\ \\ \tfrac{3}{5} & \tfrac{2}{5} \end{bmatrix}. \) If \( I \) is the \( 2 \times 2 \) identity matrix and \( 0 \) is the \( 2 \times 2 \) zero matrix, then

Let \( X \) be a random variable with the probability density function \[ f(x) = \begin{cases} \dfrac{\alpha^{p}}{\Gamma(p)} e^{-\alpha x} x^{p-1}, & x \geq 0, \, \alpha > 0, \, p > 0, \\ 0, & \text{otherwise}. \end{cases} \] If \( E(X) = 20 \) and \( \text{Var}(X) = 10 \), then \( (\alpha, p) \) is 
 

Let \( X \) be a random variable with the distribution function \[ F(x) = \begin{cases} 0, & x < 0, \\ \frac{1}{4} + \frac{4x - x^2}{8}, & 0 \leq x < 2, \\ 1, & x \geq 2. \end{cases} \] Then \[ P(X = 0) + P(X = 1.5) + P(X = 2) + P(X \geq 1) \] equals 
 

Let \( X_1, X_2 \) and \( X_3 \) be i.i.d. \( U(0,1) \) random variables. Then \( E\left( \dfrac{X_1 + X_2}{X_1 + X_2 + X_3} \right) \) equals 
 

Consider four coins labelled as 1, 2, 3 and 4. Suppose that the probability of obtaining a 'head' in a single toss of the \(i\)th coin is \( \frac{1}{4} \), \( i = 1, 2, 3, 4 \). A coin is chosen uniformly at random and flipped. Given that the flip resulted in a 'head', the conditional probability that the coin was labelled either 1 or 2 equals
Consider the linear regression model \[ y_i = \beta_0 + \beta_1 x_i + \epsilon_i, i = 1, 2, \dots, n, \text{where} \epsilon_i \text{ are i.i.d. standard normal random variables. Given that} \] \[ \frac{1}{n} \sum_{i=1}^n x_i = 3.2, \frac{1}{n} \sum_{i=1}^n y_i = 4.2, \frac{1}{n} \sum_{j=1}^n \left( x_j - \frac{1}{n} \sum_{i=1}^n x_i \right)^2 = 1.5, \] \[ \frac{1}{n} \sum_{j=1}^n \left( x_j - \frac{1}{n} \sum_{i=1}^n x_i \right) \left( y_j - \frac{1}{n} \sum_{i=1}^n y_i \right) = 1.7, \] the maximum likelihood estimates of \( \beta_0 \) and \( \beta_1 \), respectively, are

Let \( f : [-1, 1] \to \mathbb{R} \) be defined by \( f(x) = \dfrac{x^2 + [\sin\pi x]}{1 + |x|}, \text{ where } [y] \text{ denotes the greatest integer less than or equal to } y. \) Then 
 

Let \( f, g : \mathbb{R} \to \mathbb{R} \) be defined by \[ f(x) = x^2 - \frac{\cos(x)}{2}, g(x) = \frac{x \sin(x)}{2}. \] Then