List of practice Questions

Elements A and B have FCC and BCC structures respectively with unit cell edge length \( a = 3 \, \text{Å} \).
The number of atoms in 210 g of A = number of atoms in 594 g of B.
If density of A is \( 7 \, \text{g/cm}^3 \), what is the density of B?
A sphere of surface area \( 4 \, \text{m}^2 \), temperature 400 K, emissivity 0.5 is in an environment of 200 K. The net rate of energy exchange is (take \( \sigma = 5.67 \times 10^{-8} \, \text{Wm}^{-2} \text{K}^{-4} \))
An electric dipole with dipole moment \( 5 \times 10^{-7} \, \text{Cm} \) is in an electric field of \( 2 \times 10^4 \, \text{NC}^{-1} \) at an angle of \( 60^\circ \) with the direction of the field. The torque acting on the dipole is:
If \( \sin\left(x + \frac{\pi}{3}\right) + \sin\left(x - \frac{\pi}{3}\right) = 1 \), then the value of \( x \) in the interval \( [0, \pi] \) is:
If \( \cosh(x - \log 3) = \sinh x \), then \( x = \):
On the locus of the point \( P(x, y) \) equidistant from \( (3, 0) \) and \( (0, 4) \), if \( A \) and \( B \) are two points that satisfy \( 4x = 3y \) and \( x = y \) respectively, then the distance between A and B is:
The molecule which has more number of lone pair of electrons than the bond pair of electrons in its central atom is
Among the ions O$^{2-}$, Na$^+$, F$^-$, N$^{3-}$, Mg$^{2+}$ the ions with smallest and largest radii are respectively
The ratio of effective number of atoms in a unit cell of fcc and bcc lattices is
For two events $A$ and $B$, a true statement among the following is
If $ \sin \theta = -\frac{3}{4} $, then compute $ \sin 2\theta $
Identify the correct statements:
Which of the following is correct for fcc lattice?

Identify the product(s) formed in the following reaction:

If f is a relation from set of positive real numbers to the set of positive real numbers defined by \(f(x) = 3x^2-2\) then f is
Suppose ABOC is a rhombus in the first quadrant with O being the origin. If the vertices B and C of \(\triangle ABC\) lie respectively on \(y=\frac{x}{\sqrt{3}}\) and \(y=0\), and the side BC passes through \((\frac{2}{3}, \frac{2}{3})\), then the midpoint of BC is
The equation of the line which is a normal to the curve \(3x^2-xy+y^2=3\) at \((1,1)\) is
If \( \frac{d}{dx} \left( \frac{x^2+1}{(x^2+5)(x^2+9)} \right) = \frac{2x(x^2+1)}{(x^2+5)(x^2+9)} \left[ \frac{1}{f(x)} - \frac{1}{g(x)} - \frac{1}{h(x)} \right] \), then \( 2h(x)-f(x)-g(x) = \)
\( \int \frac{dx}{(1+\sqrt{x})^{2022}} = \)
A wheel undergoes a constant acceleration starting form rest at t = 0. The angular velocity of the wheel is 3.14 \(\frac{\text{rad}}{\text{sec}}\) when t = 2 s. The accelerator is abruptly ceased at t = 20 s. The number of revolutions, wheel makes in the interval t = 0 to t = 40 s is
In a triangle \( ABC \), the identity holds: \[ s \left( \frac{r_1 - r}{a} + \frac{r_2 - r}{b} + \frac{r_3 - r}{c} \right) = ? \]
When a ball is dropped from a height $h$ it takes $t$ sec to reach the ground. If the same experiment is done on a different planet having the mass 100 times the earth’s mass and radius 10 times the earth’s radius, then the time it will take to cover the same height $h$ is
Two planets, A and B, orbit around a star such that the time period of A is 8 times the time period of B. The ratio of orbital velocities of the planets A and B is
If $f(x) = \sqrt{2 - x^2}$ and $g(x) = \log(1 - x)$ are two real-valued functions, then the domain of the function $(f+g)(x)$ is:
Simplify: $\dfrac{\cos\theta}{1 - \tan\theta} + \dfrac{\sin\theta}{1 - \cot\theta}$