List of practice Questions

In a random experiment, two dice are thrown and the sum of the numbers appeared on them is recorded. This experiment is repeated 9 times. If the probability that a sum of 6 appears at least once is \( P_1 \) and a sum of 8 appears at least once is \( P_2 \), then \( P_1 : P_2 = \):

Three glass slabs A, B, C of different media are arranged as shown in the figure. Refractive index of B is 1.3 and the refractive index of C is 1.7. If the number of waves passing through B and C are equal to the total number of waves passing through B and C, then the refractive index of slab A is:

At $ T $ (K), the following data was obtained for the reaction: $ S_2O_8^{2-} + 3 I^- \rightarrow 2 SO_4^{2-} + I_3^- $.

From the data, the rate constant of the reaction (in $ M^{-1} s^{-1} $) is: 

If the roots of the equation \[ Z^3 + iZ^2 + 2i = 0 \] are the vertices of a triangle ABC, then that triangle ABC is:
If \( (r, \theta) \) denotes \( r (\cos \theta + i \sin \theta) \). If \[ x = (1, \alpha), \quad y = (1, \beta), \quad z = (1, \gamma) \] and \( x + y + z = 0 \), then \[ \sum \cos (2\alpha - \beta - \gamma) = \]
The first ionization enthalpy (IE\(_1\)) and second ionization enthalpy (IE\(_2\)) of Mg(g) are 178 and 348 kcal mol\(^{-1}\) respectively. The energy required for the reaction Mg(g) \(\rightarrow\) Mg\(^{2+}\)(g) + 2e\(^{-}\) (in kcal mol\(^{-1}\)) is:

Three blocks of masses 2 m, 4 m and 6 m are placed as shown in figure. If \( \sin 37^\circ = \frac{3}{5} \), \( \sin 53^\circ = \frac{4}{5} \), the acceleration of the system is:

When 40 J of heat is absorbed by a monatomic gas, the increase in the internal energy of the gas is

A wire of 60 cm length and mass 10 g is suspended by a pair of flexible leads in a magnetic field of 0.60 T as shown in the figure. The magnitude of the current required to remove the tension in the supporting leads is: 

Semiconductors suitable for solar cell fabrication have a band gap nearby:
A light of frequency \( x \) Hz when falls on a metal plate emits electrons that have double the kinetic energy compared to when light of frequency \( y \) Hz falls on the same plate. The threshold frequency of the metal in Hz is:
At 27 °C, \(x \, g\) of \( \text{CaCl}_2 \) was dissolved in 2.5 L of water. The osmotic pressure of the resultant solution is 0.82 atm. What is \(x\) in grams? (Given \(i = 2.5\), \(R = 0.082 \, \text{L atm mol}^{-1} \text{K}^{-1}\))
Let \( A(2, 3), B(1, -1) \) be two points. If \( P \) is a variable point such that \( \angle APB = 90^\circ \), then the locus of \( P \) is
Equation of a tangent line of the parabola \(y^2 = 8x\), which passes through the point \((1, 3)\) is:
The length of the subnormal at any point on the curve \( y = \left(\frac{x}{2024}\right)^k \) is constant if the value of \( k \) is:
An object is placed at a distance of 18 cm in front of a mirror. If the image is formed at a distance of 4 cm on the other side, then the focal length, nature of the mirror and nature of the image are respectively:
Inner shell electrons in atoms moving from one energy level to another lower energy level produce:
Which of the following amino acids possess two chiral centres?
The value of \( c \) such that the straight line joining the points \[ (0,3) \quad {and} \quad (5,-2) \] is tangent to the curve \[ y = \frac{c}{x+1} \] is:
The solution of the differential equation \[ \frac{dy}{dx} = \frac{y + x \tan \left( \frac{y}{x} \right)}{x}. \] \[ \sin\frac{y}{x} = \]
At a place the horizontal component of earth’s magnetic field is \( 3 \times 10^{-5} \) T and the magnetic declination is \( 30^\circ \). A compass needle of magnetic moment \( 18 \) A\(m^2\) pointing towards geographic north at this place experiences a torque of:
If all the letters of the word ‘SENSELESSNESS’ are arranged in all possible ways and an arrangement among them is chosen at random, then, the probability that all the E’s come together in that arrangement is:
The equation of the circle touching the circle \[ x^2 + y^2 - 6x + 6y + 17 = 0 \] externally and to which the lines \[ x^2 - 3xy - 3x + 9y = 0 \] are normal is:
The differential equation formed by eliminating arbitrary constants \( A \) and \( B \) from the equation \[ y = A \cos 3x + B \sin 3x \] is:
A particle is projected from the surface of the Earth with a velocity equal to twice the escape velocity. When the particle is far from the Earth, its speed will be: