List of practice Questions

\[\text{Let } A = \begin{bmatrix} 2 & 1 & 2 \\ 6 & 2 & 11 \\ 3 & 3 & 2 \end{bmatrix} \text{ and } P = \begin{bmatrix} 1 & 2 & 0 \\ 5 & 0 & 2 \\ 7 & 1 & 5 \end{bmatrix}.\]The sum of the prime factors of \( |P^{-1}AP - 2I| \) is equal to.

In photoelectric experiment energy of 2.48 eV irradiates a photo sensitive material. The stopping potential was measured to be 0.5 V. Work function of the photo sensitive material is :

A small ball of mass $m$ and density $\rho$ is dropped in a viscous liquid of density $\rho_0$. After some time, the ball falls with constant velocity. The viscous force on the ball is:

An element \(\triangle l=\triangle x \hat i\) is placed at the origin and carries a large current I = 10A. The magnetic field on the y-axis at a distance of 0.5 m from the elements x of 1 cm length is :

Four particles A, B, C, D of mass $\frac{m}{2}$, $m$, $2m$, $4m$ have the same momentum, respectively. The particle with maximum kinetic energy is:

A light string passing over a smooth light pulley connects two blocks of masses \(m_1\)​ and \(m_2\)​ (where \(m_2​>m_1\)​). If the acceleration of the system is $\frac{g}{\sqrt{2}}$, then the ratio of the masses $\frac{m_1}{m_2}$ is:

A sample contains a mixture of helium and oxygen gas. The ratio of root mean square speed of helium and oxygen in the sample is:

The correct truth table for the following logic circuit is :

Given below are two statements:
Statement I: In an LCR series circuit, current is maximum at resonance. 
Statement II: Current in a purely resistive circuit can never be less than that in a series LCR circuit when connected to the same voltage source. 
In the light of the above statements, choose the correct from the options given below.

The value of unknown resistance (x) for which the potential difference between B and D will be zero in the arrangement shown, is :

\(\sigma\) is the uniform surface charge density of a thin spherical shell of radius \(R\). The electric field at any point on the surface of the spherical shell is:

While measuring the diameter of a wire using a screw gauge, the following readings were noted: Main scale reading is \( 1 \, \text{mm} \) and circular scale reading is equal to 42 divisions. Pitch of the screw gauge is \( 1 \, \text{mm} \), and it has 100 divisions on the circular scale. The diameter of the wire is \( \frac{x}{50} \, \text{mm} \). The value of \( x \) is:

Which of the following phenomena does not explain by wave nature of light.
(A) reflection
(B) diffraction
(C) photoelectric effect
(D) interference
(E) polarization
Choose the most appropriate answer from the options given below :

To project a body of mass \( m \) from Earth's surface to infinity, the required kinetic energy is (assume, the radius of Earth is \( R_E \), \( g \) = acceleration due to gravity on the surface of Earth):

The ratio of the shortest wavelength of Balmer series to the shortest wavelength of Lyman series for hydrogen atom is :

A bullet of mass 50 g is fired with a speed 100 m/s on a plywood and emerges with 40 m/s. The percentage loss of kinetic energy is :

To find the spring constant (k) of a spring experimentally, a student commits 2% positive error in the measurement of time and 1% negative error in measurement of mass. The percentage error in determining value of k is :

Let P be the point (10, –2, –1) and Q be the foot of the perpendicular drawn from the point R(1, 7, 6) on the line passing through the points (2, –5, 11) and (–6, 7, –5). Then the length of the line segment PQ is equal to ________.

For \( n \in \mathbb{N} \), if \[ \cot^{-1} 3 + \cot^{-1} 4 + \cot^{-1} 5 + \cot^{-1} n = \frac{\pi}{4}, \] then \( n \) is equal to ______.

Let the first term of a series be \( T_1 = 6 \) and its \( r^\text{th} \) term \( T_r = 3T_{r-1} + 6^r \), \( r = 2, 3, \dots, n \). If the sum of the first \( n \) terms of this series is \[ \frac{1}{5} \left(n^2 - 12n + 39\right) \left(4 \cdot 6^n - 5 \cdot 3^n + 1\right), \] then \( n \) is equal to ______.

If the second, third, and fourth terms in the expansion of \( (x + y)^n \) are \( 135 \), \( 30 \), and \( \frac{10}{3} \), respectively, then \( 6\left(n^3 + x^2 + y\right) \) is equal to ______.

Let \( L_1, L_2 \) be the lines passing through the point \( P(0, 1) \) and touching the parabola \[ 9x^2 + 12x + 18y - 14 = 0. \] Let \( Q \) and \( R \) be the points on the lines \( L_1 \) and \( L_2 \) such that the \( \Delta PQR \) is an isosceles triangle with base \( QR \). If the slopes of the lines \( QR \) are \( m_1 \) and \( m_2 \), then \( 16\left(m_1^2 + m_2^2\right) \) is equal to ______.

Let \( x_1, x_2, x_3, x_4 \) be the solution of the equation \[ 4x^4 + 8x^3 - 17x^2 - 12x + 9 = 0 \] and \[ \left(4 + x_1^2\right)\left(4 + x_2^2\right)\left(4 + x_3^2\right)\left(4 + x_4^2\right) = \frac{125}{16} m.\] Then the value of \( m \) is ______.

Let \( r_k = \frac{\int_{0}^{1} (1 - x^7)^k \, dx}{\int_{0}^{1} (1 - x^7)^{k+1} \, dx}, \, k \in \mathbb{N} \). Then the value of \[ \sum_{k=1}^{10} \frac{1}{7(r_k - 1)}\]is equal to ______.