List of practice Questions

If \[ 3f(x) - 2f\left(\frac{1}{x}\right) = x, \] then \( f'(2) \) is:
If \( a \) is a common root of \( x^2 - 5x + \lambda = 0 \) and \( x^2 - 8x - 2\lambda = 0 \) (\( \lambda \neq 0 \)) and \( \beta, \gamma \) are the other roots of them, then \( a + \beta + \gamma + \lambda = \):
Identify the incorrect set from the following:
If \[ f(x) = 5 \cos^3 x - 3 \sin^3 x \quad \text{and} \quad g(x) = 4 \sin^3 x + \cos^2 x, \] then the derivative of \( f(x) \) with respect to \( g(x) \) is:
If \[ \int \log \left( 6\sin^2x + 17\sin x + 12 \right)^{\cos x} \, dx = f(x) + c \] then, \[ f\left(\frac{\pi}{2}\right) = \]
A vessel of volume \( V \) L contains an ideal gas at \( T(K) \). The vessel is partitioned into two equal parts. The volume (in L) and temperature (in K) in each part is respectively:
At 300 K, 6 g of urea was dissolved in 500 mL of water. What is the osmotic pressure (in atm) of the resultant solution? (R = 0.082 L atm K$^{-1}$ mol$^{-1}$) (C=12;N=14;O=16;H=1)
If \( y = \frac{ax + \beta}{\gamma x + \delta} \), then \( 2y_1 y_3 = \) ?
The angle of polarisation for a medium with respect to air is \( 60^\circ \). The critical angle of this medium with respect to air is:
In a common emitter amplifier, a.c. current gain is 40 and input resistance is 1 kΩ. The load resistance is given as 10 kΩ. Then the voltage gain is:
A uniform rod of length \( 2L \) is placed with one end in contact with the earth and is then inclined at an angle \( \alpha \) to the horizontal and allowed to fall without slipping at the contact point. When it becomes horizontal, its angular velocity will be:
Initially the pressure of 1 mole of an ideal gas is \( 10^5 \) Nm\(^2\) and its volume is 16 liters. When it is adiabatically compressed, its final volume is 2 liters. Work done on the gas is (molar specific heat at constant volume \( C_V = \frac{3R}{2} \)):
If \( E_1 = 4V \) and \( E_2 = 12V \), the current in the circuit and potential difference between the points P and Q respectively are:
the current in the circuit and potential difference
Which one of the following is NOT a disaccharide?

Let \( F \) and \( F' \) be the foci of the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) (where \( b<2 \)), and let \( B \) be one end of the minor axis. If the area of the triangle \( FBF' \) is \( \sqrt{3} \) sq. units, then the eccentricity of the ellipse is: 

Evaluate the integral: \[ \int \frac{3x^9 + 7x^8}{(x^2 + 2x + 5x^9)^2} \,dx= \] 

The two reactions involved in the conversion of benzene diazonium chloride to biphenyl are respectively
An inductor of inductive reactance 80 \(\Omega\) and a resistor of resistance 60 \(\Omega\) are connected in series to an ac source. The impedance and the power factor of the circuit are respectively:
Consider the point \( P(\alpha, \beta) \) on the line \( 2x + y = 1 \). If the points \( P \) and \( (3,2) \) are conjugate points with respect to the circle \( x^2 + y^2 = 4 \), then find \( \alpha + \beta \):
Let \( T_1 \) be the tangent drawn at a point \( P(\sqrt{2}, \sqrt{3}) \) on the ellipse \( \frac{x^2}{4} + \frac{y^2}{6} = 1 \). If \( (a, \beta) \) is the point where \( T_1 \) intersects another tangent \( T_2 \) to the ellipse perpendicularly, then \( a^2 + \beta^2 = \):
Path of projectile is given by the equation \( Y = Px - Qx^2 \), match the following accordingly (acceleration due to gravity = \( g \))
Path of projectile is given by the equation Y = Px−Qx2
Identify the set of molecules in which the central atom has only one lone pair of electrons in their valence shells:
A 10 L vessel contains 1 mole of an ideal gas with pressure of P(atm) and temperature of T(K). The vessel is divided into two equal parts. The pressure (in atm) and temperature (in K) in each part is respectively.

Given are two statements regarding the properties of carbon in Group 14 of the periodic table: 
Statement-I: Carbon has the highest catenation power in group 14 elements. 
Statement-II: Carbon has small size and high electronegativity compared to other elements of group 14.

A body slides down an inclined plane of angle of inclination 30\(^\circ\) with a constant velocity of 10 ms\(^{-1}\). If the body is pushed up the same plane with a velocity of 20 ms\(^{-1}\), the distance moved by the body before coming to rest is (acceleration due to gravity = 10 ms\(^2\))