List of practice Questions

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.
The following configuration of gates is equivalent to:
configuration of gates
Energy levels A, B, and C of a certain atom correspond to increasing values of energy, i.e., \( E_A<E_B<E_C \). If \( \lambda_1, \lambda_2, \) and \( \lambda_3 \) are the wavelengths of a photon corresponding to the transitions shown, then:
Energy levels A, B, and C
A charge ‘q’ is spread uniformly over an isolated ring of radius ‘R’. The ring is rotated about its natural axis with angular speed \( \omega \). The magnetic dipole moment of the ring is:
A gas absorbs 18 J of heat and work done on the gas is 12 J. Then the change in internal energy of the gas is:
The ratio of the molar specific heat capacities of monatomic and diatomic gases at constant pressure is:
The frequency of fifth harmonic of a closed pipe is equal to the frequency of third harmonic of an open pipe. If the length of the open pipe is 72 cm, then the length of the closed pipe is:
A particle of mass \( m \) at rest on a rough horizontal surface with a coefficient of friction \( \mu \) is given a velocity \( u \). The average power imparted by friction before it stops is:
The moment of inertia of a solid sphere about its diameter is 20 kg m². The moment of inertia of a thin spherical shell having the same mass and radius about its diameter is:
As shown in the figure, two blocks of masses \(m_1\) and \(m_2\) are connected to a spring of force constant \(k\). The blocks are slightly displaced in opposite directions to \(x_1, x_2\) distances and released. If the system executes simple harmonic motion, then the frequency of oscillation of the system (\(\omega\)) is:
twoblocksofmassesm1andm2
A person walks up a stalled escalator in 90s. When standing on the same moving escalator, he reached in 60s. The time it would take him to walk up the moving escalator will be:
If \( A = \int_0^{\infty} \frac{1 + x^2}{1 + x^4} dx \) and \( B = \int_0^1 \frac{1 + x^2}{1 + x^4} dx \), then:
The differential equation for which \( ax + by = 1 \) is the general solution is:
The solution of the differential equation \( e^x y dx + e^x dy + xdx = 0 \) is:
If the length of the sub-tangent at any point P on a curve is proportional to the abscissa of the point P, then the equation of that curve is (C is an arbitrary constant):
If the angle \( \theta \) between the line \( \frac{x + 1}{1} = \frac{y - 1}{2} = \frac{z - 2}{2} \) and the plane \( 2x - y + \sqrt{\lambda}z + 4 = 0 \) is such that \( \sin \theta = \frac{1}{3} \), then the value of \( \lambda \) is:
Let \( f(x) = \begin{cases 1 + \frac{2x}{a}, & 0 \le x \le 1
ax, & 1<x \le 2 \end{cases} \). If \( \lim_{x \to 1} f(x) \) exists, then the sum of the cubes of the possible values of \( a \) is: }
If \( y = \sinh^{-1} \left(\frac{1 - x}{1 + x} \right) \), then \( \frac{dy}{dx} \) is given by:
If \[ y = (x - 1)(x + 2)(x^2 + 5)(x^4 + 8), \] then \[ \lim\limits_{x \to -1} \left( \frac{dy}{dx} \right) = ? \]
If \( P \) is a point which divides the line segment joining the focus of the parabola \( y^2 = 12x \) and a point on the parabola in the ratio 1:2, then the locus of \( P \) is:
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 \( a^2 + \beta^2 = \):
If \( y = x + \sqrt{2} \) is a tangent to the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), then equations of its directrices are:
The area of the quadrilateral formed with the foci of the hyperbola \[ \frac{x^2}{16} - \frac{y^2}{9} = 1 \] and its conjugate hyperbola is (in square units):
The length of the internal bisector of angle A in \( \triangle ABC \) with vertices \( A(4,7,8) \), \( B(2,3,4) \), and \( C(2,5,7) \) is:
If the direction cosines of two lines are given by \[ l + m + n = 0 \quad \text{and} \quad mn - 2lm - 2nl = 0, \] then the acute angle between those lines is: