List of practice Questions

Given that \( \mathbf{a} \times (2\hat{i} + 3\hat{j} + 4\hat{k}) = (2\hat{i} + 3\hat{j} + 4\hat{k}) \times \mathbf{b} \), \( |\mathbf{a} + \mathbf{b}| = \sqrt{29} \), \( \mathbf{a} \cdot \mathbf{b} = ? \)
Coefficient of \( x^9 \) in the expansion of \( \left( 4 - \frac{x^2}{9} \right)^{12} \)
Evaluate the expression: \[ \frac{\sin \frac{\pi}{7} + \sin \frac{3\pi}{7}}{1 + \cos \frac{\pi}{7} + \cos \frac{2\pi}{7}} \]
Axis of parabola is \( x = 0 \). If the vertex is at a distance of 3 from the origin above the x-axis, find the vertex.
Find the integral: \[ \int \left( \sin^{-1} \sqrt{x} + \cos^{-1} \sqrt{x} \right) \, dx \]
Evaluate: \[ \sec \left( \cos^{-1} \left( \frac{2024}{2025} \right) \right) \]
Find the probability of getting at most 2 heads when 4 coins are tossed.
Find the limit: \[ \lim_{x \to 0} \frac{\sin(\pi \sin^2 x)}{x^2} \]
Find the values of: \[ \cos 75^\circ, \cos 15^\circ, \cos 45^\circ \]
Evaluate the integral: \[ \int \frac{\cos \theta}{2 - \sin^2 \theta} \, d\theta \]
Given that \( a, \frac{3}{4} a r^2, a r^3 \) are in G.P. Product of first four terms = \( \frac{3^6}{4^3} \), then find \( a \):
Integrating factor of \[ \frac{dy}{dx} - 2y = 2x - 3 \]
A first-order reaction is 75 percent completed in 6000s. What is the half-life?
Total number of orbitals when \( n = 3 \)?
Nitrogen base not present in DNA:
Alkali metal with highest first ionization enthalpy is:
Maximum magnetic moment shown by:
Molecule with 2\( \sigma \) and 2\( \pi \) bonds:
Number of valence electrons in Germanium?
Which of the following contains 2 primary –OH groups in its structure?
The tennis ball of mass 120 g moving with a velocity of 20 m/s towards gets by a rocket of velocity and the final speed becomes 30 m/s with direction reversed. If the time of contact is 0.01 s, what is the average force acting on the ball?
For an ideal gas of molar mass \( M \), the slope of the graph between the rms speed \( V \) and \( \sqrt{T} \) where \( T \) represents the temperature is...
Relation between Azimuthal quantum number and magnetic quantum number
What is the dimension of \[ \frac{M B}{K T} \] where \( M \) represents magnetic moment, \( K \) represents the Boltzmann constant, \( B \) represents the magnetic field, and \( T \) represents temperature?
What is the ratio of de Broglie wavelength of the particles if their kinetic energy are 0.002 eV and 2 eV respectively?