List of practice Questions

A particle moving in a circle of radius R with uniform speed takes time T to complete one revolution. If this particle is projected with the same speed at an angle θ to the horizontal, the maximum height attained by it is equal to 4R. The angle of projection θ is then given by

The dimensional formula of angular impulse is :

In series LCR circuit, the capacitance is changed from C to 4C. To keep the resonance frequency unchanged, the new inductance should be :

10 divisions on the main scale of a Vernier calliper coincide with 11 divisions on the Vernier scale. If each division on the main scale is of 5 units, the least count of the instrument is :

The pressure and volume of an ideal gas are related as \( PV^{3/2} = K \) (constant). The work done when the gas is taken from state A (\( P_1, V_1, T_1 \)) to state B (\( P_2, V_2, T_2 \)) is:

A ball of mass 0.5 kg is attached to a string of length 50 cm. The ball is rotated on a horizontal circular path about its vertical axis. The maximum tension that the string can bear is 400 N. The maximum possible value of angular velocity of the ball in rad/s is,:

Two moles a monoatomic gas is mixed with six moles of a diatomic gas. The molar specific heat of the mixture at constant volume is :

The reading in the ideal voltmeter (V) shown in the given circuit diagram is :

If \( \sin\left(\frac{y}{x}\right) = \log_e |x| + \frac{\alpha}{2} \) is the solution of the differential equation \[x \cos\left(\frac{y}{x}\right) \frac{dy}{dx} = y \cos\left(\frac{y}{x}\right) + x\]and \( y(1) = \frac{\pi}{3} \), then \( \alpha^2 \) is equal to

If \( R \) is the radius of the earth and the acceleration due to gravity on the surface of the earth is \( g = \pi^2 \, \text{m/s}^2 \), then the length of the second's pendulum at a height \( h = 2R \) from the surface of the earth will be:

If \( \log_e a, \log_e b, \log_e c \) are in an A.P. and \( \log_e a - \log_e 2b, \log_e 2b - \log_e 3c, \log_e 3c - \log_e a \) are also in an A.P., then \( a : b : c \) is equal to

With rise in temperature, the Young's modulus of elasticity

If the shortest distance between the lines \(\frac{x - \lambda}{-2} = \frac{y - 2}{1} = \frac{z - 1}{1}\) and \(\frac{x - \sqrt{3}}{1} = \frac{y - 1}{-2} = \frac{z - 2}{1}\) is 1, then the sum of all possible values of \( \lambda \) is:

If \( 5f(x) + 4f\left(\frac{1}{x}\right) = x^2 - 2 \), for all \( x \neq 0 \), and \( y = 9x^2f(x) \), then \( y \) is strictly increasing in:

If the mean and variance of five observations are \( \frac{24}{5} \) and \( \frac{194}{25} \) respectively and the mean of first four observations is \( \frac{7}{2} \), then the variance of the first four observations is equal to

Let 3, a, b, c be in A.P. and 3, a – 1, b + l, c + 9 be in G.P. Then, the arithmetic mean of a, b and c is :

Let \( f: \mathbb{R} \to \mathbb{R} \) be defined as: \(f(x) =  \begin{cases}  \frac{a - b \cos 2x}{x^2}, & x < 0, \\  x^2 + cx + 2, & 0 \leq x \leq 1, \\  2x + 1, & x > 1.  \end{cases}\)
If \( f \) is continuous everywhere in \( \mathbb{R} \) and \( m \) is the number of points where \( f \) is NOT differentiable, then \( m + a + b + c \) equals:  

Let \( P(3, 2, 3) \), \( Q(4, 6, 2) \), and \( R(7, 3, 2) \) be the vertices of \( \triangle PQR \). Then, the angle \( \angle QPR \) is

For \(0 < \theta < \frac{pi}{2}\), if the eccentricity of the hyperbola \( x^2 - y^2 \csc^2 \theta = 5 \) is \( \sqrt{7} \) times the eccentricity of the ellipse \( x^2 \csc^2 \theta + y^2 = 5 \), then the value of \( \theta \) is:

Let \( f: \mathbb{R} \to \mathbb{R} \) and \( g: \mathbb{R} \to \mathbb{R} \) be defined as: \(f(x) = \begin{cases} \log_e x, & \text{if } x > 0, \\ e^{-x}, & \text{if } x \leq 0, \end{cases}\) and
\(g(x) = \begin{cases} x, & \text{if } x \geq 0, \\ e^x, & \text{if } x < 0. \end{cases}\) Then \( g \circ f: \mathbb{R} \to \mathbb{R} \) is:

Let \(\vec{a} = -5\hat{i} + \hat{j} - 3\hat{k}, \quad \vec{b} = \hat{i} + 2\hat{j} - 4\hat{k},\) and  \(\vec{c} = \left( \left( \left( \vec{a} \times \vec{b} \right) \times \hat{i} \right) \times \hat{i} \right) \times \hat{i}.\) Then  \(\vec{c} \cdot (-\hat{i} + \hat{j} + \hat{k})\) is equal to:  

Let the median and the mean deviation about the median of 7 observation 170, 125, 230, 190, 210, a, b be 170 and \(\frac{205}{7}\)respectively. Then the mean deviation about the mean of these 7 observations is :

Let \( f(x) = x^3 + x^2 f'(1) + x f''(2) + f'''(3) \), \( x \in \mathbb{R} \). Then \( f'(10) \) is equal to ______.

Element not showing variable oxidation state is: