List of practice Questions

A body of mass 10 g is tied to a string of length 0.4 m and it is whirled in a horizontal plane with a speed of 6 m/s. Then the tension in the string is

The kinetic energy of a car is doubled when its velocity is increased by 1 m/s. Then the initial velocity of the car is

A car of mass 1000 kg having a velocity of 10 m/s collides a horizontally mounted spring. If the spring constant is 4000 N/m, then the maximum compression of the spring is

Three particles of each mass \( m \) are kept at the three vertices of an equilateral triangle. If \( I_1 \) is the moment of inertia of the system of the particles about an axis along one side of the triangle and \( I_2 \) is the moment of inertia of the system of the particles about an axis along the perpendicular bisector of a side, then \( I_1 : I_2 \) is

If \( u(n) = \int_0^{\frac{\pi}{2}} (1 + \sin t)^n \sin 2t \, dt,\; n \in \mathbb{N} \), then \( u(4) = \)

If a curve passes through (1, 2) and has the slope of its tangent \(1 - \frac{1}{x^2}\) at a point \((x, y)\), then the equation of that curve is

The differential equation having \( y = (a + b)e^{cx+d} \) as its general solution, where \( a, b, c, d \) are arbitrary constants, is

The integrating factor of the linear differential equation \( \frac{dy}{dx} = \frac{1}{4x + 3y} \) is

If speed of light in vacuum (\(3 \times 10^8\) m/s), acceleration due to gravity (\(10 \, \text{m/s}^2\)), and mass of electron (\(9.1 \times 10^{-31} \, \text{kg}\)) are taken as fundamental physical quantities, then the unit of time in this system is

If the velocity of a particle moving along a straight line with uniform acceleration, is given by \( V = \sqrt{196 - 16X} \, \text{m/s} \), then its acceleration is

The sum of the magnitudes of two vectors acting at a point is 18 and the magnitude of their resultant is 12. If the resultant is at \( 90^\circ \) with the vector of smaller magnitude, then the magnitudes of the vectors are

Assertion (A): \(\displaystyle \int_0^{\frac{\pi}{2}} (\sin^6 x + \cos^6 x)\, dx\) lies in the interval \(\left(\frac{\pi}{8}, \frac{\pi}{2}\right)\) Reason (R): \(\sin^6 x + \cos^6 x\) is a periodic function with period \(\dfrac{\pi}{2}\)

If \(\displaystyle \int \frac{3x+2}{4x^2+4x+5} \, dx = A \log(4x^2+4x+5) + B \tan^{-1} \left(x + \frac{1}{2}\right) + C\), then \((A, B) =\)

\(\displaystyle \int e^x \left( \log x + \frac{1}{x^2} \right) dx =\)

\(\displaystyle \int \frac{\sqrt{x^4 + x^{-4} + 2}}{x^3} dx =\)

The area bounded by the curve \( x = \log(|y|) \), the lines \( x = -1 \) and \( x = 0 \) is:

\(\displaystyle \int_{0}^{\pi} \frac{\cos x}{\sqrt{1 - \sin^2 x}} \, dx =\)

If \( \displaystyle \int_{3}^{b} \frac{x - 1}{2x - x^2} \, dx = \frac{1}{2} \), then \( (b - 1)^2 = \)

If \(u = \sin\left(\frac{x}{y}\right),\ x = e^t,\ y = t^2\), then \[ t^6 \left(\frac{du}{dt}\right)^2 \div e^{2t}(t - 2)^2 = \]

If \(g(x)\) is the inverse of the function \(f(x)\) and \(f'(x) = \dfrac{1}{h(x)}\), then what is the value of \(g'(x)\)?

\(f(x)\) is a continuous function on \(\mathbb{R}\) and \(y = f(x)\) is a curve. If \((\alpha, \beta)\) is a point such that \(\beta = f(\alpha)\) and \(p\alpha + m\beta + n = 0\ (p \ne 0, m \ne 0)\), then which one of the following is True?

The sum of the global minimum and global maximum values of the function \[ f(x) = \frac{4}{3}x^3 - 4x \quad \text{in } [0, 2] \text{ is} \]

If the slope of the tangent on a curve at any point \((x, y)\) is equal to \[ \frac{y^2 - x^2}{2xy} \] then the equation of the normal at the point \(\left(1, \frac{\sqrt{3}}{2}\right)\) is

Let \(A(1,15),\ B(3,-12),\ C(6,12)\) be three consecutive turning points of a continuous curve \(y = f(x)\). If \(f(x) = 0\) only for \(x = \alpha\) and \(x = \beta\), then \[ |\beta - \alpha| < ? \]

Evaluate the integral \[ \int \frac{\sin^{-1} \sqrt{x} - \cos^{-1} \sqrt{x}}{\sqrt{x} \left( \sin^{-1} \sqrt{x} + \cos^{-1} \sqrt{x} \right)} \, dx \]