List of practice Questions

A car is moving on a circular track banked at an angle of 45°. If the maximum permissible speed of the car to avoid slipping is twice the optimum speed of the car to avoid the wear and tear of the tyres, then the coefficient of static friction between the wheels of the car and the road is
A block of mass 0.5 kg is at rest on a horizontal table. The coefficient of kinetic friction between the table and the block is 0.2. If a horizontal force of 5 N is applied on the block, the kinetic energy of the block in a time of 4 s is (Acceleration due to gravity = 10 m/s\(^2\))
Regarding fundamental forces in nature, the correct statement is
A body is falling freely from the top of a tower of height 125 m. The distance covered by the body during the last second of its motion is \(x\%\) of the height of the tower. Then \(x\) is (Acceleration due to gravity = 10 m/s\(^2\))
If \( \left\lfloor x^2 \right\rfloor \) is the greatest integer less than or equal to \( x^2 \), then \[ \int_0^{\sqrt{2}} \left\lfloor x^2 \right\rfloor \, dx = \]
Find the value of \[ \left| \left( \frac{1+i}{\sqrt{2}} \right)^{2024} \right|. \]
The integral \( \int (x^4 - 8x^2 + 16x)(4x^3 - 16x + 16) \, dx \) is:
Let \( \left\lfloor x \right\rfloor \) be the greatest integer less than or equal to \( x \). Then \[ \lim_{x \to 0^-} \frac{x \left( \left\lfloor x \right\rfloor + |x| \right)}{|x|} \] is equal to:
If \( x = 5 \tan t \) and \( y = 5 \sec t \), then \( \frac{dy}{dx} \) at \( t = \frac{\pi}{3} \) is:
The area bounded by the curves \( y = x^2 \) and \( y = 2x \) in the first quadrant, is equal to:
The integral

\[ \int \frac{\sec x}{(\sec x + \tan x)^2} \, dx \] is:
If

\[ \int x e^{-x} \, dx = M e^{-x} + C, \quad \text{where } C \text{ is an arbitrary constant, then } M \text{ is equal to:} \]
The value of \( \int_{-4}^{-2} \left[ (x+3)^3 + 2 + (x+3)\cos(x+3) \right] \, dx \) is equal to:
Evaluate the integral \[ \int_{-500}^{500} \ln \left( \frac{1000 + x}{1000 - x} \right) dx \]
When \( y = vx \), the differential equation \[ \frac{dy}{dx} = \frac{y}{x} + \frac{f\left( \frac{y}{x} \right)}{f'\left( \frac{y}{x} \right)} \] reduces to:
The integrating factor of

\[ (1 + 2e^{-x}) \frac{dy}{dx} - 2e^{-x} y = 1 + e^{-x} \] is:
The solution of \( \frac{e^y}{dx} = x + 2 \) is:
The solution of \( \frac{dy}{\cos y} = dx \) is:
The solution of \( (y \cos y + \sin y) \, dy = (2x \log x + x) \, dx \) is:
The area enclosed by the curve \[ x = 3 \cos \theta, \quad y = 5 \sin \theta, \quad 0 \leq \theta \leq 2\pi, \] is equal to:

The limit: \[ \lim_{x \to 0} \frac{\sin \left( \pi \sin^2 x \right)}{x^2} \] is equal to:

If \[ \lim_{x \to 1} \frac{x^2 - ax - b}{x - 1} = 5, { then } a + b = ? \]
The integral

\[ \int \frac{dx}{x^8 \left( 1 + x^7 \right)^{2/3}} \] is equal to:
The value of \[ \int_0^{\frac{\pi}{2}} \frac{\cos^{2024} x}{\sin^{2024} x + \cos^{2024} x} \, dx \] is equal to:
The dihedral angles in gaseous and solid phases of H\(_2\)O\(_2\) molecule respectively are