List of practice Questions

In a hydraulic lift, if the radius of the smaller piston is \(5 \ \text{cm}\), and the radius of the larger piston is \(50 \ \text{cm}\), then the weight that the larger piston can support when a force of \(250 \ \text{N}\) is applied to the smaller piston is
The force required to stretch a steel wire of area of cross-section \(1 \ \text{mm}^2\) to double its length is
(Young’s modulus of steel \(= 2 \times 10^{11} \ \text{N/m}^2\))
The number of genes located on human Y chromosome.

Match the following:

List-I (Metal in drinking water)List-II (Maximum prescribed concentration in ppm)
A) Mn I) \( 0.05 \)
B) ZnII) \( 3.0 \)
C) CdIII) \( 0.005 \)
D) CuIV) \( 5.0 \)
When the mass attached to a spring is increased from 4 kg to 9 kg, the time period of oscillation increases by \(0.2\pi\) s. Then the spring constant of the spring is
For a particle executing simple harmonic motion, the ratio of kinetic and potential energies at a point where displacement is one half of the amplitude is
A thin uniform circular disc rolls with a constant velocity without slipping on a horizontal surface. Its total kinetic energy is
A body is projected at an angle of \(60^\circ\) with the horizontal. If the initial kinetic energy of the body is \(X\), then its kinetic energy at the highest point is
A body is moving along a straight line under the influence of a constant power source. If the relation between the displacement (s) of the body and time (t) is \(s \propto t^x\), then \(x =\)
Two balls each of mass 250 g moving in opposite directions each with a speed 16 m/s collide and rebound with the same speeds. The impulse imparted to one ball due to the other is
A train of mass \(10^6 \, \text{kg}\) is moving at a constant speed of \(108 \, \text{km/h}\). If the frictional force acting on it is \(0.5 \, \text{N per 100 kg}\), then the power of the train is
A particle crossing the origin at time \(t = 0\), moves in the xy-plane with a constant acceleration ‘a’ in y-direction. If the equation of motion of the particle is \(y = bx^2\) (where \(b\) is a constant), then its velocity component in the x-direction is
The magnitudes of two vectors are A and B (A>B). If the maximum resultant magnitude of the two vectors is ‘n’ times their minimum resultant magnitude, then \(\frac{A}{B} =\)
The ratio of the displacements of a freely falling body during second and fifth seconds of its motion is
The dimensional formula of Planck’s constant is
The general solution of the differential equation \[ y + \cos x \left( \frac{dy}{dx} \right) - \cos^2 x = 0 \] is
If the degree of the differential equation corresponding to the family of curves
\[ y = ax + \frac{1}{a} \quad (\text{where } a \neq 0 \text{ is an arbitrary constant}) \] is \(r\) and its order is \(m\), then the solution of
\[ \frac{dy}{dx} - \frac{y}{2x}, \quad y(1) = \sqrt{r + m} \] is
The area of the region lying between the curves \( y = \sqrt{4 - x^2} \), \( y^2 = 3x \) and the Y-axis is:
Evaluate the integral: \[ \left| \int_{-\pi/4}^{\pi/3} \tan\left(x - \frac{\pi}{6}\right) dx \right| \]
Evaluate the integral: \[ \int \frac{(3x - 2)\tan\left(\sqrt{9x^2 - 12x + 1}\right)}{\sqrt{9x^2 - 12x + 1}} \, dx =\ ?\]
Evaluate the integral: \[ \int \frac{x^4 - 1}{x^2 \sqrt{x^4 + x^2 + 1}} \, dx =\ ? \]
Evaluate the integral: \[ \int \frac{1}{9\cos^2 x - 24 \sin x \cos x + 16 \sin^2 x} \, dx = \]
The interval in which the curve represented by \( f(x) = 2x + \log\left(\frac{x}{2 + x}\right) \) is increasing is
If the extreme value of the function \( f(x) = \frac{4}{\sin x} + \frac{1}{1 - \sin x} \) in \(\left[0, \frac{\pi}{2}\right]\) is \(m\) and it exists at \(x = k\), then \(\cos k =\)}
The displacement \(S\) of a particle measured from a fixed point \(O\) on a line is given by \[ S = t^3 - 16t^2 + 64t - 16. \] Then the time at which the displacement of the particle is maximum is