List of practice Questions

The vector form of the universal law of gravitation:
The pressure required to decrease the volume of 4000 cc of water by 0.05%: (Bulk modulus of water = \( 2.2 \times 10^9 \, \text{Nm}^{-2} \))
A 20 g copper block is suspended by a vertical spring causing 1 cm elongation over the natural length of the spring. If a beaker of water is placed below the block so that the copper block is completely immersed in the liquid, the elongation of the spring is (Density of copper 9000 kg m\(^{-3}\), Density of water 1000 kg m\(^{-3}\), \( g = 10 \) ms\(^{-2}\)):
A 50 g ice cube at -10°C is added to 200 g of water at 30°C. The final temperature of the mixture is: (Specific heat of water = 1 cal g\(^{-1}\)°C\(^{-1}\), latent heat of fusion of ice = 80 cal g\(^{-1}\), specific heat of ice = 0.5 cal g\(^{-1}\)°C\(^{-1}\))
The relation between volume (V) and absolute temperature (T) of a gas in an adiabatic process is:
A motor boat is moving in a river with velocity $\vec{v} = 7\hat{i} + 2\hat{j} - 5\hat{k}$ kms$^{-1}$. If the flow water offers resistive force $\vec{F} = 9\hat{i} + 3\hat{j} - 3\hat{k}$ N, then the power of the boat is
The force required to stop a body of mass 10 kg moving along a straight line path with a velocity of 10 ms\(^{-1}\) in a time of 10 s is:
A truck of mass 2000 kg is moving along a circular path having a radius of curvature of 10 m. If the banking angle is 39°, then the maximum permissible speed of the truck is (Acceleration due to gravity = 10 ms\(^{-2}\), take tan 39° = 0.81):
A spring has a spring constant 200 Nm\(^{-1}\). If it is stretched by 1 cm then the potential energy stored in it is:
The work done in moving a body of mass 2 kg to a height of 4 m from the surface of the earth is: (Acceleration due to gravity = 10 ms\(^{-2}\))
Which of the following statements is true regarding the vector product of two vectors?
The general solution of \( \frac{dy}{dx} = \cos^2(x - y - 1) \) is given by \( x = \):
The degree of the differential equation $\log \left( \frac{dy}{dx} \right) = \left( 2x + 3 \frac{dy}{dx} \right)^2$ is
If $y = y(x)$ is a particular solution of $\sqrt{1 - x^2} \frac{dy}{dx} + \frac{2x}{\sqrt{1 - x^2}} y = x$, $y(0) = 1$, then $y\left(\frac{1}{2}\right) =$
The density of a substance is 4 g/cc. In a system in which the unit of length is 5 cm and the unit of mass is 20 g, the density of the substance is:
A truck of mass \( M \) and a car of mass \( \frac{M}{10} \) moving with the same momentum are brought to halt by the application of the same breaking force. The ratio of the distances travelled by the truck and car before they come to stop is:
A car is travelling at \(30 \, \text{ms}^{-1}\) speed on a circular road of radius \(300 \, \text{m}\). If its speed is increasing at the rate of \(4 \, \text{ms}^{-2}\), then its acceleration is:
\(\int \frac{8^{1+x} + 4^{1+x}}{2^{2x}} dx =\):
If $[\cdot]$ denotes the greatest integer function, then $\int_{0}^{1000} e^{x - \lfloor x \rfloor} dx =$
If $I = \int_{1}^{3} \sqrt{3 + x + x^2} dx$, then $I$ lies in the interval
The area bounded by \( y - 1 = -|x| \) and \( y + 1 = |x| \) is:
\( \int_{-4\pi}^{4\pi} \tan^9 x \sin^6 x \cos^3 x \, dx = \)
Given that $\frac{d}{dx} \int_{0}^{\phi(x)} f(t) dt = f(\phi(x)) \phi'(x)$. For all $x \in (0, \frac{\pi}{2})$, if $\int_{1}^{\cos x} t^2 f(t) dt = \cos 2x$, then $f\left(\frac{1}{\sqrt{2}}\right) =$
The equation of the normal to the curve \( y = \cosh x \) drawn at the point nearest to the origin is:
Let \( n \in (0, \infty) \). If for all the curves \( y = x^n \log x \) for distinct values of \( n \), we have \( y = x - 1 \) as the tangent at a fixed point \((\alpha, \beta)\), then \(\alpha + \beta = \):