List of practice Questions

If $\int \sqrt{x}(1-x^3)^{-1/2} dx = \frac{2}{3}g(f(x))+c$, then

$\int \frac{x^2}{(x^2-1)(x^2+1)} dx =$

Let $f:\mathbb{R} \to \mathbb{R}$ be a differentiable function such that $|f(x)-f(y)| \le 2|x-y|^{3/2} \ \forall x,y \in \mathbb{R}$. If $f(0)=1$, then $\int_0^1 f^2(x)dx=$
$\int_0^\pi x \sin^3 x \cos^2 x\ dx =$
If $\int_0^{2\pi} |x \sin x| dx = k\pi$, then $k=$

$\displaystyle \int_{\frac{1}{25}}^{1} x^{-2} e^{x^{-1/2}} dx =$

The area (in sq. units) bounded by the curves $y=\frac{8}{x}$, $y=2x$ and $x=4$ is

Which one of the following is a linear differential equation?
If $X=x+h, Y=y+k$ transforms $\frac{dy}{dx} = \frac{2x+3y-7}{3x+2y-8}$ to a homogeneous differential equation, then $(h,k)=$
If $\log y$ is an integrating factor of $\frac{dx}{dy}+P(y)x=Q(y)$, then $P(y)=$
A rtain physical quantity is calculated from the formula $\frac{\pi}{3}(a^2-b^2)hd$, where a, b and h are all lengths and d is density. The physical quantity being calculated is
If the engine of a long train moving with constant acleration crosses a tree with velocity 'u' and the last compartment of the train crosses the same tree with velocity 'v', then the velocity with which the middle compartment crosses the same tree is
A swimmer wants to cross a river which is flowing at a speed 'v'. If the swimmer can swim in still water at speed 'V', the direction he should swim to cross the river in least time is
The resultant magnitude of two vectors of same magnitude is equal to magnitude of either. The angle between the two vectors is
A bullet of mass 20 g moving with $500 \text{ ms}^{-1}$ is pierd 1 cm into a wooden block, then the retarding for experiend by the bullet is
A body of mass 10 kg is kept on a rough horizontal surfa of coefficient of friction 0.3. If a horizontal for of 50 N is applied on the body, then the acleration of the body is (Acleration due to gravity $= 10 \text{ ms}^{-2}$)
A toy car of mass 100 g is moving with velocity of $(\hat{i}+2\hat{j}-3\hat{k})$ m/s, then the kinetic energy of the car is
The heat energy released by water of mass 2 kg when it is cooled by $10~^\circ\text{C}$ is (Specific heat capacity of water $= 4200 \text{ J kg}^{-1} \text{K}^{-1}$)
Which of the following statements regarding ntre of mass is NOT true?
A particle of mass 10 g is moving towards east with a velocity of $10 \text{ ms}^{-1}$ and another particle of mass 15 g is moving towards north with a velocity of $5 \text{ ms}^{-1}$. The magnitude of the velocity of the ntre of mass of the system of the two particles is
The equation of motion of a particle executing simple harmonic motion is given by $x = 3 \sin(6t+\frac{\pi}{6})$, where x is in metres and t is in seconds. The ratio of the potential and kinetic energies of the particle at time $t=0$ is
Two particles execute simple harmonic motion along the same straight line with same amplitude and same frequency. The two particles pass one another when moving in opposite directions each time at a distan of $\frac{1{\sqrt{2}}$ times the amplitude from their common mean position. The phase differen between the two particles is
A satellite is plad in a circular orbit around the earth at an altitude of 1000 km. The time period of the satellite in minutes is approximately (mass of the earth $= 6 \times 10^{24}$ kg, radius of the earth $= 6.4 \times 10^6$ m, $G = 6.67 \times 10^{-11}$ Nm$^2$kg$^{-2}$)
The sag developed in a bar of length $l$, breadth $b$, and thickness $d$ when subjected to a load $w$ at the nter is (Y -- Young's modulus of the material of the bar)
A swimming pool has depth of 3 meters. The pressure at the bottom of the pool due to water alone is (Density of water is $1000$ kg/m$^3$, $g = 10$ m/s$^2$)