List of practice Questions

n objects are distributed at random among n persons. The number of ways in which this can be done so that at least one of them will not get any object is

A cylinder of mass m and material density ρ hanging from a string is lowered into a vessel of cross - sectional area A containing a liquid of density σ (< ρ) until it is fully immersed. The increase in pressure at the bottom of the vessel is:

A cylindrical vessel, open at the top, contains 15 liters of water. Water drains out through a small opening at the bottom. 5 liters of water comes out in time t1, the next 5 litre in further time t₂, and the last 5 litre in further time t₃. then:

A, B are fixed points with coordinates (0,a) and (0,b) (a>0,b>0). P is a variable point (x,0) referred to as the rectangular axis. If the angle \(\angle\)APB is maximum, then

A missile is fired from the ground level rises x meters vertically upwards in t sec, where \(x=100t-\frac{25}{2}t^2\). the maximum height reached is

The dimensions of four wires if the same material are given below. The increase in length is maximum in the wire of :

Let f(x)=\(\begin{Bmatrix} x+1&\,\,\,-1\leq x\leq 0\\ -x&\,\,\,\,\,0<x\leq1 \end{Bmatrix}\)

A man weighing 75 kg is standing in a lift. The weight of the man standing on a weighing machine kept in the lift when the lift is moving downwards freely under gravity is:

If a hyperbola passes through the point P(\(\sqrt{2},\sqrt{3}\)) and has foci at (±2,0), then the tangent to this hyperbola at P is

The displacement of a particle executing simple harmonic motion is given by x = 2cos(t) where t is the time in seconds then the time period of the particle is:

The ratio of the radii of two solid spheres of same mass in 2:3. The ratio of the moments of inertia of the spheres about their diameters is:

Given \(\frac{d^2y}{dx^2}=cot\,x\frac{dy}{dx}+4y\,cosec^2x=0\). Changing the independent variable x to z by the substitution z=log tan \(\frac{x}{2}\), the equation is changed to

An engine is dragging a mass of 5000 kg with a velocity of 5 ms^-1 along a smooth inclined plane of inclination 1 in 50. Then the power of the engine is

If y=\(\frac{x}{log_e|cx|}\) is the solution of the differential equation \(\frac{dy}{dx}=\frac{y}{x}+\phi(\frac{x}{y})\), then \(\phi (\frac{x}{y})\) is given by

If \(\int \frac{dx}{(x+1)(x-2)(x-3)}=\frac{1}{k}log_e\left \{ \frac{|x-3|^3|x+1|}{(x-2)^4}\right \}+c\), then the value of k is

A player can throw a ball to a maximum horizontal distance of 80 m. If he throws the ball vertically with the same velocity, then the maximum height reached by the ball is:

The number of significant figures in the measurement of a length 0.079000 m is:

The expression \(\frac{\int_{0}^{n}[x]dx}{\int_{0}^{n}\left \{ x\right \}dx}\), where [x] and {x} are respectively integral and fractional part of x and n∈N, is equal to

Let f:[1,3]\(\rightarrow\)R be continuous and be derivable in (1,3) and f'(x)=[f(x)]²+4∀x∈(1,3). Then

Let \(cos^{-1}\frac{y}{b}\)=\(log_e (\frac{x}{n})^n\), then Ay₂+By₁+Cy=0 is possible : (where y₂=\(\frac{d^2y}{dx^2}\), y₁=\(\frac{dy}{dx}\))

\(\lim_{x\rightarrow \infty}\){\(x-\sqrt[n]{(x-a_1)(x-a_2)......(x-a_n)}\)} where a₁,a₂,.....a_n are positive rational numbers.The limit