List of practice Questions

A string of density \(7.5\) g cm\(^{-3}\) and area of cross-section \(0.2\) mm\(^2\) is stretched under a tension of 20 N. When it is plucked at the mid-point, the speed of the transverse wave on the wire is

Three blocks of masses \(m_1\), \(m_2\) and \(m_3\) are connected by massless strings as shown on a frictionless table. They are pulled with a force \(T_3 = 40\) N. If \(m_1 = 10\) kg, \(m_2 = 6\) kg and \(m_3 = 4\) kg, the tension \(T_2\) will be}

Two identical cells whether connected in parallel or in series gives the same current when connected to an external resistance 1.5 \(\Omega\). Find the value of internal resistance of each cell.

If $\int e^{2x}f' \left(x\right)dx =g \left(x\right)$, then $ \int\left(e^{2x}f\left(x\right) + e^{2x} f' \left(x\right)\right)dx =$

$C_p$ and $C_v$ are specific heats at constant pressure and constant volume respectively. It is observed that $C_p-C_v$ = a for hydrogen gas $C_p-C_v$ = b for nitrogen gas The correct relation between $a$ and $b$ is :

A physical quantity $P$ is described by the relation $P = a^{1/2} b^2 c^3 d^{-4} $ If the relative errors in the measurement of $a, b, c$ and $d$ respectively, are $2\%, 1\%, 3\%$ and $5\%$, then the relative error in $P$ will be :

A car is standing $200\, m$ behind a bus, which is also at rest. The two start moving at the same instant but with different forward accelerations. The bus has acceleration $2 \, m/s^2$ and the car has acceleration $4 \, m/s^2$. The car will catch up with the bus after a time of :

Two particles $A$ and $B$ of equal mass $M$ are moving with the same speed $v$ as shown in the figure. They collide completely inelastically and move as a single particle $C$. The angle $\theta$ that the path of $C$ makes with the $X-axis$ is given by :

A conical pendulum of length $1\,m$ makes an angle $\theta = 45^{\circ}$ w.r.t. $Z-axis$ and moves in a circle in the $XY$ plane. The radius of the circle is $0.4 \,m$ and its center is vertically below $O$. The speed of the pendulum, in its circular path, will be : (Take $g = 10 \, ms^{-2}$)

A diverging lens with magnitude of focal length $25\, cm$ is placed at a distance of $15\, cm$ from a converging lens of magnitude of focal length $20\, cm$. A beam of parallel light falls on the diverging lens. The final image formed is :

A magnetic dipole in a constant magnetic field has :

A magnetic needle of magnetic moment $6.7 \times 10^{-2} \, Am^2$ and moment of inertia $7.5 \times 10^{-6} \, kg \, m^2$ is performing simple harmonic oscillations in a magnetic field of $0.01\, T$. Time taken for $10$ complete oscillations is :

A man grows into a giant such that his linear dimensions increase by a factor of $9$. Assuming that his density remains same, the stress in the leg will change by a factor of :

For any three positive real numbers a, b and c, $9(25a^2 + b^2) + 25 (c^2 - 3ac) = 15b (3a + c)$. Then :

Let $a, b, c \, \in \, R$. If $f(x) = ax^2 + bx + c$ is such that $a + b + c = 3$ and $f (x + y) = f (x) + f (y) + xy, \forall \, x, y \, \in \, R,$ then $\displaystyle\sum^{10}_{n = 1} f(n)$ is equal to :

$\displaystyle\lim_{x \to \frac{\pi}{2}} \frac{\cot x - \cos x}{\left(\pi - 2x\right)^{3}} $ equals :

If for $x \epsilon \left(0, \frac{1}{4}\right) ,$ the derivative of $ \tan^{-1} \left(\frac{6x \sqrt{x}}{1-9x^{3}}\right) $ is $\sqrt{x} . g(x)$, then $g(x)$ equals :

Twenty meters of wire is available for fencing off a flower-bed in the form of a circular sector. Then the maximum area (in $s m$) of the flower-bed, is :

Let $l_n = \int \tan^{n} x \, dx , (n > 1) . l_4 + l_6 = a \, \, \tan^5 \, x + bx^5 + C$, where $C$ is a constant of integration, then the ordered pair $(a, b)$ is equal to :

A radioactive nucleus $A$ with a half-life $T$, decays into a nucleus $B$. At $t = 0$, there is no nucleus $B$. At sometime t, the ratio of the number of $B$ to that of $A$ is $0.3$. Then, $t$ is given by :

The integral $\int\limits^{\frac{3\, \pi}{4}}_{\frac{\pi}{4}} \frac{dx}{ 1 + \cos \, x}$ is equal to :

The area (in s units) of the region $\{ (x , y) : x \geq 0 , x + y \leq 3, x^2 \leq 4 y$ and $y \leq 1 + \sqrt{x} \}$ is

If $(2 + \sin \, x ) \frac{dy}{dx} + (y + 1) \cos \, x = 0$ and $y(0) = 1,$ then $y \left( \frac{\pi}{2} \right)$ is equal to :

A metal crystallises in a face centred cubic structure. If the edge length of its unit cell is $'a'$, the closest approach between two atoms in metallic crystal will be :

A particle is executing simple harmonic motion with a time period $T$. At time $t = 0$, it is at its position of equilibrium. The kinetic energy - time graph of the particle will look like :