List of practice Questions

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 :

A steel rail of length $5\,m$ and area of cross section 40 $cm^2$ is prevented from expanding along its length while the temperature rises by $10^{\circ}C$. If coefficient of linear expansion and Young?s modulus of steel are $1.2 \times 10^{-5} \, K^{-1}$ and $2 \times 10^{11} \, Nm^{-2}$ respectively, the force developed in the rail is approximately :

A uniform wire of length $l$ and radius $r$ has a resistance of $100 \, \Omega$ . It is recast into a wire of radius $\frac{r}{2}$. The resistance of new wire will be :

Addition of sodium hydroxide solution to a weak acid (HA) results in a buffer of pH 6. If ionisation constant of HA is $10^{-5}$, the ratio of salt to acid concentration in the buffer solution will be :

An observer is moving with half the speed of light towards a stationary microwave source emitting waves at frequency $10\, GHz$. What is the frequency of the microwave measured by the observer ? (speed of light $= 3 \times 10^8 \, ms^{-1}$)

Among the following, the essential amino acid is :

An electric dipole has a fixed dipole moment $\vec{p}$, which makes angles $\theta$ with respect to x-axis. When subjected to an electric field $\vec{E}_1 = E \hat{i}$, it experiences a torque $\vec{T}_1 = \tau \hat{k}$. When subjected to another electric field $\vec{E}_2 = \sqrt{3 E_1} \hat{j}$ it experiences a torque $\vec{T}_2 = - \vec{T}_1$. The angle $\theta$ is

An electron beam is accelerated by a potential difference $V$ to hit a metallic target to produce $X-$rays. It produces continuous as well as characteristic $X$-rays. If $\lambda_{\text{min}}$ is the smallest possible wavelength of $X$-ray in the spectrum, the variation of $\log \, \lambda_{\text{min}}$ with log $V$ is correctly represented in :

An engine operates by taking $n$ moles of an ideal gas through the cycle $ABCDA$ shown in figure. The thermal efficiency of the engine is : (Take $C_v = 1.5 \, R,$ where $R$ is gas constant)

At $300\, K$, the density of a certain gaseous molecule at $2$ bar is double to that of dinitrogen $(N_2)$ at $4$ bar. The molar mass of gaseous molecule is :

Both lithium and magnesium display several similar properties due to the diagonal relationship; however, the one which is incorrect, is :

Given $E _{ Cl _{2} / Cl ^{-}}^{\circ}=1 . 36\, V,\, E _{ Cr ^{3+} / Cr }^{\circ}=-7 \cdot 74\, V$ $E _{ Cr _{2} O _{7}^{2-} / Cr ^{3+}}^{\circ}=1 .33\, V,\, E _{ MnO _{4}^{-} / Mn ^{2}+}^{\circ}=1. 51\, V$ Among the following, the strongest reducing agent is: