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A simple pendulum with bob (mass $m$ and charge $q$) is in equilibrium in presence of horizontal electric field $E$ then tension in thread is .

Six charges (four $+q$, two $-q$) are present at circle of radius $r$ and centred at origin as shown. Electric field at origin is .

In a potentiometer null point for two resistance $R_{1}$ and $R_{2}$ is at $40~cm$ as shown. If $16~\Omega$ is connected in parallel to $R_{2}$ then null point is at $50~cm$ then $R_{1}$ and $R_{2}$ are respectively .

Match the following : \begin{tabular}{|l|l|} \hline List -1 & List-II \\ \hline (A) Spring constant & (i) $ML^{2}T^{-2}K^{-1}$ \\ \hline (B) Thermal conductivity & (ii) $ML^{0}T^{-2}$ \\ \hline (C) Boltzmann constant & (iii) $ML^{2}T^{-2}A^{-2}$ \\ \hline (D) inductance & (iv) $ML^{2}T^{-3}K^{-1}$ \\ \hline & (v) $M^{1}L^{-1}T^{-2}$ \\ \hline & (vi) $ML^{2}T^{-2}$ \\ \hline \end{tabular}

A projectile is thrown upward at an angle $60^{\circ}$ with the horizontal. The speed of the projectile is $20~m/s$ when its direction of motion is $45^{\circ}$ with horizontal. The initial speed of particle is .

If radius and density of planet $A$ are $R$ and $\rho$ respectively and escape velocity of planet $A$ is $10~km/h$. If radius of planet $B$ is $10\%$ of planet $A$ and $\rho_B = 10\%$ planet of $A$. Calculate of escape velocity of planet $B$ .

In the given situation force at center on $1~kg$ mass is $F_{1}$. Now if $4~m$ and $3~m$ is interchanged the force is $F_{2}$ Given: $\frac{F_{1}}{F_{2}}=\frac{2}{\sqrt{\alpha}}$ Find $\alpha$ . (Diagram shows masses $4m$, $3m$, $m$, $2m$ arranged around a center point) .

Given $\vec{E}=10x\hat{i}+5y\hat{j}$. Potential at $(10, 20)$ is $500~V$. Find potential at origin .

In series R-L circuit, voltage of battery is $10~V$. Resistance and inductance are $10~\Omega$ and $10~mH$ respectively. Find energy stored in the inductor when current reaches $\frac{1}{e}$ times of maximum value :

Two disc having same moment of inertia about their axis. Thickness is $t_{1}$ and $t_{2}$ and they have same density. If $R_{1}/R_{2}=1/2$, then find $t_{1}/t_{2}$ :

Chord joining at two points $(x_{1},y_{1})$ and $(x_{2},y_{2})$ on the parabola $y^{2}=12x$ intersect at right angle at the vertex, then $x_{1}x_{2}-y_{1}y_{2}=$ .

Let $A$ satisfy the equation $\lambda\begin{bmatrix}1\\2\\3\end{bmatrix}=\begin{bmatrix}-1\\0\\3\end{bmatrix}$, and $|2\text{adj}(A+I)|=2^{\alpha}3^{\beta}11^{\gamma}$ then $\alpha+\beta+\gamma$ is equal to .

If $\int(\cos x)^{-5/2}(\sin x)^{-11/2}dx=\frac{P_{1}}{q_{1}}(\cot x)^{9/2}+\frac{P_{2}}{q_{2}}(\cot x)^{5/2}+\frac{P_{3}}{q_{3}}(\cot x)^{1/2}-\frac{P_{4}}{q_{4}}(\cot x)^{-3/2}+C$, then $\frac{15P_{1}P_{2}P_{3}P_{4}}{q_{1}q_{2}q_{3}q_{4}}$ is equal to :

If the area of the region $\{(x,y):x^{2}+1\le y\le3-x\}$ is divided by the line $x=-1$ in the ratio $m : n$, where $\gcd(m,n)=1$. Then the value of $m+n$ is .

If a line $ax+y=1$ does not intersect the hyperbola $x^{2}-9y^{2}=9$, then possible value of $a$ is :

If $\frac{\cos^{2}48^{\circ}-\sin^{2}12^{\circ}}{\sin^{2}24^{\circ}-\sin^{2}6^{\circ}}=\frac{\alpha+\sqrt{5}\beta}{2}$, then $(\alpha+\beta)$ is :

Let $Q(5,b,c)$ be the mirror image of $P(1,3,a)$ with respect to the line $\frac{x-1}{3}=\frac{y-3}{2}=\frac{z-2}{2}$, then the value of $a^{2}+b^{2}+c^{2}$ is .

Two numbers '$a$' and '$b$' are chosen at random from the set $\{1,2,3,...,50\}$. The probability that $ab$ is divisible by 3, is :

Value of $\int_{-\pi/2}^{\pi/2}\frac{dx}{[x]+4}$: is $-\pi/2$ where $[.]$ is GIF .

Let $\int_{1}^{x}f(t)dt=3xf(x)-x^{3}+4$, then $f(2)-f(3)$ is .