Class 12th

A two point charges $4q$ and $-q$ are fixed on the $x$ -axis at $x=-\frac{d}{2}$ and $x=\frac{d}{2}$ , respectively. If a third point charge ' $q$ ' is taken from the origin to $x=d$ along the semicircle as shown in the figure, the energy of the charge will:

Take the breakdown voltage of the zener diode used in the given circuit as $6\mathrm{V}$ . For the input voltage shown in figure below, the time variation of the output voltage is: (Graphs drawn are schematic and not to scale)

A Tennis ball is released from a height $h$ and after freely falling on a wooden floor it rebounds and reaches height $\frac{h}{2}$ . The velocity versus height of the ball during its motion may be represented graphically by:

(graphs are drawn schematically and on not to scale)

The value of the integral ${\displaystyle \underset{0}{\overset{\frac{\pi }{2}}{\int }}60\frac{sin\left(6x\right)}{sinx}}$ dx is equal to…….

A bag contains 4 white and 6 black balls. Three balls are drawn at random from the bag. Let X be the number of white balls, among the drawn balls. If ${\sigma }^2$ is the variance of X, then $100{\sigma }^2$ is equal to………

Let z = a ib, b $\ne$ 0 be complex numbers satisfying z² = $\overline{z}.2^{1-\left|z\right|}.$ Then the least value of n $\in$ N, such that zn = ${\left(z+1\right)}^n,$ is equal to………

Let $S=[-\pi ,\frac{\pi }{2})-\left\{-\frac{\pi }{2},-\frac{\pi }{4},-\frac{3\pi }{4},\frac{\pi }{4}\right\}.$ Then the number of elements in the set $|A=\left\{\theta \in S:tan\theta \left(1+\sqrt[]{5}tan\left(2\theta \right)\right)=\sqrt[]{5}-tan\left(2\theta \right)\right\}$ is……….

If $\frac{6}{3^{12}}+\frac{10}{3^{11}}+\frac{20}{3^{10}}+\frac{40}{3^9}+...+\frac{10240}{3}=2^n.$ m, where m is odd, then m. n is equal to……….

Two tangent lines ${\mathcal{l}}_{1}and{\mathcal{l}}_2$ are drawn from the point (2, 0) to the parabola 2y² = x. If the lines ${\mathcal{l}}_{1}and{\mathcal{l}}_2$ are also tangent to the circle (x – 5)² + y²= r, then 17r is equal to……….