For a group of $100$ students, the mean ${\overline{x}}_1$ and the standard deviation ${\sigma }_1$ of their marks were found to be $40$ and $15$ respectively. Later it was observed that the scores $40$ and $50$ were misread as $30$ and $60$ respectively. If the mean and the standard deviation with the corrected observations of the scores are ${\overline{x}}_2$ and ${\sigma }_2$ respectively, then

If \(y=\log _e \tan \left(\frac{\pi}{4}+\frac{x}{2}\right)\), then \(\tanh \left(\frac{y}{2}\right)=\)

If \(\mathrm{X}\) is random variable with probability distribution \(P(X=k)=\frac{(k+1) c}{2^k}, k=0,1,2, \ldots\), then \(P(X \geq 3)=\)

The solution of \(\cos y+(x \sin y-1) \frac{d y}{d x}=0\) is

The minimum and maximum values of \(\cos \left(x+\frac{\pi}{3}\right)+2 \sqrt{2} \sin \left(x+\frac{\pi}{3}\right)\) are respectively

If \(\sec \theta \cosh y=\operatorname{cosec} x\) and \(\operatorname{cosec} \theta \sinh y=\sec x\), then \(\sinh ^2 y=\)

The number of normals drawn to the parabola \(y^2=4 x\) from the point \((1,0)\) is

A galvanometer has a deflection of 25 divisions for a current of 0.1 A flowing through it. Then the current sensitivity of the galvanometer is

Let ' \(\mathrm{O}\) ' be the origin, \(\mathrm{A}\) and \(\mathrm{B}\) be two points with position vectors \(-3 \hat{i}-3 \hat{j}+4 \hat{k}\) and \(4 \hat{i}-4 \hat{j}-3 \hat{k}\) respectively. Let \(P\) be a point such that the line drawn through \(P\) parallel to \(\overrightarrow{\mathrm{OB}}\) meets \(\mathrm{OA}\) in \(\mathrm{L}\) and another line through \(\mathrm{P}\) parallel to \(\overrightarrow{\mathrm{OA}}\) meets \(\mathrm{OB}\) in \(\mathrm{M}\). If \(\mathrm{L}\) divides \(\mathrm{OA}\) in the ratio \(2: 3\) and \(\mathrm{M}\) divides \(\mathrm{OB}\) in the ratio \(3: 2\), then the distance from 0 to \(\mathrm{P}\) is

The energy stored in a capacitor is W. To double the charge on the plates of the capacitor, the additional work to be done is

Which of the following polymer is biodegradable?

Let \(f\) be a polynomial function defined on \([2,7]\). If \(f(2)=3\) and \(f^{\prime}(x) \leq 5\) for all \(x\) in \((2,7)\), then the maximum possible value attained by \(f\) at \(x=7\) is

One second after projection, a projectile is travelling in a direction inclined at \(45^{\circ}\) to horizontal. After two more seconds it is travelling horizontally. Then the magnitude of velocity of the projectile is \(\left(\mathrm{g}=10 \mathrm{~ms}^{-2}\right)\)