If the volume of spherical ball is increasing at the rate of $4\pi$ $c{m}^3/\sec$ then the rate of change of its surface area when the volume is $288\pi c{m}^3$ is

If \(\bar{a}=a_1 \hat{i}+a_2 \hat{j}+a_3 \hat{k}, \quad \bar{b}=b_1 \hat{i}+b_2 \hat{j}+b_3 \hat{k}\), \(\overline{\mathrm{c}}=\mathrm{c}_1 \hat{i}+\mathrm{c}_2 \hat{\mathrm{j}}+\mathrm{c}_3 \hat{\mathrm{k}}\) and \(\left[\begin{array}{lll}3 \overline{\mathrm{a}}+\overline{\mathrm{b}} & 3 \overline{\mathrm{~b}}+\overline{\mathrm{c}} & 3 \overline{\mathrm{c}}+\overline{\mathrm{a}}\end{array}\right]\) \(=\lambda\left|\overline{\mathrm{a}} \cdot \hat{\mathrm{i}} \overline{\mathrm{a}} \cdot \hat{\mathrm{j}} \overline{\mathrm{a}} \cdot \hat{\mathrm{k}} \ \overline{\mathrm{b}} \cdot \hat{\mathrm{i}} \overline{\mathrm{b}} \cdot \hat{\mathrm{j}} \overline{\mathrm{b}} \cdot \hat{\mathrm{k}} \ \overline{\mathrm{c}} \cdot \hat{\mathrm{i}} \overline{\mathrm{c}} \cdot \hat{\mathrm{j}} \overline{\mathrm{c}} \cdot \hat{\mathrm{k}}\right|\) then the value of \(\lambda\) is

\(\mathrm{ABCD}\) is a parallelogram, \(\mathrm{P}\) is the mid-point of \(\mathrm{AB}\). If \(\mathrm{R}\) is the point of intersection
of \(\mathrm{AC}\) and \(\mathrm{DP}\), then \(\mathrm{R}\) divides \(\mathrm{AC}\) internally in the ratio

The particular solution of the differential equation, \(x y \frac{\mathrm{~d} y}{\mathrm{~d} x}=x^2+2 y^2\) when \(y(1)=0\) is

The distance of the point \((1,6,2)\) from the point of intersection of the line \(\frac{x-2}{3}=\frac{y+1}{4}=\frac{z-2}{12}\) and the plane \(x-y+z=16\) is

The inverse of the statement
"If the surface area increase, then the pressure decreases.", is

The value of \(\int_{\frac{\pi}{6}}^{\frac{\pi}{4}} \frac{1}{\sin 2 x\left(\tan ^5 x+\cot ^5 x\right)} \mathrm{dx}\) is

Which one of the following blood cells produce antibodies as immune response of the body?

What is the formal charge on sulfur in following Lewis structure?

If \(\bar{a}=2 \hat{i}+3 \hat{j}+2 \hat{k}, \bar{b}=2 \hat{i}+\hat{j}-\hat{k}\) and \(\bar{c}=3 \hat{i}-\hat{j} \quad\) are such that \(\bar{a}+\lambda \bar{b}\) is perpendicular to \(\bar{c}\), then the value of \(\lambda\) is

\(y=m x+\frac{2}{m}\) is the general solution of

A first order reaction is \(75 \%\) completed in 60 minutes, the time required for it's \(50 \%\) completion is.

If \(f(x)=\frac{\left(e^{2 x}-1\right) \sin x^{0}}{x^{2}}, x \neq 0\) is continuous at \(x=0\), then \(f(0)=\)