If \(\sqrt{1-x^6}+\sqrt{1-y^6}=a\left(x^3-y^3\right)\), then \(y^2 \frac{d y}{d x}=\)

Let \(\vec{a}=3 \hat{i}+\hat{j}-2 \hat{k}, \vec{b}=-5 \hat{i}+7 \hat{j} \quad\) and \(\quad \vec{c}=3 \hat{i}+y \hat{j} \quad\) be three vectors such that \(|\vec{a}-\vec{b}+\vec{c}|=\sqrt{141}\). If \(y_1\) and \(y_2\) are the values of \(y\) satisfying the given condition, then \(\left|\mathrm{y}_1-\mathrm{y}_2\right|=\)

If \(\bar{a}=2 \bar{i}+\bar{j}-3 \bar{k}, \bar{b}=\bar{i}-2 \bar{j}+\bar{k}, \bar{c}=-\bar{i}+\bar{j}-4 \bar{k}, \bar{d}=\bar{i}+\bar{j}+p \bar{k}\), where \(p\) is a positive real number and \(|(\bar{a} \times \bar{b}) \times(\bar{c} \times \bar{d})|=5 \sqrt{114}\), then a value of \(p\) is

It is given that the discrete random variable is \(X \sim B(n, p)\) and \(P(X=2)=P(X=3)\).

The square root of independent term in the expansion of \(\left(\frac{2 x^2}{5}+\sqrt{\frac{5}{x}}\right)^{10}\) is

Choose the correct option about the matrices given below
\(\begin{aligned} & A=\left[\begin{array}{ccc}\cos \frac{\pi}{4} & \sin \frac{\pi}{4} & 0 \\ -\sin \frac{\pi}{4} & \cos \frac{\pi}{4} & 0 \\ 0 & 0 & 1\end{array}\right] \\ & B=\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & \cos \frac{\pi}{3} & \sin \frac{\pi}{3} \\ 0 & -\sin \frac{\pi}{3} & \cos \frac{\pi}{3}\end{array}\right] \\ & C=\left[\begin{array}{ccc}\cos \frac{\pi}{6} & 0 & \sin \frac{\pi}{6} \\ 0 & 1 & 0 \\ -\sin \frac{\pi}{6} & \cos \frac{\pi}{6} & 0\end{array}\right]\end{aligned}\)
\(D=\left[\begin{array}{ccc}
\cos \frac{\pi}{2} & \sin \frac{\pi}{2} & 0 \\
-\sin \frac{\pi}{2} & \cos \frac{\pi}{2} & 0 \\
0 & 0 & 1
\end{array}\right]\)

If \(\bar{a}, \bar{b}\) are two non-parallel unit vectors and the vector \(\alpha \bar{a}+\bar{b}\) bisects the internal angle between \(\bar{a}\) and \(\bar{b}\), then \(\alpha\) is equal to

Which of the following is / are ionic in nature?
(i) \(\mathrm{GeF}_4\) (ii) \(\mathrm{SnF}_4\) (iii) \(\mathrm{PbF}_4\)

A car is moving along a circular path having coefficient of friction 0.5 and radius of curvature \(16.2 \mathrm{~m}\). Then the maximum velocity of the car that can travel without falling outwards is
(Acceleration due to gravity \(=10 \mathrm{~ms}^{-2}\) )

If a point \(P\) moves such that its distances from the point \(A(1,1)\) and the line \(x+y+2=0\) are equal, then the locus of \(P\) is

The half life of a radioactive nuclide is

If a force \(F\) is applied on a body and it moves with a velocity \(v\), the power will be

Identify the starting compound ' \(\mathrm{P}\) ' in the following reaction.
\(\mathrm{P} \stackrel{\mathrm{Na} / \text { dry ether }}{\longrightarrow}\left(\mathrm{H}_3 \mathrm{C}\right)_2 \mathrm{CHCH}_2 \mathrm{CH}_2 \mathrm{CH}\left(\mathrm{CH}_3\right)_2\)