If the roots of the equation $\sqrt{\frac{x}{1-x}}+\sqrt{\frac{1-x}{x}}=\frac{5}{2}$ are $p$ and $q\left(p>q\right)$ and the roots of the equation $\left(p+q\right)x^4-pq{x}^2+\frac{p}{q}=0$ are $\alpha , \beta , \gamma , \delta$, then ${\left(\Sigma \alpha \right)}^2-\Sigma \alpha \beta +\alpha \beta \gamma \delta =$

Let \(a, b, c\) be three vectors such that \(\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{a}}=\overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{b}}=\overrightarrow{\mathrm{c}} \cdot \overrightarrow{\mathrm{c}}=5\) and \(|\vec{a}+\vec{b}-\vec{c}|^2+|\vec{b}+\vec{c}-\vec{a}|^2+|\vec{c}+\vec{a}-\vec{b}|^2=50\) then \(\vec{a} \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a}=\)

A plane \(\Pi\) is passing through the points \(A=(0,0,2), B=(1,0,1)\) and \(C=(\beta, 1,1)\). If the plane \(\Pi\) makes angle \(\alpha\) and \(\beta\) with the \(X Y\) and \(X Z\)-coordinate planes respectively, then \(\sin ^2 \alpha+\sin ^2 \beta=\)

If \(X\) is a poisson variate such that \(\alpha=P(X=1)=P(X=2)\), then \(P(X=4)\) is equal to

Let \(\overrightarrow{\mathbf{a}}=a_1 \hat{\mathbf{i}}+a_2 \hat{\mathbf{j}}+a_3 \hat{\mathbf{k}}\) Assertion (A) : The identity \(|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{i}}|^2+|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{j}}|^2+|\overrightarrow{\mathbf{a}} \times \hat{\mathbf{k}}|^2=2|\overrightarrow{\mathbf{a}}|^2\) holds for \(\vec{a}\). Reason (R) : \(\overrightarrow{\mathbf{a}} \times \hat{\mathbf{i}}=a_3 \hat{\mathbf{j}}-a_2 \hat{\mathbf{k}}\), \(\overrightarrow{\mathbf{a}} \times \hat{\mathbf{j}}=a_1 \hat{\mathbf{k}}-a_3 \hat{\mathbf{i}}, \overrightarrow{\mathbf{a}} \times \hat{\mathbf{k}}=a_2 \hat{\mathbf{i}}-a_1 \hat{\mathbf{j}}\) Which of the following is correct?

If \(|\vec{a}|=1,|\vec{b}|=2,|\vec{a}-\vec{b}|^2+|\vec{a}+2 \vec{b}|^2=20\) then \((\vec{a}, \vec{b})=\)

The radius of a circular plate is increasing at the rate of \(0.01 \mathrm{~cm} / \mathrm{s}\) when the radius is \(12 \mathrm{~cm}\). Then, the rate at which the area increases, is

A ball is let fall from a height \(h_0\). It makes \(n\) collisions with the earth. After \(n\) collisions it rebounds with a velocity \(v_n\) and the ball rises to a height \(h_n\), then coefficient of restitution is given by

The number of sigma and pi \((\pi)\) bonds present in benzene respectively are :

Momentum before collision,
\(
p_1=4 m v_1-2 m v_2
\)

Momentum after collision,
\(
p_2=6 m\left(-5 v_2\right)=-30 m v_2
\)


Conservation of momentum gives,
\(
\begin{gathered}
p_1=p_2 \\
\Rightarrow \quad 4 m v_1-2 m v_2=-30 m v_2 \quad \Rightarrow 4 v_1=-32 v_2 \\
\Rightarrow \quad \frac{v_1}{v_2}=\frac{-32}{4}=-8 \text { or }\left|\frac{v_1}{v_2}\right|=8
\end{gathered}
\)

What are the suitable conditions for the following transformation?

If an angular bisector of the coordinate axes is one of the lines of \(x^2+2 a x y+3 y^2=0\), then sum of all possible values of \(a\) is

Consider the following statements \(A\) and \(B\) and identify the correct answer A. In an elastic collision, if a body suffers a head on collision with another of same mass at rest, then the first body comes to rest while the other starts moving with the velocity of the first one. B. Two bodies of equal mass suffering a head on elastic collision merely exchanges their velocities