The function \(f(x)=2 x^3-9 a x^2+12 a^2 x+1(a>0)\) attains its maximum and minimum at \(p\) and \(q\) respectively and \(p^2=q\). Then, \(a=\)

The values of \(\theta\), for which \(\frac{3+2 i \sin \theta}{1-2 i \sin \theta}\) is real are

If \(x_1, x_2, \ldots x_n\) are ' \(n\) ' observations and \(x\) is their mean. If \(\sum_{i=1}^n\left(x_1-\bar{x}\right)^2\) is almost zero, then a true statement among the following is

If \(\alpha, \beta\) and \(\gamma\) are length of the altitudes of a \(\triangle A B C\) with area \(\Delta\), then \(\frac{\Delta^2}{R^2}\left(\frac{1}{\alpha^2}+\frac{1}{\beta^2}+\frac{1}{\gamma^2}\right)\) is equal to

If \(A=\left[\begin{array}{ccc}1 & -3 & 2 \\ -2 & 1 & 3 \\ 3 & 2 & -1\end{array}\right]\) then \(A^2 \operatorname{Adj} A=\)

The number of solutions of the equations \(x+y+z=1 ; x^2+y^2+z^2=1 ; x^3+y^3+z^3=1\) is

Let \(\alpha \neq 1\) be a real root of the equation \(x^3-a x^2+a x-1=0\), where \(a \neq-1\) is a rea number. Then, a root of this equation, among the following, is

The growth of fish gets inhibited, if the concentration of dissolved oxygen is below \(\underline{X}\) ppm. The value of \(\mathrm{X}\) is

Observe the following statements :
a. The thermal stability of hydrides of group 16 elements follow the order :
\(\mathrm{H}_2 \mathrm{O}>\mathrm{H}_2 \mathrm{~S}>\mathrm{H}_2 \mathrm{Se}>\mathrm{H}_2 \mathrm{Te}\)
b. Acidic nature of hydrides of group 16 elements follow the order :
\(\mathrm{H}_2 \mathrm{O}>\mathrm{H}_2 \mathrm{~S}>\mathrm{H}_2 \mathrm{Se}>\mathrm{H}_2 \mathrm{Te}\)
c. The reducing nature of \(\mathrm{H}_2 \mathrm{~S}, \mathrm{H}_2 \mathrm{Se}\) and \(\mathrm{H}_2\) Te follow the order :
\(\mathrm{H}_2 \mathrm{~S} < \mathrm{H}_2 \mathrm{Se} < \mathrm{H}_2 \mathrm{Te}\)
The correct statement are:

The sum of bond order values of \(\mathrm{C}_2\) and \(\mathrm{O}_2^{2+}\) is \(x\), which is equal to sum of bond order values of \(\mathrm{a}, \mathrm{b}\) and c. What are \(\mathrm{a}, \mathrm{b}\) and c ?

A bullet of mass \(m\) is fired from a rifle of mass \(M\). If \(\mathbf{v}\) be the velocity of the bullet, velocity acquired by the rifle is

A ray of light incidents at an angle of \(60^{\circ}\) on the first face of a prism. The angle of the prism is \(30^{\circ}\) and its second face is silvered. If the light ray inside the prism retraces its path after reflection from the second face, then the refractive index of the material of the prism is

The displacement of a particle in an oscillatory motion at a time ' \(t\) ' is given by \(\mathrm{x}=8 \sin \frac{\pi \mathrm{t}}{4} \mathrm{~cm}\), then its displacement in the time interval \(\mathrm{t}=0 \mathrm{~s}\) to \(\mathrm{t}=2 \mathrm{~s}\) is