If the area of an equilateral triangle is increasing at the rate of \(4 \sqrt{3} cm^2 / sec\), then the rate of increase of its perimeter when the side is 4 cm , is

The value of \(\left|\begin{array}{ccc}7! & 8! & 9! \\ 8! & 9! & 10! \\ 9! & 10! & 11!\end{array}\right|\) is :

In the following matrix equation, the value of \(x\) is :
\(\left[\begin{array}{cc}x+y & 4 \\ -5 & 3 y\end{array}\right]=\left[\begin{array}{cc}3 & 4 \\ -5 & 6\end{array}\right]\)

For the following probability distribution,

X	1	2	3	4
p(x)	\(\frac{1}{10}\)	\(\frac{1}{5}\)	\(\frac{3}{10}\)	\(\frac{2}{5}\)

The value of \(E\left(X^2\right)-[E(X)]^2\)

If \(A=\left[\begin{array}{cc}1 & 0 \\ -1 & 7\end{array}\right]\) and \(A^2-8 A+K I=O\), then the value of \(K\) is:

The maximum value of \(\frac{\log x^3}{3 x}\) occurs at \(x=\)

The effective rate, which is equivalent to a nominal rate of \(12 \%\) compounded semi-annually, is

Select the appropriate reagent for the synthesis of tert-Butyl ethyl ether

An ordinary light of intensity \(I_0\) is passed through a polaroid \(P_1\) which is crossed with another polaroid \(P _{ 2 }\). Now a third polaroid is placed in between \(P _1\) and \(P _2\) with its pass aris making an angle \(\beta\) with the pass axis of \(P_1\). The value of \(\beta\) for which the intensity of light from \(P_2\) is \(\frac{I_0}{8}\) is :

Read the passage carefully and answer
When a metal is placed in a solution of its ions, the metal can lose electron and can go in the solution as ion or the metal ion in the solution can take electron from the electrode and get deposited as metal. Thus, the electrode acquires either positive or negative charge with respect to the solution leading to the development of a potential difference between the metal electrode and the solution. This potential difference is called the electrode potential of the metal. The electrode potential of a metal is a measure of relative tendency to undergo oxidation (loss of electron) or reduction (gain of electron). When the concentration of the ions in the solution is unity, the electrode potential is called standard electrode potential. The magni of potential depends upon the nature of electrode, concentration of ions in solution and the temperature. In a galvanic cell, the electrode at which oxidation occurs is known as anodic half-cell and the electrode at which reduction occurs is known as cathodic half cell. The emf of the cell in terms of standard reduction potential is given by \(E_{\text {cell }}^{\circ}=E_{\text {cathode }}^{\circ}-E_{\text {anode. }}^{\circ}\).The standard free energy change of the redox reaction taking place in the cell is related to \(E_{\text {cell }}^{\circ}\) by \(\Delta G^{\circ}=-n F E^{\circ}\). A redox reaction would occur spontaneously if the free energy change is negative. The equilibrium constant of the reaction is related to the standard free energy change by \(\Delta G^{\circ}=-R T \ln K\).
The half call reaction is the one that

Match List-I with List-II
Consider the function \(f(x)=2 x^3-21 x^2+36 x+80, x \in[0,6]\).

List-I	List-II
(A) one of its critical points is at \(x=\)	(I) \(-28\)
(B) Its absolute maximum value is	(II) \(-42\)
(C) Its absolute minimum value is	(III) 97
(D) Its second derivative at \(x=0\)	(IV) 6

Choose the correct answer from the options given below :

An object is placed before a convex lens and its real image is obtained. Now if half of the surface is covered with paper. The result will be :

Pipe A can fill a tank in 24 hours and pipe B in 32 hours. If both the pipes are opened in an empty tank at the same time, then the time taken to fill the tank is: