If the objective function Z = px + qy is maximum at (2,-2) and the maximum value is 8, such that p = 3q, then the values of p and q respectively are:

The refractive index of the material of an equilateral prism is \(\sqrt {2}\). The angle of minimum deviation of that prism is

The probability that a leap year selected at random will have 53 Mondays is

The general solution of the differential equation \(ydx + xdy = 0\) is:

If the vectors \(\vec{a}=3 \hat{i}-p \hat{j}+5 \hat{k}\) and \(\vec{b}=-6 \hat{i}+14 \hat{j}+q \hat{k}\) are collinear, then the value of \(p\) and \(q\) are :

If the elementary column transformation \(C_1 \rightarrow 2 C_1\) is carried out on the following matrix equation.
\(\left[\begin{array}{ll}1 & 3 \\ 2 & 4\end{array}\right]\left[\begin{array}{cc}-1 & 2 \\ 5 & 0\end{array}\right]=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]\) Then the transformed equation will be:

Area bounded by the curve \(y=\cos ^{-1} x\) and the lines \(x=0, x=1\) is:

Amniocentesis should not be used for:

A long solenoid carrying a current produces a magnetic field B along its axis. If the value of current is halved and the number of turns per unit length is doubled then the new value of magnetic field is

Read the following passage carefully and answer the given questions.
Productivity in the Ecosystem The flow of energy through a terrestrial ecosystem starts with the harnessing of sunlight by autotrophs. The radiant energy is converted by photosynthesis to organic compounds is known as productivity. The productivity is expressed in units of energy per unit of area per unit time. The amount of organic matter present at any given time is biomass. The productivity of any ecosystem is affected by many parameters like weather conditions, nutrient availability etc. The productivity of any ecosystem does vary seasonally and yearly.
The Net Primary Productivity of an ecosystem are represented by:

If \(y=e^{a \cos ^{-1} x},-1< x< 1\), then \(\left(1-x^2\right) \frac{d^2 y}{d x^2}-x \frac{d y}{d x}\) is equal to

The probability distribution of a random variable X is

X	0	1	2	3	4
P(X)	0.2	k	k	2k	k

Match List-I with List-II

List-I	List-II
(A) value of \(k\)	(I) \(\frac{16}{25}\)
(B) \(P(x \geq 2)\)	(II) \(\frac{9}{25}\)
(C) \(P(X=3)\)	(III) \(\frac{4}{25}\)
(D) \(P(X<2)\)	(IV) \(\frac{8}{25}\)

Choose the Correct answer from the options given below :

To remove an electron from the first orbit to infinity of hydrogen atom, the amount of energy required will be :