The corner points of the bounded feasible region determined by a set of constraints (linear inequalities) are \(A(0,5), B(3,5), C(5,0), D(4,1)\) and the objective function is \(z=p x+2 q y\) where \(p, q>0\).
The condition on \(p\) and \(q\) such that the maximum \(z\) occurs at B and D , is

\(\int \sin x \sin 2 x \sin 3 x d x\) is equal to:

If the corner points of the bounded feasible region for a linear programming problem (LPP) are (0, 2), (3, 0), (6, 0), (6, 8) and (0, 5), then which of the following are correct for the objective function \(Z=4 x+6 y \) ?
(A) The minimum value of the objective function occurs at (0, 2) and (3, 0) only.
(B) The minimum value of the objective function occurs at the mid-point of the line segment joining the points (0, 2) and (3, 0) only.
(C) The minimum value of the objective function occurs at every point of the line segment joining the points (0, 2) and (3,0).
(D) The difference between the maximum value and minimum value of the objective function is 60.
Choose the correct answer from the options given below :

The area of the region bounded by the curve y = x + 1, x-axis and the lines x = 2 and x = 3 is

The probability distribution of a random variable X is given below:
\(\begin{array}{lllll} X & 0 & 1 & 2 & 3 \\ P ( X ) & k & & \\ & & \frac{k}{2} & \frac{k}{4} & \frac{k}{8}\end{array}\)
then the value of \(P(X \leq 2)+P(X>2)\) is :

Match List-I with List-II

List-I	List-II
(A) Degree of this differential equation \(\frac{d^4 y}{d x^4}+2 \log _e\left(\frac{d^3 y}{d x^3}\right)=0\)	(I) 1
(B) Order of this differential equation \(e^{\left(\frac{dy}{dx}\right)^3}+3 y\left(\frac{d^2 y}{d x^2}\right)^3=0\)	(II) 4
(C) Degree of \(\frac{d^4 y}{d x^4}+\left(\frac{d y}{d x}\right)^2=0\)	(III) not defined
(D) Order of the differential equation \(2 \frac{d^4 y}{d x^4}+\left(\frac{d^2 y}{d x^2}\right)^5=0\)	(IV) 2

Choose the Correct answer from the options given below :

If A is a skew-symmetric matrix of order \(n \times n\) and \(a_{i j}\) is the \((i, j)^{\text {th }}\) elements, then :

The properties which depend on the number of solute particles irrespective of their nature relative to the total number of particles in the solution are called colligative properties. These are relative lowering of vapour pressure of the solvent, depression of freeaing point of the solvent, elevation of boiling point of the solvent and osmotic pressure of the solution. All of these properties help in calculating the observed molar mass of the solute.
Determination of osmotic pressure is considered best for the determination of molecular mass of the solute.
Which ssolution has the highest boiling point ?

Arrange the following steps of retrovirus replication sequentially.
A. Production of viral DNA by reverse transcriptase
B. Introduction of viral RNA in host cell
C. Production of new viral RNA by host cell
D. Incorporation of viral DNA into host genome
Choose the correct answer from the options given below:

The particle that can not be accelerated by van de Graaff generator is :

Solubility of a gas in a liquid:

Which of the following statements is INCORRECT about Genetic Engineering Approval Committee?

Which of the following cannot be explained by the open chain structure of glucose?