If the random variable X has the following probability distribution :

X	0	1	2	Otherwise
P(X)	K	3K	5K	0

then \(K\) is equal to

The value of the integral \(\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left(x+x^3+x^5\right) d x\) is:

Probabilities to solve a specific problem by A, B and C are \(\frac{1}{2}, \frac{1}{3}\) and \(\frac{1}{4}\ respectively. Probability that at least one will solve the problem is:

Let \(A=\left[\begin{array}{ccc}1 & \sin \theta & 1 \\ -\sin \theta & 1 & \sin \theta \\ -1 & -\sin \theta & 1\end{array}\right]\),where \(0 \leq \theta \leq 2 \pi\).
Then \(\operatorname{det}(A)\) lies in the interval :

Match List-I with List-II (where \(c\) is an arbitrary constant)

List-I	List-II
(A) \(\int \tan x d x\)	(I) \(\log |\sec x+\tan x|+c\)
(B) \(\int \cot x d x\)	(II) \(\log |\sec x|+c\)
(C) \(\int \sec x d x\)	(III) \(\log |\sin x|+c\)
(D) \(\int \operatorname{cosec} x d x\)	(IV) \(\log |\operatorname{cosec} x-\cot x|+c\)

Choose the correct answer from the options given below :

The solution X of the matrix equation AX = B, where A is an invertible matrix, is given by

If \(A=\left[\begin{array}{ll}1 & 2 \\ 0 & 3\end{array}\right]\), then \(|A \operatorname{adj} A|\) is

Arrange the following steps regarding HIV infection in correct sequence
A. Viral RNA produced by the infected cell
B. Virus infects macrophages.
C. Viral DNA is produced by reverse transcriptase.
D. Viral RNA is introduced into the cell.
E. Viral DNA incorporates into the host genome.
Choose the correct answer from the options given below:

Equation of a line is \(\frac{x-3}{2}=\frac{4-y}{3}=\frac{2-z}{4}\) then its direction cosines are:

Read the passage carefully and answer the questions
The osmotic pressure is one of the four colligative properties which depends on the number of particles of solute in the solution, irrespective of their nature. The osmotic pressure is equal to the product of concentration, gas constant and temperature. It is the most commonly used property for the determination of the molar mass of biomolecules. In many situations, the molecules get dissociated or associated when they are dissolved in a solvent, thereby changing the number of molecules in the solution. As a result, an abnormal molecular mass is obtained. The extent of dissociation or association is given by van't Hoff factor (i), which is the ratio of normal molar mass to abnormal molar mass. It is related with the degree of association or ionization. The introduction of van't Hoff factor in the osmotic pressure equation modifies the equation.
How is the osmotic pressure of a concentrated solution related to the osmotic pressure of a dilute solution of the same substance at a given temperature?

A and B throw a die alternatively till one of them gets a number more than 4 and wins the game. Then the probability of winning the game by B , if A starts first :

Which of the following is a barrier method of contraception?

Match List-I with List-II

List-I	List-II
(A) Low mass number nuclei	(I) Medium penetrating power
(B) Electron microscope	(II) Nuclear reactor
(C) Beta decay	(III) Diffracted electron
(D) \( {}^{235}_{92}\text{U} \)	(IV) Fusion

Choose the correct answer from the options given below :