A random variable X has the following probability distribution where k is some number :

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

Find the value of k.

Consider the equation of the line \(\vec{r}=-\hat{i}+2 \hat{k}+\mu(4 \hat{i}-\hat{j}+2 \hat{k})\).
Match List-I with List-II

List-I	List-II
(A) It passes through the point	(I) \(4,-1,2\)
(B) Its direction ratios are	(II) \(\frac{4}{\sqrt{21}}, \frac{-1}{\sqrt{21}}, \frac{2}{\sqrt{21}}\)
(C) Its Cartesian form is	(III) \((-1,0,2)\)
(D) Its direction cosines are	(IV) \(\frac{x+1}{4}=\frac{y}{-1}=\frac{z-2}{2}\)

Choose the correct answer from the options given below :

If \(\left|\begin{array}{ccc}1 & -2 & 5 \\ 2 & a & -1 \\ 0 & 4 & 2a\end{array}\right|=86\), then product of all values of \(a\) is :

If \(\left[\begin{array}{lll}1 & 2 & 1\end{array}\right]\left[\begin{array}{lll}1 & 2 & 0 \\ 2 & 0 & 1 \\ 1 & 0 & 2\end{array}\right]\left[\begin{array}{l}0 \\ 2 \\ x\end{array}\right]=0\), then value of x is

For which values of \(a, f(x)=a(x+\sin x)\) is an increasing function?

The interval in which the function \(g(x)=x^2 e^{-x}\) is increasing is :

\(x\)-axis, \(y=\cos x\) and \(y=\sin x, 0 \leq x \leq \frac{\pi}{2}\) is:

Read the passage and answer the question given below.
When a host is exposed to antigens, which may be in the form of living or dead microbes or other proteins, antibodies are produced in the host body. This type of immunity is called active immunity. Active immunity is slow and takes time to give its full effective response. Injecting the microbes deliberately during immunisation or infectious organisms gaining access into body during natural infection induce active immunity. When antibodies are directly given to protect the body against foreign agents, it is called passive immunity.
Match List I with List II

LIST I	LIST II
A. Physical Barrier	I. Saliva
B. Cellular Barrier	II. Interferons
C. Physiological Barrier	III. Skin
D. Cytokine Barrier	IV. Polymorpho-nuclear leukocytes

Choose the correct answer from the options given below:

"Toddy" is a traditional drink which is made by-

A conducting ring of radius r in a magnetic field varying at rate x has electric field intensity at any ring point as :

Four point charges -2Q, +q, -2q and +Q are placed at four corners of a square A, B, C and D, respectively, as shown in the figure and OA = OB = \( a/\sqrt{2} \). If the potential at the center of a square is zero, what is the relation between Q and q?

Match List I with List II

LIST I	LIST II
A. \(\lim_{x \to 0} \frac{(1 - \cos 2x) \sin 5x}{x^2 \sin 3x}\)	I. 18
B. \(\lim_{x \to \infty} \frac{(3x-5)(2x-7)}{(4x-9)(5x-3)}\)	II. \(\frac{10}{3}\)
C. \(\lim_{x \to 0} \frac{2 \sin^2 3x}{x^2}\)	III. \(\frac{3}{4}\)
D. \(\lim_{x \to \frac{\pi}{4}} \frac{1 - \cos^3 x}{2 \cot x - \cot^3 x}\)	IV. \(\frac{3}{10}\)

Choose the correct answer from the options given below:

Choose the correct statement/statements for Cannizaro reaction.
A. Takes place in aldehydes and ketones having absence of \(\alpha\)-hydrogen
B. Takes place in aldehydes having absence of \(\alpha\)-hydrogen
C. It is a disproportionation reaction
D. Get a mixture of carboxylic acid salt and ethers
Choose the correct answer from the options given below: