If \(\tan ^{-1}(2 x)+\tan ^{-1}(3 x)=\frac{\pi}{4}\), then the value of \(x\) is:

Match List-I with List-II

List-I (Equationof Lines)	List-II (Information)
(A) \(\vec{r}=(3 \hat{i}-2 \hat{j}+\hat{k})+\lambda(\hat{j}-2 \hat{k})\)	(I) Direction ratios are \(2,4,-1\)
(B) \(\frac{2-x}{1}=\frac{2 y+1}{4}, z=2\)	(II) Perpendicular to \(2 \hat{i}-\hat{j}+\hat{k}\)
(C) \(\frac{x}{1}=\frac{y-3}{2}=\frac{3-4 z}{2}\)	(III) Passing through the point \((3,-2,1)\)
(D) \(\vec{r}=(3 \hat{i}+2 \hat{j}+\hat{k})+\lambda(2 \hat{i}+\hat{j}-3 \hat{k})\)	(IV) Direction ratios are \(-1,2,0\)

Choose the correct answer from the options given below :

The probability distribution function of a normal variate with mean \(\mu\) and variance \(\sigma^2\) is given by :
\(f(x)=\frac{1}{\sigma \sqrt{2 \pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2},-\infty< x< \infty,-\infty < \mu < \infty, \sigma>0\)
If \(y=f(x)\) be the normal probability curve, then which of the following is correct?
(A) The normal curve is symmetrical about the line \(x=\mu\).
(B) Mean, median and mode of the distribution coincide.
(C) \(Y\)-axis is an asymptote to the normal curve.
(D) If \(x\) increases numerically, \(f(x)\) decreases rapidly.
Choose the correct answer from the options given below :

If an investment value gets doubled in 10 years then its compound annual growth rate (CAGR) is:
(Given \(2^{1 / 10}=1.0729\) )

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.

If \(y=\frac{1}{2} x \sqrt{a^2-x^2}+\frac{a^2}{2} \sin ^{-1}\left(\frac{x}{a}\right)\), then \(\frac{d y}{d x}=\)

The value of integral \(I=\int \frac{1-\sin x}{\cos ^2 x} d x\) is:

If the system of equations
\(x-3 y+5 z=3\)
\(x-2 y+4 z=4\)
\(2 x-7 y+\lambda z=5\)
has infinite number of solutions, then the value of \(\lambda\) is:

The relation R in the set {1, 2, 3} given by R = {(1,1), (2, 2), (3, 3), (1, 2), (1, 3), (2, 3)} is

Which one of the following statements is incorrect about ecological pyramids?

Match List I with List II

List I (Drug)	List ll (Plant)
(A) Morphine	(I) Tobacco
(B) Cannabinoids	(II) Erythroxylum coca
(C) Cocaine	(III) Papaver somniferum
(D) Nicotine	(IV) Cannabis sativa

Choose the correct answer from the options given below:

When the angle of incidence on a material is \(60^{\circ}\), the reflected light is completely polarised. The velocity of the refracted ray inside the material in cm/s is :

In the equation \( P = \frac{E}{c} \), where \( P \) is momentum, \( E \) is energy and \( c \) is the speed of light, this equation is valid in which of the following cases: