The simplest form of \(\tan ^{-1}\left(\frac{\sqrt{1+x^2}-1}{x}\right), x \neq 0\) is:

The solution of the differential equation \(\frac{d y}{d x}=\frac{6}{x^2} ; y(1)=3\) is :

Suppose \(10 \backslash \%\) of men and \(0.5 \backslash \%\) of women have grey hair. A grey haired person is selected at random. If there are equal number of males and females, then the probability of the selected person being a female is :

Match List I with List II

LIST I	LIST II
A. \(\int \sin x \cos x d x\)	I. \(\log _e\left(1+\sin ^2 x\right)+C\)
B. \(\int \frac{d x}{\sin ^2 x \cos ^2 x}\)	II. \(\tan \frac{x}{2}+C\)
C. \(\int \frac{\sin 2 x}{1+\sin ^2 x} d x\)	III. \(-\frac{1}{4} \cos 2 x+C\)
D. \(\int(1-\cos x)\operatorname{cosec}^2 x d x\)	IV. \(\tan x-\cot x+C\)

If \(A=\left[\begin{array}{cc}2 & -3 \\ -4 & 7\end{array}\right]\) and \(2 A^{-1}=K I-A\), where \(K\) is a real number and \(I\) is the identity matrix of order 2 , then the value of \(K\) is :

A parallelogram is constructed on the vectors \(\vec{a}=3 \vec{\alpha}-\vec{\beta}, \vec{b}=\vec{\alpha}+3 \vec{\beta}\). If \(|\vec{\alpha}|=|\vec{\beta}|=2\) and the angle between \(\vec{\alpha}\) and \[\vec{\beta}\] is \(\frac{\pi}{3}\), then length of the diagonal of the parallelogram is:

Match List-I with List-II

List-I	List-II
(A) If A is a non singular matrix of order n, then \(| A (\operatorname{adjA})|\) is equal to	(I) \(|A|^{n-1}\)
(B) If A is a non singular matrix of order n, then |adj(adj A)| is equal to	(II) \(|A|^{n-2} A\)
(C) If A is a non singular matrix of order n, then \(\operatorname{adj}(\operatorname{adj} A )\) is equal to	(III) \(|A|^n\)
(D) if A is a non singular matrix of order n, then \(|(\operatorname{adj} A)|\) is equal to	(IV) \(|A|^{(n-1)^2}\)

Choose the correct answer from the options given below :

When an electron of hydrogen atom is raised from the ground state to an excited state.

The magnetic field at a distance r from a long wire carrying current I is 0.4 T. The magnetic field at a distance 2r from the wire is

What will be the product of the following reaction :
\(C_6H_5ONa + C_2H_5Cl \rightarrow ?\)

Read the passage carefully and answer the questions based on the passage:
Pollen pistil interaction involves all events from the landing of pollen grains on the stigma until the pollen tube enters the embryo sac (when the pollen is compatible). When a pollen tube grows through the style and enters into the ovules, it finally discharges two male gametes in one of the synergids. Syngamy and triple fusion are two fusion events occur in angiosperms. Thus, angiosperms exhibit double fertilization. The products of these fusions are the diploid zygote and triploid primary endosperm nucleus. Zygote develops into embryo and primary endosperm cell forms the endosperm tissue. The developing embryo passes through different stages before maturation.
The coconut water from tender coconut, a good source of nutrition is nothing but:

Introduction of which of the following on Galopagos islands caused extinction of Abingdon tortoise ?

In an electromagnetic wave, the ratio of energy densities of electric and magnetic fields is :