Let \(x=t^2, y=t^3\). Then \(\frac{d^2 y}{d x^2}\) is equal to

The value of \(\tan ^{-1} 1+\cos ^{-1}\left(-\frac{1}{2}\right)+\sin ^{-1}\left(-\frac{1}{2}\right)\) is

For the differential equation \((x+y) d y+(x-y) d x=0\), which of the following is/are correct?
(A) Differential equation is homogeneous
(B) Order of differential equation is 1
(C) Integrating factor of differential equation is \(e ^{ x }\)
(D) Degree of the equation is not defined
Choose the correct answer from the options given below:

When two independent small samples of sizes \(n_1\) and \(n_2\) with means \(\bar{x}_1\) and \(\bar{x}_2\) respectively are drawn from populations with identical population variances, the test-statistic is computed as :

A unit vector perpendicular to the vectors \(\hat{i}-\hat{j}\) and \(\hat{i}+\hat{j}\) is

\(\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{d x}{1+\cos 2 x}\) is equal to

The general solution of the differential equation \(\frac{d y}{d x}=x y+x+y+1\) is

The higher stability of metal carbonyls is very evident on the basis of spectrochemical series. Identify the incorrect statement(s) about metal carbonyls.
A. The metal-carbon bond has both \(\sigma\) and \(\pi\) characters.
B. Metal-carbon \(\sigma\) bond is formed by donation of lone pair of electrons on carbonyl carbon into vacant \(d\)-orbital of metal.
C. Metal-ligand bond has synergic effect.
D. Transition metals never have zero oxidation state in carbonyls.
E. Metal-carbon \(\pi\) bond is formed by donation of electrons from antibonding molecular orbital of CO to metal.
Choose the correct answer from the options given below :

The Hardy - Weinberg Principle says:
(A) Allele frequency in a population is stable and constant.
(B) Allele frequency in a population is unstable.
(C) The gene pool remains constant. It is called genetic equilibrium.
(D) The sum total of all the Allelic frequencies is 1.
Choose the correct answer from the options given below:

The mean number of heads in three tosses of a fair coin is:

In a reaction A and B reacts to form product. The initial rate of reaction was determined using different initial concentration of A and B.

A/molL-1	0.10	0.10	0.20
\(B / mol L ^{-1}\)	0.30	0.10	0.30
\(r_o / mol L ^{-1} s^{-1}\)	\(6.81 \times 10^{-4}\)	\(2.27 \times 10^{-4}\)	\(13.62 \times 10^{-4}\)

What is the initial rate of reaction (\(r_o\)) when critical concentration of A and B is \(0.50\ mol/L\) and \(0.50\ mol/L\) respectively.

Identify the correct statements for Ethylene dichloride and Ethylidene chloride
(A) They represent structural isomers.
(B) Both are formed by reaction of ethene with chlorine in presence of\(CCl _4\)
(C) Both the compounds belong to a class of dihalocompounds
(D) Both the compounds are optically inactive.
Choose the correct answer from the options given below:

\[\int e^x\left(1+x^2\right) d x=\]