The value of the integral \(\int_2^4 \frac{x}{x^2+1} d x\) is :

If \(x=\frac{1-t}{1+t}\) and \(y=\frac{2 t}{1+t}\), then \(\frac{d^2 y}{d x^2}\) is equal to :

Pipe A can fill the tank 3 times faster than pipe B. If both pipes A and B running together can fill the tank in 15 minutes, then the time taken by pipe B alone to fill the tank is :

Identify correctly:
A. \(\begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 5 \\ 3 & 5 & 6 \end{bmatrix}\) is a symmetric matrix
B. \(\begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}\) is a null matrix
C. \(\begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{bmatrix}\) is an Identity matrix
D. \(\begin{bmatrix} 0 & 1 & 2 \\ -1 & 0 & 3 \\ -2 & -3 & 0 \end{bmatrix}\) is a skew symmetric matrix
E. \(\begin{bmatrix} \sqrt{3} & 0 & 0 \\ 0 & \sqrt{3} & 0 \\ 0 & 0 & \sqrt{3} \end{bmatrix}\) is a scalar matrix
Choose the correct answer from the options given below:

A company is shut down due to unavailability of electricity due to non payment of electricity Bill because of some unavoidable circumstances. Under which component of time series does this situation fall?

Find the value of k making the following function continuous at x = 5
\(f(x)=\left\{\begin{array}{ll}2 k x-1, & x \geq 5 \\ 2 x+3, & x<5\end{array}\right.\)

If \(A\) and \(B\) are matrices of same order, then ( \(A B^{ T }-B A^{ T }\) ) is always:

Match List-I with List-II

List-I	List-II
(A) Catalysts alter the rate of reaction	(I) cannot be fraction or zero
(B) Molecularity	(II) by lowering the activation energy
(C) Second half life of a first order reaction	(III) is the same as the first
(D) \(e^{-E_a / R T}\)	(IV) refers to the fraction of molecules with energy equal to or greater than activation energy

Choose the correct answer from the options given below:

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 :

A company has been producing steel tubes of mean inner diameter 2 cm.
A sample of 10 tubes gives a mean inner diameter of 2.01 cm and a variance of \(0.0004 cm^2\).
The value of the test statistic is :

The inverse of the matrix \(A=\left[\begin{array}{ll}2 & 3 \\ 1 & 2\end{array}\right]\) is:

A compound is formed by 2 elements P and Q. Atoms of element Q (as anions) makes ccp and those of P (as cations) fill all tetrahedral voids and also occupy half of the octahedral voids. What is the correct formula of this compound?

If \(\left|\begin{array}{lll}a+x & a-x & a-x \\ a-x & a+x & a-x \\ a-x & a-x & a+x\end{array}\right|=0\), then the values of \(x\) are: