Match List-I with List-II (where \(c\) is an arbitrary constant)

List-I	List-II
(A) \(\int \tan x d x\)	(I) \(\log |\sec x+\tan x|+c\)
(B) \(\int \cot x d x\)	(II) \(\log |\sec x|+c\)
(C) \(\int \sec x d x\)	(III) \(\log |\sin x|+c\)
(D) \(\int \operatorname{cosec} x d x\)	(IV) \(\log |\operatorname{cosec} x-\cot x|+c\)

Choose the correct answer from the options given below :

The least non-negative remainder, when \(5^{61}\) is divided by 7 , is

Which of the following statements are TRUE?
(A) The variable \(t\) of \(t\)-distribution ranges from \(-\infty\) to \(\infty\).
(B) The probability curve of the \(t\)-distribution is symmetric about the line \(t=0\)
(C) The variance of the \(t\)-distribution is greater than one.
(D) As the number of degrees of freedom decreases, the \(t\)-distribution curve moves closer to the standard normal probability curve.
Choose the correct answer from the options given below :

If \(A=\left(\begin{array}{cc}1 & 0 \\ -1 & 7\end{array}\right), I=\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right)\), and \(A^2=8 A+K I\) the value of \(K\) is :

The general solution of the differential equation \(y d x-\left(x+2 y^2\right) d y=0\) is:

The function \(f:[0, \infty) \rightarrow R\) defined by, \(f(x)=2 x^2+3\), is:

If \(A=\left[\begin{array}{ccc}\cos \theta & \sin \theta & 0 \\ -\sin \theta & \cos \theta & 0 \\ 0 & 0 & 1\end{array}\right]\) and B is a square matrix of order 3, then |AB| is equal to:

Which of the following statements is incorrect?

The volume of a cube is increasing at a rate of 9 cubic centimeter per second. Then rate of change of surface area when the length of an edge is 10 centimeter is:

An LCR series circuit with L = 100 mH, C = 100 \( \mu \)F, R = 10 \( \Omega \) is connected to an AC source of emf 220 V. The resonant frequency of the circuit will be nearly

Read the following passage given below and answer the question.
Aldehydes and ketones undergo nucleophilic addition reactions. A nucleophile attacks the electrophilic carbon atom of polar carbonyl group from perpendicular to plane of \(sp^2\) hybridised orbitals of carbonyl carbon. The hybridisation of carbon changes from \(sp^2\) to \(sp^3\) and a tetrahedral alkoxide intermediate is formed. This captures \(H^+\) to give neutral product. In carboxylic acids, carbonyl carbon is less electrophilic than carbonyl carbon in aldehydes and ketones.How will you convert acetaldehyde to lactic acid ?

In double-slits experiment using light of wavelength 550 nm, the angular width of a fringe formed on a distant screen is 0.2°, What is the spacing between the two slits?

"Answer the question on basis of passage given below:
\(G = \frac{\kappa A}{l}\)=k (both \(A\) and \(l\) are unity in their appropriate units in m or cm)
Molar conductivity of a solution at a given concentration is the conductance of the volume \(V\) of solution containing one mole of electrolyte kept between two electrodes with area of cross section \(A\) and distance of unit length (\(l\)). Therefore,
\(\Lambda_m = \frac{\kappa A}{l}\)
Since \(l = 1\) and \(A = V\) (volume containing 1 mole of electrolyte),
\(\Lambda_m = \kappa V\)
If the length \(l\) in the column of an electrolytic cell is halved and its area of cross-section is tripled, then conductance (\(G\)) would become:"