If A and B are two non-singular matrices of order n, then which of the following statement / statements is/are not correct?
(A) AB is non-singular.
(B) AB is singular.
(C) \((A B)^{-1}=A^{-1} B^{-1}\)
(D) \((A B)^{-1}\) does not exist.
Choose the correct answer from the options given below :

The area of the region bounded by the lines x = 2y + 3, x = 0, y = 1 and y = - 1 is:

The order and degree of the differential equation \(x\left(\frac{d y}{d x}\right)^3+\left(\frac{d^2 y}{d x^2}\right)^2-3 \frac{d y}{d x}+y=0\) respectively, are:

A box contains 3 white and 2 red balls. If we draw one ball and without replacing the first ball, the probability of drawing red ball in the second draw is:

Let A and B be square matrices of order 3, \(\left[\left(A-A^T\right)+\left(B-B^T\right)\right]\) then det is equal to

The vector equation of the line passing through the point \((-1,5,4)\) and perpendicular to the plane \(z=0\) is :

If \(C(x)=a x^ 2-bx-c\) represents the total cost function then the slope of the tangent to the marginal cost curve at the point \((x, y)\) is:

A measure of an average yearly growth of an investment over a certain time period when returns are reinvested is

An emf \( E = 200 \sin 300t \text{ volt} \) is applied across an inductor coil having a resistance of \( 1.0 \text{ } \Omega \). If the maximum current is found to be 10 A, then the value of inductance is approximately:

If \(z=3 x+4 y\) be the objective function of a linear programming problem (LPP) and (3,1), (2,4), (0,4), (5,0) be corner points of the bounded feasible region. Then the maximum value of objective function is

Organic compounds derived from NH3 by replacing one or more H-atoms with alkyl or aryl groups will form primary, secondary and tertiary amines. The common methods which include their preparations are reduction of nitro group, amide group and nitrile group, Gabriel phthalimide synthesis and ammonolysis of alkyl halides. Amines are basic in nature due to the presence of a lone pair of electrons on the nitrogen atom. Important chemical properties of amines include acylation, alkylation and their interaction with nitrous acid. There are various tests which can distinguish different types of amines from each other, for e.g. carbylamine test, Hinsberg's test, coupling reaction etc. Another important class of compounds which can be derived from primary amines is azo compounds. These compounds are intensely colored and widely used as dyes in textile, leather and food industries.
Correctly predict the final product and the type of reaction for the following reaction:
\(C _6 H _5- N _2^{+} Cl ^{-}+ C _6 H _5- OH \xrightarrow{ HO ^{-}}\)?

If \(y=e^{\frac{1}{2} \log \left(1+\tan ^2 x\right)}\), then \(\frac{d^2 y}{d x^2}\) is equal to :

A telescope has an eye piece of focal length 5.0 cm. The objective lens has focal length 12 times that of the eye-piece. In normal adjustment, what is the separation between the two lenses?