Let \(A=\left[a_{i j}\right]_{n \times n}\) be a matrix, then match List-I with List-

List-I	List-II
(A) \(|A|=0\)	(I) A is a symmetric matrix
(B) \(|A| \neq 0\)	(II) A is a skew-symmetric matrix
(C) \(A^T=A\)	(III) A is a singular matrix
(D) \(A^T=-A\)	(IV) A is a non-singular matrix

Choose the correct answer from the options given below:

\(\int_2^5|x-3| d x\) equals

Let \(\theta\) be the angle between two vectors \(\vec{a}\) and \(\vec{b}\). Then match List-I with List-II

List-I	List-II
(A) \(\sin \theta\)	(I) \(\frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}\)
(B) \(\cos \theta\)	(II) \(|\vec{a} \times \vec{b}|\)
(C) Area of the parallelogram with adjacent sides represented by \(\vec{a}\) and \(\vec{b}\)	(III) \(\frac{\vec{a} \cdot \vec{b}}{|\vec{a}|}\)
(D) Projection of \(\vec{a}\) on \(\vec{b}\)	(IV) \(\frac{|\vec{a} \times \vec{b}|}{|\vec{a}||\vec{b}|}\)

Choose the Correct answer from the options given below :

Choose the wrong statement.

The following system of equations:
x + y - z = 7
\(4 x+\lambda y-\lambda z=3\)
3x + 2y - 4x = 5
does not possess a solution if the value of \(\lambda\) is:

Let \(A=\left[a_{i j}\right]\) be a square matrix of order 3 with each entry either 0 or 1 .
Then the number of all such possible matrices is :

The value of \(-\operatorname{cosec}^2\left(\cot ^{-1} y\right)+\sec ^2\left(\tan ^{-1} x\right)\) is equal to :

Read the passage carefully and answer:
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.
Conversion of nitro group into primary amine is preferred using

Which of the following is true for species area relationship equation log S = log C + Z log A

If the molar conductivities at infinite dilution for \(\mathrm{CH}_3 \mathrm{COOK}, \mathrm{K}_2 \mathrm{SO}_4\) and \(\mathrm{H}_2 \mathrm{SO}_4\) are respectively, \(\mathrm{z}, \mathrm{y}\) and \(\times\left(\mathrm{S} \mathrm{cm}^2 \mathrm{~mol}^{-1}\right)\), the molar conductivity of \(\mathrm{CH}_3 \mathrm{COOH}\) at infinite dilution will be:

Read the passage carefully and answer the Questions
Haloalkanes undergo nucelophilic substitution reactions owing to the polarity of CX bond. The nucleophile reacts with the haloalkane on the carbon possessing a partial positive charge holding the halogen atom. The halogen atom \(X\) is replaced by a nucleophile. Depending on the kinetics and mode of bond breaking, the mechanism can be either SN1 or SN2 reaction. The rate of SN1 reaction is governed by the stability of carbocation and in SN2 reaction, the rate of reaction is governed by steric factor. Chirality is the main factor in both SN1and SN2. In SN1 reaction, the chirality of alkyl halide is accompanied by racemization of the product, while in SN2 reaction, the product is characterized by inversion of configuration. The structure of alkyl halide and the nature of the solvent also governs the mechanism of the substitution.
Which of the following bromo isomer of butane will react at a faster rate towards SN2 reaction?

The area bounded by \(x = 1, x = 2, xy = 1\) and x-axis is :

DNA fingerprinting works on the principle of