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:

The value of the integral \(\int_0^\pi 2 x \sin ^3 x\ d x\) is:

Match List-I with List-II

List-I (Example)	List-II (Time-series component)
(A) Labour strike	(I) Secular-trend
(B) Continuous decline in death rate	(II) Seasonal
(C) Rise in prices before Diwali	(III) Cyclical
(D) Rise and fall of the share-market	(IV) Irregular

Choose the correct answer from the options given below :

Let \(A\) be a square matrix of order 3 then \(|3 A|\) is equal to

Solution of the differential equation \(y \log _e y d x-x d y=0\) is (Where \(c\) is an arbitrary constant)

If a function \(f: R \rightarrow R\) is defined by \(f(x)=x^2+1\), then the pre-images of 17 and -3 respectively are:

Angle between tangents to the curve \(y=x^2-5 x+6\) at the points \((2,0)\) and \((3,0)\) is :

\(\text{CH}_3-\text{CH}(\text{Br}) - \text{CH}_2 - \text{CH}_3 \xrightarrow{\text{alc. KOH}} \text{A} \xrightarrow{\text{Br}_2} \text{CH}_3-\text{CH}(\text{Br}) - \text{CH}(\text{Br}) - \text{CH}_3 \text{(B)}\)
\(\text{(B)} \xrightarrow{\text{alc. KOH}} \text{CH}_3-\text{C} = \text{C} – \text{CH}_3 \text{(C)}\)
\(\text{(C)} \xrightarrow{\text{Hg}^{2+}/\text{H}^+\atop \text{H}_2\text{O}} \text{D}\)
A and D respectively are :

Which of the following statements are correct?
(A) In dispersion through a prism of glass, the red color ray deviates most.
(B) In simple microscope, to get a magnified image, the object is placed within the focus of a convex lens.
(C) In compound microscope, total magnification is m = (magnification due to eye lens) x (magnification due to objective)
(D) Magnifying power of telescope M = focal length of eye lens / focal length of objective lens
Choose the correct answer from the options given below:

Read the following passage carefully and answer the given questions.
Insulin consists of two short polypeptide chains- chain A and chain B, that are linked together by disulphide bridges. Some strains of Bacillus thuringiensis produce proteins that kill certain insects such as lepidopterans, coleopterans and dipterans. Bt toxin protein is converted into an active form of toxin due to the alkaline pH of the gut which solubilise the crystals. Transgenic animals that produce useful biological products can be created by the introduction of the portion of DNA (or genes) which codes for a particular product such as human protein (alpha-1-antitrypsin) used to treat emphysema. Plants, bacteria, fungi and animals whose genes have been altered by manipulation are called Genetically Modified Organisms (GMO). GM plants have been useful in many ways. Genetic modification has: (i) made crops more tolerant to abiotic stresses. (ii) reduced reliance on chemical pesticides (pest-resistant crops). (iii) helped to reduce post harvest losses. (iv) increased efficiency of mineral usage by plants (this prevents early exhaustion of fertility of soil). (v) enhanced nutritional value of food.
Which one of the following species is not killed by insecticidal protein produced by Bacillus thuringiensis?

The linear programming problem:
Maximize \(Z = x + y\),
subject to the constraints \(x-y \leq-1,-x+y \leq 0, x, y \geq 0\) has

An electric dipole of dipole moment \(5 \times 10^{-6} \mathrm{Cm}\) is placed in a uniform electric field of \(10^{-2} \mathrm{~N} / \mathrm{C}\) making an angle of \(30^{\circ}\) with the direction of the field. The torque exerted by the electric field on the dipole is:

Read the passage carefully and answer the questions
Carboxylic acids are the organic compounds which comprise of carboxyl functional group, -COOH. These are generally found in nature, some higher members are known as fatty acids as they occur in natural fats as esters of glycerol. They can be prepared from oxidation of alcohols and aldehydes, hydrolysis of nitriles and amides and Grignard reagents. They are stronger acids than simple phenols and alcohols. but weaker than mineral acids. Substituents affect the acidity of aliphatic and aromatic carboxylic acids. They undergo reactions involving cleavage of -OH and C-OH bond. Carboxylic acids having an alpha hydrogen get halogenated in the presence of red phosphorus to form alpha halo carboxylic acids. The reaction is known as Hell Volhard Zelinsky reaction. Besides these aromatic carboxylic acids undergo electrophilic substitution reactions. They are versatile and are employed in the manufacture of various useful compounds. For example, methanoic acid is used in rubber, textile, dyeing, leather and electroplating industries. Higher fatty acids are used in the manufacture of soaps and detergents. Esters of Benzoic acid are used in perfumery.
Esterification of carboxylic acid with alcohols or phenols is which type of reaction?