Match List-l with List-ll.Find the derivatives from List-I.

List-l	List-II
(A) \(y=\sqrt{\sin x+y}\)	(I) \(\frac{-\sin x}{1+\cos y}\)
(B)\(\sin y=x \sin (a+y)\)	(II) \(\frac{\cos x}{2 y-1}\)
(C) \(y+\sin y=\cos x\)	(III) \(\frac{-1}{\sin ^2(x+y)}\)
(D) \(y=\tan (x+y)\)	(IV) \(\frac{\sin ^2(a+y)}{\sin a}\)

Choose the correct answer from the options given below:

If \(A\) be a square matrix of order 3 such that \(|A|=2\), then \(|\operatorname{adj}(2 A)|\) is equal to

The value of \(\tan ^{-1} 5+\tan ^{-1} 3-\cot ^{-1} \frac{4}{7}\) is :

A function \(f:[0,2] \rightarrow R\) is strictly increasing in \([0,1]\) and strictly decreasing in \([1,2]\). Then which statement is TRUE?

The product of order and degree of the differential equation \(\left(\frac{d^3 y}{d x^3}\right)^2+x^2 y\left(\frac{d^2 y}{d x^3}\right)^3=2 x^5\) is:

The general solution of the differential equation \(\frac{d y}{d x}+4 x y^2=0\) is :

For which values of \(a, f(x)=a(x+\sin x)\) is an increasing function?

The window of the house is in the form of a rectangle surmounted by a semi-circular portion, having a perimeter of 10 m as shown in figure.

If \(x\) and \(y\) represent the length and breadth of rectangular region the whole area of window is maximum, then value of \(x\) is :

Identify A and B in the following reaction.

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.
Benzoic acid in the presence of conc. \(HNO _3\) and conc. \(H _2 SO _4\) gives major product as:

For which value of \(k\), inverse does not exist for the matrix\(
\left[\begin{array}{ll}
1 & 2 \\
k & 6
\end{array}\right]
\)

What mode of propagation is used for the Ultra high frequency range?

In the month of January, the number of cases diagnosed of influenza epidemic are given in the following table:

Date	1	2	3	4	5	6	7
Number of cases	2	0	5	12	20	27	46

The 3-day moving averages are: