The differential equation \(x \frac{d y}{d x}-y=x^2\) has the general solution:

Let R be the relation in the set \(\{a, b, c\}\) given by \(R=\{(a, b),(b, b),(a, a),(a, c),(c, c),(c, b)\}\) Choose the correct answer from the options below :

A Linear Programming Problem (LPP) consists of which of the following components?
(A) Decision variables
(B) The graphical compliment
(C) The objective function
(D) The linear constraints
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Solution of differential equation \(\frac{d y}{d x}=\cos (x+y+3)\) is:

A furniture trader deals in only two items - chairs and tables.
He has Rs. 50, 000 to invest and a space to store at most 35 items.
A chair costs Rs. 1000 and a table costs Rs. 2000.
The trader earns a profit of Rs. 150 on a chair and Rs. 250 on a table.
Choose the correct option that describes the given linear programming problem (LPP) to maximize the profit, where \(x\) and \(y\) are the number of chairs and tables.

If \(\vec{a}\) and \(\vec{b}\) are two non-zero vectors such that \(|\vec{a} \cdot \vec{b}|=|\vec{a} \times \vec{b}|\), then the angle \(\theta\) between \(\vec{a}\) and \(\vec{b}\) is

If \(y=e^{a \cos ^{-1} x},-1< x< 1\), then \(\left(1-x^2\right) \frac{d^2 y}{d x^2}-x \frac{d y}{d x}\) is equal to

The IUPAC name for the following complex compound
\(\text{K[Au(Br)(ONO) (SO}_4\text{)(NH}_3\text{)]}\) is

Match List - I with List - II.

List – I (Conversion)	List - II (Reagent)
(A) Conversion of phenol to salicylaldehyde	(1) Zinc dust
(B) Conversion of phenol to benzene	(II) \(Na _2 Cr _2 O _7 / H _2 SO _4\)
(C) Conversion of phenol to benzoquinone	(III) NaOH and \(CO _2 / H ^{+}\)
(D) Conversion of phenol to o-hydroxybenzoic acid	(IV) \(CHCl _3+\) aq. NaOH

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Read the passage carefully and answer the Questions
Aldehydes and ketones are important organic compounds prepared mainly by oxidation of alcohols. Primary alcohols give aldehyde and secondary alcohols yield ketones. Aldehydes can also be formed by partial reduction of acid chlorides, reduction of esters or by Gattermann-Koch reaction. Ketones are prepared by Friedel-Craft's acylation, hydration of alkyne and reaction of Grignard's reagent with nitriles. Chemically, aldehydes and ketones are reactive in nature due to the carbonyl group and undergo nucleophilic addition reactions. Aldehydes are more reactive than ketones and can be oxidized to carboxylic acids, while both can be reduced to alcohols or hydrocarbons and exhibit condensation reaction
The correct IUPAC name of Vanillin is

The corner points of the feasible region determined by the system of linear constraints are \((0,10),(5,5),(15,15),(0,20)\). Let \(Z=a x+b y\), where \(a, b>0\). Condition on \(a\) and \(b\) so that the maximizing value of \(Z\) occurs at both the points \((15,15)\) and \((0,20)\) is:

What is the IUPAC name of resorcinol?

Match List - I with List - II.

List - I	List - II
(A) Range of \(y=\operatorname{cosec}^{-1} x\)	(I) \(\quad R-(-1,1)\)
(B) Domain of \(\sec ^{-1} x\)	(II) \(\quad(0, \pi)\)
(C) Domain of \(\sin ^{-1} x\)	(III) \(\quad[-1,1]\)
(D) Range of \(y=\cot ^{-1} x\)	(IV) \(\left[\frac{-\pi}{2}, \frac{\pi}{2}\right]-\{0\}\)

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