If corner points of a feasible region are \((0,0),(2,0),\left(\frac{20}{19}, \frac{45}{19}\right)\) and \((0,3)\), then
(A) Maximum value of \(z=5 x+3 y\) is 10
(B) Minimum value of \(z=5 x+3 y\) is 0
(C) Maximum value of \(z=5 x+3 y\) is \(\frac{235}{19}\) and minimum value is 0
(D) Maximum value of \(z=5 x+3 y\) is 10 and minimum value is 0
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If \(y=\frac{1}{2} x \sqrt{a^2-x^2}+\frac{a^2}{2} \sin ^{-1}\left(\frac{x}{a}\right)\), then \(\frac{d y}{d x}=\)

If \(A\) and \(B\) are square matrices of the same order 3 , such that \(\operatorname{det}(A)=3\) and \(A B=3 I\), where \(I\) is an identity matrix of order 3. Then the value of \(\operatorname{det}(B)\) 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 function \(f(x)=2 x^3+3 x^2-12 x+1\) is strictly increasing in :

If the equation of a floor of a room is given by \(x+y-z+4=0\) and the equation of roof is given by \(x+y-z+5=0\). Then, the height of the room is :

If \(\cos y=x \cos (a+y)\), then \(\frac{d y}{d x}=\)

In a potentiometre, the balanced point is obtained when :

A closed circuit in the form of a regular hexagon of side a carries a current I. What is the magnetic field at the centre of the hexagon?

If the interval in which \(f(x)=\frac{x}{4}+\frac{4}{x}, x \neq 0\) is strictly increasing is \((-\infty, a) \cup(b, \infty)\), then

"Answer the question on basis of passage given below :
Carboxylic acids are weaker than mineral acids, but they are stronger acids than alcohols and many simple phenols (\(\text{pKa}\) is ~ 16 for ethanol and 10 for phenol). The higher acidity of carboxylic acids as compared to phenols can be understood similarly. The conjugate base of carboxylic acid, a carboxylate ion, is stabilised by two equivalent resonance structures in which the negative charge is at the more electronegative oxygen atom. The conjugate base of phenol, a phenoxide ion, has non-equivalent resonance structures in which the negative charge is at the less electronegative carbon atom. Further, the negative charge is delocalised over two electronegative oxygen atoms in carboxylate ion whereas it is less effectively delocalised over one oxygen atom and less electronegative carbon atoms in phenoxide ion. Thus, the carboxylate ion is more estabilised than phenoxide ion, so carboxylic acids are more acidic than phenols.
Effect of substituents on the acidity of carboxylic acids: substituents may affect the stabliity of the conjugate base and thus, also affect the acidity of the carboxylic acids. Electron withdrawing group increase the acidity of carboxylic acids by stabilising the conjugate base through delocalisation of the negative charge by inductive and/or resonance effects. Conversely, electron donating groups decrease the acidity by destabilising the conjugate base.
Which of the following compounds would have the smallest value of \(\text{pKa}\)?"

Answer the question on the basis of passage given below :
Aldehydes and ketones represent two essential classes of organic compounds
which are extensively used in the synthesis of a wide variety of other organic molecules.
Due to the inherent polarity of the central carbonyl group, these substances readily undergo nucleophilic addition reactions.
Broadly speaking, these addition reactions can be categorized into two types:
one in which simple addition of the nucleophilic reagent occurs across the \(> C = O\) bond, and the second type where the initial addition is subsequently followed by the elimination of a water molecule.
A variety of ammonia derivatives, including hydroxylamine, hydrazine, phenylhydrazine,
2, 4-dinitroplhenylhdrazine, and semicarbagide, all belong to this second category of reactions.
These specific derivatives are fundamentally used for the identification and definitive characterization of aldehydes and ketones. Significantly, both types of nucleophilic addition reactions are strongly influenced by both steric and electronic factors.
As a general rule, aliphatic aldehydes exhibit higher reactivity compared to aromatic aldehydes.
The aliphatic aldehydes demonstrate the ability to reduce both Fehling's solution and Tollen's reagent, whereas aromatic aldehydes are limited to reducing only Tollen's reagent.
In a similar manner to aldehydes, aromatic ketones are observed to be less reactive than aliphatic ketones towards all nucleophilic addition reactions.
In a separate context, phenols and carboxylic acids both exhibit acidic properties in nature.
Both classes of compounds readily dissolve in NaOH solution and cause blue litmus to turn red.
However, carboxylic acids stand out as much stronger acids than phenols and thus can decompose sodium bicarbonate \(\left( NaHCO _3\right)\) with the distinct evolution of \(CO _2\) gas, a reaction which phenols do not undergo.
It is important to note that both electron- donating and electron- withdrawing substituents significantly influence the acid strength of both aliphatic as well as aromatic carboxylic acids.
Furthermore, unlike all other common aliphatic acids, formic acid uniquely possesses reducing properties and simultaneously does not display the typical reactions characteristic of the alkyl group.
Fehling Solution A and B respectively, are :

Match List-I with List-II

List-I (Diseases)	List-II (Causative organisms)
(A) Pneumonia	(I) Microsporum
(B) Ringworms	(II) Wuchereria bancrofti
(C) Elephantiasis	(III) Entamoeba histolytica
(D) Amoebiasis	(IV) Haemophilus influenzae

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