If under pure competition demand and supply functions are given by \(p=\sqrt{10-x}\) and \(p=\frac{1}{2}(x-2)\) respectively, where \(p\) is price per unit and \(x\) is quantity, then the consumer surplus is :

Year (t)	2010	2011	2012	2013	2014	2015	2016
Sales (in Rs. crores) (y)	6	8	9	11	13	17	20

In reference to the above data, which of the following statements are correct?
(where \(x=t-2013\) )
(A) If the equation of the straight line trend is \(y=12+2.29 x\), then the trend value for the year 2017 is 21.16.
(B) If the equation of the straight line trend is \(y=12+2.29 x\), then the trend value for the year 2013 is 11.
(C) If the equation of the straight line trend is \(y=12+2.29 x\), then the trend value for the year 2015 is 17.
(D) If \(\left(t_1, y_1\right),\left(t_2, y_2\right), \ldots,\left(t_n, y_n\right)\) denote the time series and \(y_t\) are the trend values of \(y\), then \(\sum\left(y-y_t\right)=0\).
Choose the correct answer from the options given below :

There are two values of a for which the determinant, \(\Delta = \begin{vmatrix} 1 & -2 & 5 \\ 0 & a & -1 \\ 0 & 4 & 2a \end{vmatrix} = 86\), then the sum of these values of a is:

The value of the determinant \(\left|\begin{array}{lll}a^2 & a & 1 \\ b^2 & b & 1 \\ c^2 & c & 1\end{array}\right|\) is:

Let \(y=\cos \left(\sin x^2\right)\), then the value of \(\frac{d y}{d x}\) at \(x=\frac{\sqrt{\pi}}{2}\) is equal to

If the system of equations
\(x-3 y+5 z=3\)
\(x-2 y+4 z=4\)
\(2 x-7 y+\lambda z=5\)
has infinite number of solutions, then the value of \(\lambda\) is:

The area of the region bounded by the curves \(y=x^2+2, y=x, x=0\) and \(x=2\) is

Identify the correct statements from the following for an electrical circuit.
(A) An ammeter of very low resistance is connected in series in the circuit.
(B) A Voltmeter of high resistance is connected in series with a resistance in the circuit.
(C) A current I coming at a point P of a circuit, divides into 4 small currents flowing through different branches and all the currents again combine to give current I at the next point S of the circuit. The branches are in parallel.
(D) The internal resistance of the batteries can always be neglected
Choose the correct answer from the options given below:

Which one of the following is not true for a molecule that can act as a genetic material?

A compound forms hexagonal closed packed structure. What is the number of tetrahedral voids in \(0.8\text{ mol}\) of it?

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, phenylhydraaine, 2, 4-dinitroplhenylhdrazine, and semicarbazide, 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.
Carboxylic acid do not give the characteristic reaction of carbonyl group because:

For the cell represented by the following reaction :
\(Mg ( s )+2 Ag ^{+}(0.0001 M ) \rightarrow Mg ^{2+}(0.2 M )+2 Ag ( s )\)
If \(E_{\text {cell }}^{\circ}=3.17 V\), its \(E_{\text {cell }}\) at 298 K will be:
(Given: \(\log _{10} 2=0.301\) )

When a primary amine reacts with \(\text{CHCl}_3\) in presence of ethanolic KOH to produce a foul-smelling compound, the reaction is called: