The difference of two different skew-symmetric matrices is :

If \(A=\left[\begin{array}{lll}1 & 2 & 1 \\ 2 & 3 & 1 \\ 0 & 0 & 1\end{array}\right]\), then \(\left|\operatorname{adj}\left(3 A^T\right)\right|^2\) is equal to

If \(\int_0^a 3 x^2 d x=27\), then value of \(a\) is :

If \(\sin ^{-1} x+\sin ^{-1} y=\frac{2 \pi}{3}\), then the value of \(\cos ^{-1} x+\cos ^{-1} y\) is:

Match List-I with List-II
[.] denotes the greatest integer function.

List-I	List-II
(A) \(\int_0^3[x] d x\)	(I) \(\frac{1}{2}\)
(B) \(\int_0^1[2 x] d x\)	(II) 1
(C) \(\int_0^1[3 x] d x\)	(III) \(\frac{3}{2}\)
(D) \(\int_0^1[4 x] d x\)	(IV) 3

Choose the correct answer from the options given below :

Which of the following is a statistic?

The value of the determinant \(\left|\begin{array}{ccc}85 & 15 & 30 \\ 1 & 3 & 4 \\ 17 & 3 & 6\end{array}\right|\) is:

For some constant 'k', if the system of linear equations
\(2 x-y+3 z=1\)
\(x-2 y+z=3\)
\(k x+y-z=0\)
has a unique solution, then

Which of the following is found in symbiotic association with leguminous plants ?

Select the correct relation between the electrochemical cell and Gibbs free energy for the following reaction
\(Zn (s)+ Cu ^{2+}(a q) \rightarrow Zn ^{2+}(a q)+ Cu (s)\)

Which of the following statements are correct?
(A) the mid-point between the two similar and equal point charges located nearby, is called neutral point.
(B) There are maximum number of field lines at the neutral point.
(C) The work done to bring any positive charge from infinity to a point in an electric field is called the electric potential at that point.
(D) The work done to move a charge from one point to another point on the equipotential surface will be zero.
Choose the correct answer from the options given below:

Read the passage carefully and answer the Questions
Carbonyl compounds such as aldehydes, ketones, and carboxylic acids play an important role in the biochemical process of plants and animal kingdom.
Many aldehydes and ketones find applications in dye, perfumery and other industries.
The carbonyl carbon atom is \(s p^2\)-hybridized forming three sigma and one pi-bond with oxygen.
C-O bonds are polar bond due to high electronegativity of oxygen than carbon.
Hence the carbonyl carbon behaves as an electrophilic center and oxygen as a nucleophilic centre.
Primary and secondary alcohols yield aldehydes and ketones respectively upon oxidation with variety of oxidizing agents.
The boiling points of aldehydes and ketones differ from hydrocarbons and ethers.
Aldehydes and ketones undergo nucleophilic addition reaction due to presence of electrophilic carbonyl carbon.
Nucleophile and a proton add across the carbon oxygen double bond.
The alcohol which cannot be used to prepare carbonyl compound through oxidation is

In the given reaction \(CH_3CH_2OH + HCl \xrightarrow{Anhy ZnCl_2} X \xrightarrow{KCN(alc)} Y \xrightarrow{Na/Alcohol} Z\) \(Y\) and \(Z\) are: