If order of matrices \(A, B\) and \(C\) are \(4 \times 3,5 \times 4\) and \(3 \times 7\) respectively, then order of \(C^{\prime} \times\left(A^{\prime} \times B^{\prime}\right)\) is :

Two runners A and B complete a 150 meters race in 24 seconds and 36 seconds respectively. By how many meters will A defeat B?

Probability that a man speaks truth is \(\frac{3}{4}\). He throws a die and reports that it is a six. The probability that it is actually a

If the difference of two unit vectors is again a unit vector, then the angle between them is ______________

If \(y=\sin x+e^x\), then \(\frac{d^2 x}{d y^2}\) is equal to:

Let \(M_{i j}\) and \(A_{i j}\) denote respectively minors and co-factors of the element in the ith row and jth column of the matrix \(A=\left[\begin{array}{ccc}1 & 2 & -1 \\ 3 & 2 & 3 \\ 4 & -1 & 0\end{array}\right]\). Then
(A) \(M_{32}=6\)
(B) \(M_{23}=9\)
(C) \(A_{32}=-6\)
\((D) A_{23}=-9\)
Choose the correct answer from the options given below :

If \(x, y\) and \(z\) are real numbers such that \(x+y+z=0\), then the value of \(\left|\begin{array}{ccc} 3 x & -x+y & -x+z \\ x-y & 3 y & z-y \\ x-z & y-z & 3 z \end{array}\right| \) is

In an influenza epidemic, the number of cases diagnosed in a week were as follows:

Day	Monday	Tuesday	Wednesday	Thursday	Friday	Saturday	Sunday
Number of Cases	10	12	8	16	9	14	4

The 3-day moving averages are :

If \(A=\left[\begin{array}{ll}1 & 0 \\ 0 & 0\end{array}\right], B=\left[\begin{array}{ll}0 & 0 \\ 3 & 0\end{array}\right]\) 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.
Arrange the following acids in order of their increasing \(\text{pKa}\) value:
(A) Benzoic acid
(B) 2-nitrobenzoic acid
(C) 4-nitrobenzoic acid
(D) 4-methoxybenzoic acid
Choose the correct answer from the options given below:"

How many aldol condensation products can be obtained when two aldehydes, both with a-hydrogen atoms undergo reaction?

In a Young's double slit experiment, two slits are separated by 3 mm and the screen is placed 1.5 m away. The distance between two consecutive minima for a light of wavelength 420 nm is

A system consists of two protons and one electron. They are separated by a distance of 1 \(\text{\AA}\) from each other. If the potential energy at infinity is taken to be zero, the potential energy of the system is