If A and B are two independent events then,
Match List I with List II

List - I	List - II
(A) \(P(A \cup B)\)	(I) \(P\left(A^{\prime}\right) P\left(B^{\prime}\right)\)
(B) P(not A or not B)	(II) \(P(A) ; P(B) \neq 0\)
(C) \(P(A / B)\)	(III) \(1-P\left(A^{\prime}\right) P\left(B^{\prime}\right)\)
(D) P(neither A nor B)	(IV) \(1-P(A) P(B)\)

Choose the correct answer from the options given below:

Mr. X invested Rs. 4, 00, 000 in shares for 5 years.
The value of this investment was Rs. \(4,50,000\) at the end of the second year, Rs. \(4,90,000\) at the end of the third year and on maturity, the final value stood at Rs. \(6,00,000\).
The compound annual growth rate (CAGR) of this investment is :
[Given that \((1.5)^{1 / 5}=1.084\) ]

If A and B are symmetric matrices of the same order, then which of the following are true?
(A) AB - BA is a skew symmetric matrix
(B) AB is a symmetric matrix
(C) AB is a scalar matrix
(D) AB + BA is a symmetric matrix
Choose the correct answer from the options given below:

If \(x^2-y^2=t-\frac{1}{t}\), and \(x^4+y^4=t^2+\frac{1}{t^2}\), then which of the following is correct?

If \(x_1, x_2, x_3,\)........,\(x_n\) are \(n\) observations in a sample then which of following is/are TRUE?
(A) The mean \(\bar{x}\) has \(n\) degree of freedom
(B) The mean \(\bar{x}\) has ( \(n-1\) ) degree of freedom
(C) The standard deviation of the sample has ( \(n-1\) ) degree of freedom
(D) The standard deviation of the sample has \(n\) degree of freedom
Choose the correct answer from the options given below :

The projection of the vector \(\vec{a}=2 \hat{i}+3 \hat{j}+2 \hat{k}\) on the vector \(\vec{b}=\hat{i}+2 \hat{j}+\hat{k}\) is:

Match List - I with List - II.

LIST-I	LIST-I
(A) \(\csc ^{-1} x\)	(I) \((0, \pi)\)
(B) \(\sec ^{-1} x\)	(II) \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]-\{0\}\)
(C) \(\tan ^{-1} x\)	(III) \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\)
(D) \(\cot ^{-1} x\)	(IV) \([0, \pi]-\left\{\frac{\pi}{2}\right\}\)

Choose the correct answer from the options given below:

A circular loop is placed in the plane of the paper in a magnetic field B. A current is induced in the loop in an anticlockwise direction. This is because the magnetic field B is

Which of the following reagent is used in Hinsberg test of amines?

If the matrix \(A=\left[\begin{array}{ccc}0 & x+y & 1 \\ 3 & z & 2 \\ x-y & -2 & 0\end{array}\right]\) is skew-symmetric, 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:"

A proton and an alpha particle enter a uniform magnetic field with the same velocity and move along circular paths. The time period of the alpha particle will be:

In a step-down transformer, 22 kV is converted into 220 V. The number of turns in the primary coil is 60000. The number of turns in the secondary coil will be