If A is a matrix of order \(m \times n\) and B is a matrix such that \(A B^T\) and \(B^T A\) are both well-defined matrices, then order of matrix B is

A fair coin is tossed a fixed number of times. If the probability of getting 11 heads is equal to the probability of getting 13 heads, then the probability of getting 2 heads is:

The slope of the normal to the curve \(y=2 x^2\) at \(x=1\) is:

Mr. Mittal invested Rs. 20, 000 in a mutual fund in the year 2019.
The value of the mutual fund increased to Rs. 32, 000 in the year 2024. The compound annual growth rate (CAGR) of his investment is :
[Given (1.6) \({ }^{1 / 5}=1.098\) ]

Which of the following statements are correct?
(A) The mean and variance of the Poisson distribution are equal.
(B) The mean and variance of a Binomial distribution are equal.
(C) An unbiased die is thrown again and again until two sixes are obtained, then the probability of obtaining the second six in the 3rd throw is \(\frac{5}{108}\).
(D) If the variance of a Poisson distribution is 2 , then \(P(X=2)=2 e^{-2}\).
Choose the correct answer from the options given below :

The minimum value of the function f(x) = 3sinx - 4cosx, x ∈ [- 4π, 4π] is equal to

If A and B are two square symmetric matrices of same order, then AB-BA is

Match List - I with List - II.

List - I	List - II
(A) \( \oint \vec{B} \cdot d\vec{A} = 0 \)	(I) Gauss's Law of electricity
(B) \( \oint \vec{E} \cdot d\vec{A} = \frac{Q}{\epsilon_0} \)	(II) Gauss's Law of magnetism
(C) \( \oint \vec{B} \cdot dl = \mu_0 i_c + \mu_0 \epsilon_0 \frac{d\phi_E}{dt} \)	(III) Faraday's Law
(D) \( \oint \vec{E} \cdot dl = -\frac{d\phi_B}{dt} \)	(IV) Ampere-Maxwell Law

Choose the correct answer from the options given below :

Select the correct statement.

The area of triangle with vertices \(P, Q, R\) is given by (where \(\overrightarrow{A B}=\) position vector of point \(B-\) position vector of point \(A\) )

In a linear programming problem (LPP), the maximum value of the objective function \(Z=2 x+5 y\) subjected to the constraints :
\(\begin{array}{l}2 x+3 y \leq 6 \\2 x+y \leq 4 \\x, y \geq 0\end{array}\)
is

What is true about Gauss's "Competitive Exclusion Principle?"
A. Closely related spicies competing for same resources cannot exist indefinitely
B. Totally unrelated species do not compete for the same resource
C. Among the competiting species, competitively inferior species will be eliminated
D. Species facing competition, evolve organs to co-exist
Choose the correct answer from the options given below:

"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.
The weakest among the following acid is :"