Let \(\vec{a}\) and \(\vec{b}\) be two unit vectors and \(\theta\) is the angle between them. If \(\vec{a}+\vec{b}\) is a unit vector, then \(\theta\) is:

\(\int \frac{d x}{2 \sin ^2 x+5 \cos ^2 x}\) is equal to

Which of the following matrices is symmetric as well as skew-symmetric?

The marginal cost of production of \(X\) units of a commodity is \(56+\frac{3}{2} x\).
It is known that fixed costs are Rs.115.
Then the total cost of producing 50 units is :-

If A is a square matrix and I is the identity matrix of same order such that \(A^2=I\), then \((A-I)^3+(A+I)^3-3 A\) is equal to:

Which of the following statements are correct?
(A) If elements of a row (or column) of a determinant are multiplied with cofactors of any other row (or column), then their sum is zero.
(B) The value of determinant is equal to the sum of the product of elements of any row (or column) with their corresponding cofactors.
(C) If any two rows (or columns) of a determinant are interchanged, their sign of determinant does not change.
(D) The value of determinant remains unchanged if its rows and columns are interchanged.
(E) A square matrix A has inverse if and only if A is singular.
Choose the correct answer from the options given below:

Let \(A =\left[a_{i j}\right]\) be a \(2 \times 2\) matrix such that \(a_{i j}=\frac{|-3 i+j|}{2}\) then \(a_{21}\) is :

The ammonia derivatives of carbonyl compounds obtained as colorless precipitate are
(A) Oxime
(B) Hydrazone
(C) Phenyl hydrazone
(D) Semicarbazone
Choose the correct answer from the options given below:

Observe the figures and answer the question.

The spent slurry in bio-gas plant is used.

In angiosperms, the outer wall of ovary develops into :

A random variable X has the following probability distribution :

x	1	2	3	4	5
\(P ( X = x )\)	k	k	3k	\(2 k^2\)	\(4 k^2\)

Based on the above information, the value of ' \(K\) ' is :

Let \(P\) and \(Q\) be any two invertible matrices of the same order. Then Match List-I with List-II

List - I (Matrix)	List - II (Equivalent matrix)
(A) \((P Q)^{-1}\)	(I) \(Q^{-1} P\)
(B) \(\left(P^{-1} Q\right)^{-1}\)	(II) \(Q P^{-1}\)
(C) \(\left(P Q^{-1}\right)^{-1}\)	(III) \(Q^{-1} P^{-1}\)
(D) \(\left(P^{-1} Q^{-1}\right)^{-1}\)	(IV) \(Q P\)

Choose the Correct answer from the options given below :

If the length of a wire is doubled while keeping its material and cross-sectional area same, the resistance of the wire