Value of \(\int_2^3 \frac{\sqrt{x}}{\sqrt{x}+\sqrt{5-x}} d x\) is

The number of equivalence relation on the set {1, 2, 3} containing (1, 2) and (2, 1) is

The integrating factor of the differential equation \(x \frac{d y}{d x}+2 y=x^2\) is:

The function \(f(x)=x e^x\) increases only for:

The approximate volume of a cube of side a meters on increasing the side by 4% is:

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:

Out of the given statement, choose the correct statement.
(A) The direction ratios of the vector \(\vec{a}=3 \hat{i}-\hat{j}+4 \hat{k}\) is \(3,-1,4\).
(B) If \(\theta\) is the angle between two vectors \(\vec{a}\) and \(\vec{b}\), then their cross product is given as \(\vec{a} \times \vec{b}=|\vec{a}||\vec{b}| \cos \theta\).
(C) The unit vector in the direction of vector \(\vec{a}=\hat{i}+2 \hat{j}-2 \hat{k}\) is \(\hat{a}=\frac{1}{3}(\hat{i}+2 \hat{j}-2 \hat{k})\).
(D) If \(\vec{a}=3 \hat{i}\) and \(\vec{b}=4 \hat{j}\) then \(\vec{a} \cdot \vec{b}=12\).
(E) If \(\vec{a}\) and \(\vec{b}\) represent the adjacent sides of a triangle then its area is given of \(\frac{1}{2}|\vec{a} \times \vec{b}|\).
Choose the correct answer from the options given below :

If \(y=e^{a \cos ^{-1} x},-1< x< 1\), then \(\left(1-x^2\right) \frac{d^2 y}{d x^2}-x \frac{d y}{d x}\) is equal to

The correct decreasing order of basic strength of following amines in aqueous medium is.
(A) \((C_2H_5)_3N\)
(B) \((C_2H_5)_2NH\)
(C) \(C_2H_5NH_2\)
(D) \(NH_3\)
Choose the \textbf{correct} answer from the options given below:

Four point charges -2Q, +q, -2q and +Q are placed at four corners of a square A, B, C and D, respectively, as shown in the figure and OA = OB = \( a/\sqrt{2} \). If the potential at the center of a square is zero, what is the relation between Q and q?

Two coils have a mutual inductance 0.005 H . The current changes in the first coil according to the equation \(I=I_0 \sin \omega t\), where \(I_0=10 A\), and \(\omega=100 \pi rad / s\). The maximum value of emf in the second coil is (in V) :

The effective rate of return equivalent to a nominal rate of 12% per annum compounded quarterly is:
[Given that \((1.03)^4=1.1255\) ]

Which of the following primates was more like an ape?