Two events A and B are independent, if
(A) A and B are mutually exclusive
(B) \(P\left(A^{\prime} \cap B^{\prime}\right)=[1-P(A)][1-P(B)]\)
(C) \(P\left(A^{\prime}\right)=P\left(B^{\prime}\right)\)
(D) P(A) + P(B) = 1
(E) P(A) = P(B)

Let box I contains 3 black and 4 white balls, box II contains 2 black and 2 white balls, box III contains 4 black and 3 white balls. A. box is selected at random and then a ball is randomly drawn from the selected box. If the color of the ball is black, then the probability that the ball is drawn from box III, is:

Suppose that A, B and C are matrices of order \(m \times n, n \times 5\) and \(5 \times q\) respectively. The restriction on m, n and q so that AB - BC is defined are

Let \(f(x)=\log _e(\sin x), x \in(0, \pi)\), then which of the following statements is/are TRUE?
(A) \(f(x)\) is increasing on \((0, \pi / 2)\)
(B) \(f ( x )\) is decreasing on \(( \pi / 2 , \pi)\)
(C) \(f(x)\) is increasing on \((0, \pi)\)
(D) \(f(x)\) is decreasing on \((0, \pi)\)
Choose the correct answer from the options given below :

The relation R in the set \(\mathbb{R}\) of real numbers defined as \(R = \{(a, b) : a \le b^2\}\) is:

Let \(A=\left[a_{i j}\right]\) be a square matrix of order 3 with each entry either 0 or 1 .
Then the number of all such possible matrices is :

The value of \(\int_{-3}^2 x^2|2 x| d x\) is:

The value of the determinant \(\left|\begin{array}{ccc}a+b & b+c & c+a \\ c & a & b \\ 1 & 1 & 1\end{array}\right|\) is :

Which of the following statements are correct?
(A) If \(\vec{a}\) and \(\vec{b}\) represent the adjacent sides of a triangle, then its area is \(\frac{1}{2}|\vec{a} \times \vec{b}|\)
(B) If \(\vec{a}\) and \(\vec{b}\) represent the adjacent sides of a parallelogram, then its area is \(|\vec{a} \times \vec{b}|\)
(C) \(|\vec{a} \times \vec{b}|=|\vec{a}||\vec{b}| \cos \theta\)
(D) If \(\vec{a}\) and \(\vec{b}\) represent the 'diagonals' of a parallelogram, then its area is \(\frac{1}{2}|\vec{a} \times \vec{b}|\)
Choose the correct answer from the options given below :

Activation energy of a chemical reaction can be determined by

MTP is considered safe up to ___________ weeks of pregnancy.

Read the passage carefully and answer the Questions
The concentration dependence of the rate of a reaction is called the differential rate equation. It is not always convenient to determine the instantaneous rate, as it is measured by calculating the slope of the tangent at point 't' in the concentration vs time plot. This makes it difficult to determine the rate and hence the order of the reaction. This difficulty is overcome by integrating the differential rate equation. This integrated rate equation gives a direct relation between concentrations at different times and the rate constant. The integrated rate equations are different for the reactions having different reaction orders.
Choose the correct statement regarding kinetics of a chemical reaction.

Many features of the nuclear binding force are summarised below : Choose the correct statement.
(A) Coulomb force acting between two protons inside the nucleus is much weaker than Nuclear force between them
(B) Between proton-proton inside nucleus gravitational force is stronger than nuclear force
(C) Nuclear force between two nucleons falls rapidly to zero as their distance is more than a few femtometer
(D) Nuclear forces between proton-proton, neutron - proton and neutron-neutron is approximately same when distance between them is same.
Choose the correct answer from the options given below :