A coin is tossed twice and outcomes are recorded. If the random variable \(X\) represents the number of heads in the experiment, then the expectation of X will be :

The value of \(\tan ^{-1} 5+\tan ^{-1} 3-\cot ^{-1} \frac{4}{7}\) is :

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)

The corner points of the bounded feasible region determined by the system of linear constraints are \((0,0), (5,0), (6,5)\), \((6,8),(4,10), (0,8)\). Let \(Z=3 x-4 y\) be the objective function. The minimum value of \(Z\) occurs at

Let \(A=\left[\begin{array}{ccc}1 & -2 & 3 \\ 1 & 2 & 1 \\ \lambda & 2 & -3\end{array}\right]\). If \(A^{-1}\) does not exist, then \(\lambda=\)

The value of \(\tan ^{-1}\left(\tan \frac{3 \pi}{4}\right)+2 \cos ^{-1}\left(\cos \frac{7 \pi}{6}\right)\) is:

If \(\left|\begin{array}{lll}a & b & 0 \\ 0 & a & b \\ b & 0 & a\end{array}\right|=0\) then :

Which of the following statements are true?
(A) Rheostat can be used both as a variable resistance and a potential divider.
(B) An electrical circuit is functional only if all the components of the circuit are connected in proper order, assuming that all circuit components/ devices are in working condition and key is closed.
(C) An open circuit means a break in some part of a circuit which could be deliberate such as a key in open position or a fault such as broken wire or burnt out component(s) or loose connection.
(D) Rheostat as a potential divider gives variable resistance.
Choose the correct answer from the options given below:

Which of the following is the correct expression for force due to a magnetic field (\( \vec{B} \)) on a straight conductor of length (\( l \)) carrying current \( I \)?

If the points \(( a , b ),( c , d )\) and \(( a + c , b + d )\) are collinear, then

If the points P, Q, R with position vectors \(5 \hat{i}+\lambda \hat{j}, 20 \hat{i}-\hat{j}\) and \(15 \hat{i}- 6\hat{j}\) respectively are colinear, then the value of \(\lambda\) is

Match List-I with List-II

List-I (Integral)	List-II (Solution: C is an arbitrary constant)
(A) \(\int \frac{d x}{x^2+25}\)	(I) \(\frac{1}{10} \log \left|\frac{5+x}{5-x}\right|+C\)
(B) \(\int \frac{d x}{x^2-25}\)	(II) \(\log \left|x+\sqrt{x^2-25}\right|+C\)
(C) \(\int \frac{d x}{25-x^2}\)	(III) \(\frac{1}{5} \tan ^{-1}\left(\frac{x}{5}\right)+C\)
(D) \(\int \frac{d x}{\sqrt{x^2-25}}\)	(IV) \(\frac{1}{10} \log \left|\frac{x-5}{x+5}\right|+C\)

Choose the correct answer from the options given below:

If \(x, y\) and \(z\) are non zero real numbers, the inverse of matrix \(A=\left[\begin{array}{lll}x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z\end{array}\right]\) is: