If the events A and B are independent, then which of the following statements are true?
(A) \(P\left(A^{\prime} \cap B\right)=[1-P(A)] P(B)\)
(B) A and B are mutually exclusive
(C) P(A) = P(B)
(D) \(P\left(A^{\prime} \cap B^{\prime}\right)=[1-P(A)][1-P(B)]\)
Choose the correct answer from the options given below:

The system of equations
\(\begin{aligned} & x-3 y-8 z=-10 \\ & 2 x+5 y+\lambda z=13 \\ & 3 x+y-4 z=0\end{aligned}\)
has infinite number of solutions if the value of \(\lambda\) is equal to:

Match List 1 with List 2.

List - 1 (Matrix)	List - 2 (Determinants)
(A) \(\operatorname{Matrix}(A)=\left[\begin{array}{cc}1 & 7 \\ -3 & 5\end{array}\right]\)	(I) 24
(B) \(\operatorname{Matrix}(B)=\left[\begin{array}{cc}-2 & 5 \\ -3 & -3\end{array}\right]\)	(II) 32
(C) \(\operatorname{Matrix}(C)=\left[\begin{array}{ll}-12 & 8 \\ -16 & 8\end{array}\right]\)	(III) 21
(D) \(\operatorname{Matrix}(C)=\left[\begin{array}{ll}-12 & 8 \\ -16 & 8\end{array}\right]\)	(IV) 26

Choose the correct answer from the options given below :

Difference between the maximum and minimum value of the objective function \(Z=5 x+3 y\), subject to the constraints: \(3 x+5 y \leq 15 ; 5 x+2 y \leq 10 ; x, y \geq 0\) is:

The maximum value of \(f(x)=\frac{\log x}{x}\) for \(x\) in \([2, \infty]\) is

The set of value of \(x\) for which the angle between the \(\vec{a}=2 x^2 \hat{i}+4 x \hat{j}+\hat{k}\) and \(\vec{b}=7 \hat{i}-2 \hat{j}+x \hat{k}\) is obtuse is :

If the random variable \(X\) is normally distributed with mean 16 and standard deviation 4,
then the standard normal variable \(Z\) corresponding to \(X=17\) is

Read the passage carefully and answer the questions
Replacement of a hydrogen atom in a hydrocarbon by an alkoxy or carboxyl group yields a class of compounds known as ethers. Ethers are classified as symmetrical or unsymmetrical on the basis of groups attached to the oxygen atoms. Diethyl ether, a symmetrical ether, has been widely used as an inhalation anesthetic. Ethers can be prepared by acid catalyzed intermolecular dehydration of alcohols and Williamson's synthesis. Acid catalyzed dehydration of alcohols is not generally preferred as it gives a mixture of elimination and substitution products. In Williamson's synthesis, an alkyl halide is allowed to react with sodium alkoxide. Ethers containing substituted Alkyl groups may also be prepared by this method. The C-O bond in ether is weakly polar and is cleaved under drastic conditions with excess of hydrogen halides. In electrophilic substitution, the alkoxy group deactivates the aromatic ring and directs the incoming group to ortho and para positions.
The major product in the reaction of anisole with bromine in ethanoic acid, is :

For \(x \in[-1,1]\), if \(4 \sin ^{-1} x+\cos ^{-1} x=\pi\) then \(x\) is equal to :

The function \(f\) is given by
\(f(x)=\left\{\begin{array}{ll}x^3+3, & \text { if } x \neq 0 \\ 4, & \text { if } x=0\end{array}\right.\)
Then number of points of discontinuity for this function is :

"The random variable \(X\) has a probability distribution \(P ( X )\) of the following form, where \(k\) is some number.
\(P(X=x)=\left\{\begin{array}{ll}k, & \text { if } x=0 \\2 k, & \text { if } x=1 \\3 k, & \text { if } x=2 \\0, & \text { otherwise }\end{array}\right.\)
Then \(P(x \leq 2)\) is :"

Which of the following alcoholic drink is not produced by distillation of fermented broth?

For finding absolute maximum and minimum values of a function \(f\) given by
\(f(x)=2 x^3-15 x^2+36 x+1 \text { on }[1,5]\)
A. the critical points in \((1,5)\) are 2 and 3 .
B. the absolute maximum value of \(f\) on \([1,5]\) is 29 .
C. the absolute minimum value of \(f\) on \([1,5]\) is 24 .
D. the absolute minimum value of \(f\) on \([1,5]\) is 20 .
E. we evaluate the value of \(f\) at critical points and at the end points of the interval \([1,5]\).
Choose the correct answer from the options given below: