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

If \(P(A)=\frac{6}{11}, P(B)=\frac{5}{11}\) and \(P(A \cup B)=\frac{7}{11}\), then
A. \(P(A \cap B)=\frac{4}{11}\)
B. \(P(A \mid B)=\frac{ 4 }{ 5 }\)
C. \(P(B \mid A)=\frac{2}{5}\)
D. \(A\) and \(B\) are independent events
E. \(P(\) neither \(A\) nor \(B)=\frac{4}{11}\)
Choose the correct answer from the options given below:

The probability distribution of a discrete random variable x is given below:

\(x\)	\(2\)	\(3\)	\(4\)	\(5\)
\(p(x)\)	\(\frac{5}{k}\)	\(\frac{7}{k}\)	\(\frac{9}{k}\)	\(\frac{10}{k}\)

The value of \(k\) is

The variance of number of heads in three tosses of a coin is:

Consider the Linear Programming Problem
Maximize \(z=x+y\)
Subject to the constraints \(x-y \leq-1, x \geq y, x \geq 0, y \geq 0\)
Then which one of the following is TRUE?

The projection vector of the vector \(2 \hat{i}+3 \hat{j}+\hat{k}\) on \(2 \hat{i}+\hat{j}-2 \hat{k}\) is

If \(x^2-y^2=1\), then which of the following is correct?
(A) \(\left(x^2-1\right)\left(\frac{d y}{d x}\right)^2=x^2\)
(B) \(\left(x^2-1\right)\left(\frac{d^2 y}{d x^2}\right)^2=x^2\)
(C) \(\left(x^2-1\right)^3\left(\frac{d^2 y}{d x^2}\right)^2=x^2\)
(D) \(\left(x^2-1\right)^3\left(\frac{d^2 y}{d x^2}\right)^2=1\)
Choose the correct answer from the options given below :

Match List - I with List - II

List - I (Example)	List - II (Component of Time Series)
(A) Lockouts and strikes	(I) Secular trend
(B) Rise and fall of share market	(II) Seasonal trend
(C) Continuous decline in death rate	(III) Irregular trend
(D) The rise in prices before Diwali	(IV) Cyclical trend

Choose the correct answer from the options given below:

Read the following paragraph and answer the given questions.
Cancer detection is based on biopsy and histopathological studies. Techniques like radiography (use of X-rays), CT (computed tomography) and MRI (magnetic resonance imaging) are very useful to detect cancers of the internal organs. Do you know why mother's milk is considered very essential for the new-born infant? The yellowish fluid colostrum secreted by mother during the initial days of lactation has abundant antibodies (IgA) to protect the infant. Different species of Plasmodium (P. vivax, P. malaria and P. falciparum) are responsible for different types of malaria. Of these, malignant malaria caused by Plasmodium falciparum is the most serious one and can even be fatal. The exaggerated response of the immune system to certain antigens present in the environment is called allergy. The use of drugs like anti-histamine, adrenalin and steroids quickly reduce the symptoms of allergy. Drugs like barbiturates, amphetamines, benzodiazepines, and other similar drugs, that are normally used as medicines to help patients cope with mental illnesses like depression and insomnia, are often abused. Morphine is a very effective sedative and painkiller, and is very useful in patients who have undergone surgery.
The drug, which is very effective sedative and painkiller and useful for patients who have undergone surgery is -

Match List-l with List-ll

List-l (Organisms)	List-II (Products/Bioactive molecule)
(A) Trichoderma polysporum	(I) Citric acid
(B) Monascus purpureus	(II) Cyclosporin A
(C) Aspergillus niger	(III) Ethanol
(D) Saccharomyces cerevisiae	(IV) Statins

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From the following, choose the one which are not secondary haloalkanes :
(A) 2-Bromopentane
(B) 1-Bromo-3-methylbutane
(C) 3-Bromopentane
(D) 2-Bromo-2- methylbutane
(E) 2-Bromo-3-methylbutane
Choose the correct answer from the options given below :

The probability that X can take the values \(x_i\) has the following form:
\(P\left(X=x_i\right)=\left\{\begin{array}{ll}0.2 & \text { if } x_i=0 \\ k x_i & \text { if } x_i=1 \text { or } 2 \\ k\left(5-x_i\right) & \text { if } x_i=3 \\ 0 & \text { otherwise }\end{array}\right.\)
The value of P(X = 2) is:

Arrange the following in the increasing order of their solubility in n-octane on the basis of their solute-solvent interaction:
(A) KCI
(B) CH3OH
(C) CH3CN
(D) Cyclohexane
Choose the correct answer from the options given below: