A random variable has the following probability distribution:\(\begin{array}{llllll}X= x_{-} i & 2 & 3 & 4 & 5 \\P\left(X=x \_i\right) & 4 k & k & 5 k & 2 k\end{array}\)The value of \(P(X<3)\) is

Evaluate \(\int_0^{\frac{\pi}{4}} \frac{\sin 2 x}{\cos ^4 x+\sin ^4 x} d x=\)

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

Match List - I with List - II

List - I (terms)	List - II (Definitions)
(A) POPULATION	(I) Measurable characteristics of the population such as mean, variance, standard deviation etc. of population
(B) SAMPLE	(II) Measurable characteristics of the sample such as mean, variance, standard deviation etc. of a sample
(C) PARAMETER	(III) Finite set of statistical individuals drawn from a population for investigation
(D) STATISTIC	(IV) Collection of objects having the same characteristics

Choose the correct answer from the options given below:

Two pipes A and B can fill a tank in 5 hours and 6 hours respectively. Pipe C can empty the tank in 12 hours. If all three pipes are opened together, find the time taken to fill the tank

The unit vector in the xy-plane, making an angle of \(60^{\circ}\) with the positive direction of x-axis is:

At what rate of interest will the present value of a perpetuity of Rs. 1000 payable at the end of every six months be Rs. 20000?

Match List I with List II and choose the correct answer from the options given below:

List - I	List - II
(A) \(\sin ^{-1} x+\cos ^{-1} x, x \in[-1,1]\)	(I) \(-\frac{\pi}{2}\)
(B) \(\tan ^{-1} \sqrt{3}-\cot ^{-1}(-\sqrt{3})\)	(II) \(-\frac{\pi}{6}\)
(C) \(\cos ^{-1}\left(\cos \frac{13 \pi}{6}\right)\)	(III) \(\frac{\pi}{2}\)
(D) \(\sin ^{-1}\left(-\frac{1}{2}\right)\)	(IV) \(\frac{\pi}{6}\)

Read the following passage carefully and answer the given questions.
During anaerobic sludge digestion, bacteria produce a mixture of gases such as methane, hydrogen sulphide and carbon dioxide. These gases form biogas and can be used as source of energy as it is inflammable. The bioactive molecule, cyclosporin A, that is used as an immunosuppressive agent in organ-transplant patients, is produced by the fungus Trichoderma polysporum. You must have noticed that bottled fruit juices bought from the market are clearer as compared to those made at home. This is because the bottled juices are clarified by the use of pectinases and proteases. Statins produced by the yeast Monascus purpureus have been commercialised as blood-cholesterol lowering agents. It acts by competitively inhibiting the enzyme responsible for synthesis of cholesterol. The BOD test measures the rate of uptake of oxygen by micro-organisms in a sample of water and thus, indirectly, BOD is a measure of the organic matter present in the water.
Statins, used as blood-cholesterol lowering agents are produced by -

A \(95 \%\) confidence interval for a population mean was reported to be 152 to 160 , if sample standard deviation \(\sigma=15\), then sample size used in this study is (Given \(Z_{0.025}=1.96\) )

In a concave mirror, an object is placed at a distance \( x \) from the focus and the image is formed at a distance \( y \) from the focus. What is the focal length of the mirror?

Maximize \(z=2 x-y\)
Subject to constraints
\(x+y \leq 4, \quad x+3 y \leq 6, \quad x \geq 0, \quad y \geq 0 .\)
The corner points of the feasible region are:

If the relation \(R: A \rightarrow B\), where \(A=\{2,3,4\}\) and \(B=\{3,4,5\}\) is defined by \(R=\{(x, y): x < y, x \in A, y \in B\}\), then