A random variable X has the following distribution:

X	0	1	2	3	4	5	6	7
P(X)	0	3k	k	2k	\(3k^2\)	\(k^2\)	\(6k^2\)	3k

The value of k is:

\[\int e^x\left(1+x^2\right) d x=\]

If corner points of a feasible region are \((0,0),(2,0),\left(\frac{20}{19}, \frac{45}{19}\right)\) and \((0,3)\), then
(A) Maximum value of \(z=5 x+3 y\) is 10
(B) Minimum value of \(z=5 x+3 y\) is 0
(C) Maximum value of \(z=5 x+3 y\) is \(\frac{235}{19}\) and minimum value is 0
(D) Maximum value of \(z=5 x+3 y\) is 10 and minimum value is 0
Choose the correct answer from the options given below :

\(\int \frac{e^x(1+x) d x}{\cos ^2\left(e^x x\right)}\) is equal to:

The radius of a spherical ball is increasing at the rate of 1 m/sec. At the radius equal to 3m, the volume of the ball is increasing at the rate given by :

Let \(A, B, C, D\) and \(E\) be matrices of \(2 \times n, 3 \times k, 2 \times p\),
\(2 \times p\) and \(3 \times 3\) respectively.
Choose the correct statement(s) from the following :
(A) \(E B+D B\) will be defined if \(k=3, p=n\).
(B) \(E(3 \times 3) \cdot B(3 \times k) \rightarrow(3 \times k) \rightarrow n \times 3\).
(C) If \(n=p=2\), then the order of the matrix \(5 A^2-3 C\) is \(2 \times 2\).
(D) If \(n=p\), then the order of the matrix \(5 A^2-3 C\) is \(2 \times 3\).
Choose the correct answer from the options given below :

If \(f(x)=\left\{\begin{array}{l}m x+1, x \geq \pi / 2 \\ \sin x+n, x \leq \pi / 2\end{array}\right.\) is continuous at \(x=\pi / 2\), where \(m \in Z\) (set of integers), then \(\sin 2 n=\)

Read the passage carefully and answer the given questions.
Recombinant DNA Technology Asexual reproduction preserves the genetic information, while sexual reproduction permits variation. Traditional hybridization procedures used in plants and animals breeding very often lead to inclusion and multiplication of undesirable genes along with the desired genes. The techniques of genetic engineering, which include creation of recombinant DNA, use of gene cloning and gene transfer, overcome this limitation and allow us to isolate and introduce only one or a set of desirable genes without introducing genes into the target organism.
Which one of the following options is correct for working of restriction enzymes?

In alcohols, C-O-H bond angle is:

Read the passage carefully and answer the questions based on the passage:
Pollen pistil interaction involves all events from the landing of pollen grains on the stigma until the pollen tube enters the embryo sac (when the pollen is compatible). When a pollen tube grows through the style and enters into the ovules, it finally discharges two male gametes in one of the synergids. Syngamy and triple fusion are two fusion events occur in angiosperms. Thus, angiosperms exhibit double fertilization. The products of these fusions are the diploid zygote and triploid primary endosperm nucleus. Zygote develops into embryo and primary endosperm cell forms the endosperm tissue. The developing embryo passes through different stages before maturation.
The coconut water from tender coconut, a good source of nutrition is nothing but:

Assume the earth to be a spherical conductor of radius 6400 km. Calculate the capacitance of earth.

Which of the following statements are TRUE ?
(A) The sales of woolen clothes, gold, silver etc. exhibit seasonal trends.
(B) The price of stocks in the share market repeats after a definite time interval.
(C) The rise and fall of the share market is an example of a cyclic trend.
(D) The rise in prices before festivals is an example of a irregular trend.
Choose the correct answer from the options given below :

An astronomical telescope uses two lenses of power 10 D and 2 D. The distance between the objective and the eye lens in normal adjustment would be