The value of the integral \(I=\int_0^1 \frac{1}{\sqrt{1+3 \sqrt{x}}} d x\) is:

Match List I With List II

List I	List II
A. \(\frac{d^2 y}{d x^2}=\left(\frac{d y}{d x}\right)^{\frac{3}{2}}\)	I. order + degree \(=2\)
B. \(2\left(\frac{d^3 y}{d x^3}\right)^2+3\left(\frac{d^2 y}{d x^2}\right)+y\left(\frac{d y}{d x}\right)^2=e^x\)	II. order + degree \(=3\)
C. \(\frac{d y}{d x}+\frac{1}{d y / d x}=3\)	III. order + degree \(=4\)
D. \(\frac{d y}{d x}+x^2=5\)	IV. order + degree \(=5\)

Match List-I with List-II

List-I	List-II
(A) \(\int \frac{d x}{x^2-16}\)	(I) \(\frac{1}{8} \log \left|\frac{4+x}{4-x}\right|+c\)
(B) \(\int \frac{d x}{x^2+16}\)	(II) \(\log \left|x+\sqrt{x^2-16}\right|+c\)
(C) \(\int \frac{d x}{16-x^2}\)	(III) \(\frac{1}{8} \log \left|\frac{x-4}{x+4}\right|+c\)
(D) \(\int \frac{d x}{\sqrt{x^2-16}}\)	(IV) \(\frac{1}{4} \tan ^{-1}\left(\frac{x}{4}\right)+c\)

Choose the Correct answer from the options given below :

The corner points of the feasible region associated with the \(LPP\): Maximise \(Z=p x+q y, p, q>0\) subject to \(2 x+y \leq 10, x+3 y \leq 15, x, y \geq 0\) are \((0,0),(5,0),(3,4)\) and \((0,5)\). If optimum value occurs at both \((3,4)\) and \((0,5)\), then:

If the integral \(I=\int\left\{\log _e\left(\log _e x\right)^2+\frac{\alpha}{\log _e x}\right\} d x=x \log _e\left(\log _e x\right)^2+C\), where \(C\) is constant of integration. Then the value of \(\alpha\) is :

The maximum profit that a company can make if the profit function is
\(P(x)=41+24 x-18 x^2\) is :

If \(A=\left[\begin{array}{cc}3 & 1 \\ -1 & 2\end{array}\right]\), then \(A^2-5 A+7 I=\)

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

A nucleus with mass number 240 and binding energy per nucleon 7.6 MeV splits into two fragments each of mass number 120 with binding energy per nucleon 8.5 MeV. The amount of energy released in this process will be :

Consider the following L.P.P minimize \(z=x-7 y+190\) subject to \(x+y \leq 8, x+y \geq 4, x \leq 5, y \leq 5\) and \(x, y \geq 0\). Then which of the following is/are true?
(A) It's feasible region is unbounded
(B) It's feasible region is bounded
(C) It's feasible region has 5 corner polnts
(D) It's feasible region has 6 corner polnts
Choose the correct answer from the options given below :

The number of moles of hydrogen oxidized in the oxidation of 67.2 L of H 2 at STP is:

Read the passage and answer the following
Microbes are organisms that cannot be seen with the naked eye. They include bacteria, archaea, protists, viruses, prions, and fungi. They account for most of the diversity of life on earth. Far more microbes inhabit a single person's body than there are people on the entire planet. Microbes also dominate in terms of their ability to live and thrive in extreme environments, including clouds, deep-sea volcanoes, springs, ice caps, and animal intestines. Most animals, including humans, exhibit numerous associations with microbes. Our "human microbiome" includes our skin, hair, mouth, and gut.
Select the chemical substances which are produced by some microbes and they can kill or prevent growth of other microbes :

Which of following is a correct statement ?