If \(y=2 x(x-1)(2-x)\); then \(\frac{d^2 y}{d x^2}\) is:

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 \(R=\left\{(x, y) \mid x, y \in Z , x^2+y^2 \leq 4\right\}\) is a relation in \(Z\), then domain of \(R\) is :

(A) \(\frac{d}{d x} a^x=\frac{a^x}{\log _e a}\)
(B)\(\frac{d}{d x} e^{a x}=a e^{a x}\)
(C) \(\frac{d}{d x} x^x=x \cdot x^{x-1}\)
(D) \(\frac{d}{d x} x^a=a x^{a-1}\)
(E) \(\frac{d}{d x} a^a=0\)
Choose the correct answer from the options given below:

If A and B are symmetric matrices of the same order, then

Let \(\vec{a}=\hat{i}+2 \hat{j}+3 \hat{k}\) and \(\vec{b}=-2 \hat{i}+3 \hat{j}-4 \hat{k}\), then which of the following statements are correct?
(A) \(|\vec{a}|=\sqrt{14}\)
(B) \(|\vec{b}|=29\)
(C) \(\vec{a} \cdot \vec{b}=8\)
(D) Angle between \(\vec{a}\) and \(\vec{b}\) is \(\cos ^{-1}\left(\frac{-8}{\sqrt{406}}\right)\)
Choose the correct answert from the option given below :

Out of students in a school, it is known that \(65 \%\) reside in hostel and \(35 \%\) are day scholars. Previous year results report that \(40 \%\) of the students who reside in hostel attain A grade and \(20 \%\) of day scholars attain A grade in their annual examination.
At the end of the year, one student is chosen at random from the class.
Probability of the selected student attaining grade \(A\) 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.
Match List - I with List - II.

List - I	List - II
(A) Methanobacterium	(I) Swiss cheese
(B) Lactobacillus	(II) Ethanol
(C) Propionibacterium sharmanii	(III) Biogas
(D) Saccharomyces cerevisiae	(IV) Curd

Choose the correct answer from the options given below:

Match List - I with List - II.

List - I (Polymer)	List - II (Monomer)
(A) Nylon 2-nylon 6	(I) Ethylene glycol and Benzene-1,2-dicarboxylic acidne
(B) Neoprene	(II) Glycine and Amino caproic acid
(C) Glyptal	(III) Adipic acid and Hexamethylene diam
(D) Nylon 6, 6	(IV) 2-chlorobuta-1,3-diene

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Bone structure of forelimbs of vertebrates like whale, bat, cheetah and human is evidence of:

The function \(f:[-1,1] \rightarrow \mathbb{R}\) (set of real numbers) given by \(f(x)=\frac{x}{x+3}\) is

A dipositive transition metal ion with atomic number 25 in aqueous solution; what is the magnetic moment of it?

Match List-l with List-ll

List - l (Inverse Trigonometric function)	List - ll (Principal values of arguments)
(A) \(\sin ^{-1}\left(-\frac{1}{2}\right)\)	(I) \(-\frac{\pi}{3}\)
(B) \(\cos ^{-1}\left(-\frac{1}{2}\right)\)	(II) \(\frac{3 \pi}{4}\)
(C) \(\tan ^{-1}(-\sqrt{3})\)	(III) \(-\frac{\pi}{6}\)
(D) \(\sec ^{-1}(-\sqrt{2})\)	(IV) \(\frac{2 \pi}{3}\)

Choose the correct answer from the options given below :