Let \(\vec{a}=4 \hat{i}-\hat{j}+3 \hat{k}\) and \(\vec{b}=-2 \hat{i}+\hat{j}-2 \hat{k}\). Then
(A) \(\vec{a}\) is a unit vector
(B) \(\vec{a} \times \vec{b}=-\hat{i}+2 \hat{j}+2 \hat{k}\)
(C) \(\vec{a}\) and \(\vec{b}\) are parallel vectors
(D) \(\vec{a}\) and \(\vec{b}\) are neither parallel nor perpendicular vectors
Choose the \(\backslash\) textbf\{correct\} answer from the options given below :

The integral \(\int \frac{x^4+2}{x^2+1} d x\) is equal to

If the maximum value of the function \(f(x)=\frac{2 \log _e x}{x}, x>0\) occurs at \(x=e\), then \(e^3 f^{\prime \prime}(e)\) is equal to

Match List-I with List-II

List-I	List-II
(A) Degree of this differential equation \(\frac{d^4 y}{d x^4}+2 \log _e\left(\frac{d^3 y}{d x^3}\right)=0\)	(I) 1
(B) Order of this differential equation \(e^{\left(\frac{dy}{dx}\right)^3}+3 y\left(\frac{d^2 y}{d x^2}\right)^3=0\)	(II) 4
(C) Degree of \(\frac{d^4 y}{d x^4}+\left(\frac{d y}{d x}\right)^2=0\)	(III) not defined
(D) Order of the differential equation \(2 \frac{d^4 y}{d x^4}+\left(\frac{d^2 y}{d x^2}\right)^5=0\)	(IV) 2

Choose the Correct answer from the options given below :

The maximum value of \(z=2 x+3 y\) subject to the constraints \(3 x-3 y \geq 0,2 x+2 y \leq 12, x \geq 0, y \geq 0\) occurs at the point:

The degree of the differential equation \(\left(\frac{d y}{d x}\right)^3+\left(\frac{d^2 y}{d x^2}\right)^2=0\) is:

If \(A\) is a square matrix such that \(A^2=A\), then \((I+A)^3-8 A\) is equal to:

The set of positive integers less than 50 forming the equivalence class of 6 modulo 9 is given by :

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 :

Read the following passage carefully and answer the given questions.
Restriction endonuclease on finding its specific recognition sequence, bind to the DNA and cut each of the two strands of the double helix at specific points in their sugar -phosphate backbones. Each restriction endonuclease recognises a specific palindromic nucleotide sequences in the DNA. Since the DNA is enclosed within the membranes, we have to break the cell open to release DNA along with other macromolecules such as RNA, proteins, polysaccharides and also lipids. This can be achieved by treating the bacterial cells/plant or animal tissue with enzymes such as lysozyme (bacteria), cellulase (plant cells), chitinase (fungus). Now a days the most commonly used matrix is agarose which is a natural polymer extracted from sea weeds. The DNA fragments separate (resolve) according to their size through sieving effect provided by the agarose gel. The first restriction endonuclease-Hind II, whose functioning depended on a specific DNA nucleotide sequence was isolated and characterised five years later. It was found that Hind II always cut DNA molecules at a particular point by recognising a specific sequence of six base pairs. In addition to 'ori', the vector requires a selectable marker, which helps in identifying and eliminating non-transformants and selectively permitting the growth of the transformants. Transformation is a procedure through which a piece of DNA is introduced in a host bacterium. Normally, the genes encoding resistance to antibiotics such as ampicillin, chloramphenicol, tetracycline or kanamycin, etc., are considered useful selectable markers for E. coli.
Transformation is a procedure through which:

Internal acid base reaction of amino acids is due to formation of:

Two cards are drawn simultaneously from a well shuffled pack of 52 cards. Then variance of the number of kings is:

An insect ingests the inactive toxin, it is converted into an active form of toxin due to