The length of perpendicular from the point (1,0,1) to the plane x - y + z = 4 is:

For the objective function \(Z = 4x + 6y\) subject to the constraints
\(3x + 2y ≥ 5, 7x + 2y ≤ 9, x ≥ 0, y ≥ 0\),
the maximum value of Z occurs at (a, b)
and the minimum value of Z occurs at (p, q)
then the value of \(\frac{a}{p}+\frac{b}{q}\) is:

Which of the following statements are correct ?
(A) Determinant is a square matrix
(B) Determinant of a matrix of order \(3 \times 3\) is \(a_{11} c_{11}+a_{12} c_{21}+a_{13} c_{31}\left(a_{i j}=\right.\) element, \(c_{i j}=\) cofactor \()\)
(C) Determinant is a number associated with a matrix
(D) Determinant is a number associated with a square matrix
(E) Determinant of a matrix of order \(3 \times 3\) is \(a_{11} c_{11}+a_{12} c_{12}+a_{13} c_{13}\), where \(a_{i j}\) is the element, while \(c_{i j}\) is its corresponding cofactor
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Which of the following statements is/are true?
(A) The vector sum of the three sides of a triangle in order is \(\vec{c}\)
b (B) The magnitude (r), direction ratios (a, b, c) and direction cosines (l, m, n) of any vector \(\vec{r}=a \hat{i}+b \hat{j}+c \hat{k}\) are related as \(l=\frac{a}{r}, m=\frac{b}{r}, n=\frac{c}{r}\)
(C) If θ is the angle between two vectors \(\vec{a}\) and \(\vec{b}\), then their cross product is given as \(\vec{a} \times \vec{b}=|\vec{a}||\vec{b}| \sin \theta\)
(D) The cross product of two vectors is commutative
Choose the correct answert from the option given below :

Match List-I with List-II

List-I (Equationof Lines)	List-II (Information)
(A) \(\vec{r}=(3 \hat{i}-2 \hat{j}+\hat{k})+\lambda(\hat{j}-2 \hat{k})\)	(I) Direction ratios are \(2,4,-1\)
(B) \(\frac{2-x}{1}=\frac{2 y+1}{4}, z=2\)	(II) Perpendicular to \(2 \hat{i}-\hat{j}+\hat{k}\)
(C) \(\frac{x}{1}=\frac{y-3}{2}=\frac{3-4 z}{2}\)	(III) Passing through the point \((3,-2,1)\)
(D) \(\vec{r}=(3 \hat{i}+2 \hat{j}+\hat{k})+\lambda(2 \hat{i}+\hat{j}-3 \hat{k})\)	(IV) Direction ratios are \(-1,2,0\)

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If A is a square matrix of order \(3 \times 3\) and \(\left|A^T\right|=-3\), then \(\left|A A^T\right|\) is equal to

Match List-I with List-II

List-I (Integral)	List-II (Solution: C is an arbitrary constant)
(A) \(\int \frac{d x}{x^2+25}\)	(I) \(\frac{1}{10} \log \left|\frac{5+x}{5-x}\right|+C\)
(B) \(\int \frac{d x}{x^2-25}\)	(II) \(\log \left|x+\sqrt{x^2-25}\right|+C\)
(C) \(\int \frac{d x}{25-x^2}\)	(III) \(\frac{1}{5} \tan ^{-1}\left(\frac{x}{5}\right)+C\)
(D) \(\int \frac{d x}{\sqrt{x^2-25}}\)	(IV) \(\frac{1}{10} \log \left|\frac{x-5}{x+5}\right|+C\)

Choose the correct answer from the options given below:

Match List - I with List - II.

List - I	List - II
(A) Mutualism	(I) Abingdon tortoise in Galapagos island and goat
(B) Commensalism	(II) Crow and Cuckoo
(C) Brood Parasitism	(III) Sea anemone and Clown fish
(D) Competition	(IV) Lichen

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The value of \(\lambda\) and \(\mu\) if \((2 \hat{i}+6 \hat{j}+27 \hat{k}) \times(\hat{i}+\lambda \hat{j}+\mu \hat{k})=\overrightarrow{0}\) are respectively :

Solution of differential equation \(\frac{d y}{d x}=\cos (x+y+3)\) is:

Five dice are thrown simultaneously. If the occurrence of an even number in a single dice is considered a success, then the probability of at most 3 successes is:

There are two kinds of nucleases - exonucleases and endonucleases. Exonucleases remove nucleotides from the ends of the DNA whereas, endonucleases make cuts at specific positions within the DNA. The cutting of DNA by restriction endonucleases results in the fragments of DNA. These fragments can be separated by a technique known as gel electrophoresis. Since DNA fragments are negatively charged molecules they can be separated by forcing them to move towards the anode under an electric field through a medium/matrix. Small volume cultures cannot yield appreciable quantities of products. To produce in large quantities, the development of bioreactors, where large volumes (100 - 1000 litres) of culture can be processed, was required. Thus, bioreactors can be thought of as vessels in which raw materials are biologically converted into specific products, individual enzymes, etc., using microbial plant, animal or human cells. PCR stands for Polymerase Chain Reaction. In this reaction, multiple copies of the gene (or DNA) of interest are synthesised in vitro using two sets of primers (small chemically synthesised oligonucleotides that are complementary to the regions of DNA) and the enzyme DNA polymerase.
Identify the correct enzyme to release the DNA from the fungal cells.?

Let \(\vec{a}=2 \hat{i}-\hat{j}, \vec{b}=-4 \hat{j}+\hat{k}\) and \(\vec{c}=\hat{i}+2 \hat{k}\). If \(\vec{d}\) is a vector perpendicular to both \(\vec{a}\) and \(\vec{b}\) such that \(\vec{c} \cdot \vec{d}=34\), then \(|\vec{d}|\) is equal to