The objective function of an LPP is \(Z=a x+b y\). If the maximum value of the objective function is 180 , which occurs at two points \((15,15)\) and \((0,20)\), then which one of the following is true?

The equation of the normal to the curve \(y=\frac{x-7}{(x-2)(x-3)}\) at (7,0) is:

In a Linear Programming problem, the objective function is always :

The equation of the normal to the curve \(y=2 \sin x\) at \((0,0)\) is:

Relation \(R\) on the set \(A=\{1,2,\)...,\(15\}\) defined as \(R=\{(x, y): y-4 x=0\}\) is

If the binomial distribution \(X \sim B(n, p)\) has mean 3 and variance \(\frac{3}{2},(p+q)=1\), then which of the following is/are TRUE?
(A) \(q=\frac{1}{2}, n=6\)
(B) \(P(X \leq 5)=\frac{63}{64}, p=\frac{1}{2}\)
(C) \(q=\frac{1}{3}, p=\frac{2}{3}\)
(D) \(P(X=4)=\frac{15}{64}, n=6\)
Choose the correct answer from the options given below:

Consider a L.P.P, Maximize \(Z=3 x+5 y\) subject to constraints \(2 x+6 y \leq 6, x-y \geq 0, x \geq 0, y \geq 0\). Then which of the following are true?
(A) The feasible region of L.P.P is bounded region
(B) The corner points of the feasible region are \((0,0),(2,2),(0,1)\)
(C) Maximum value of Z is 9
(D) Point \((1,3)\) lies in the feasible region
Choose the correct answer from the options given below :

The function \(f(x)=x e^x\) increases only for:

After how many half lives a radioactive substance will decay to 6.25% of its original value :

Two point charges of 1 C each are separated from each other by a distance of 1 m in vacuum.
The force of interaction between them is:

Match List - I with List - II.

List - I (Organism)	List - II (Product)
(A) Aspergillus niger	(I) Lactic Acid
(B) Acetobacter aceti	(II) Citric Acid
(C) Lactobacillus	(III) Butyric Acid
(D) Clostridium butylicum	(IV) Acetic Acid

Choose the correct answer from the options given below:

A car starts from a point \(P\) at time \(t=0\) seconds and stops at \(Q\).
The distance \(x\), in meters, covered by it in \(t\) seconds is given by \(x(t)=t^3\left(3-\frac{t^2}{5}\right)\).
The distance between \(P\) and \(Q\) is:

DNA recombinant technology involves the use of restriction endonucleases. After having cloned the gene of interest and having optimised the conditions to induce the expression of the target protein, one has to consider producing it on a large scale. The cells harbouring cloned genes of interest may be grown on a small scale in the laboratory. In order to link the alien DNA, the vector needs to have very few, preferably single, recognition sites for the commonly used restriction enzymes. Presence of more than one recognition sites within the vector will generate several fragments, which will complicate the gene cloning. In Biolistics or gene gun method cells are bombarded with high velocity micro particles of gold or tungsten coated with DNA. The ligation of alien DNA is carried out at a restriction site present in one of the two antibiotic resistance genes. For example, you can ligate a foreign DNA at the BamH I site of tetracycline resistance gene in the vector pBR322. 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. The RNA can be removed by treatment with ribonuclease whereas proteins can be removed by treatment with protease. Other molecules can be removed by appropriate treatments and purified DNA ultimately precipitates out after the addition of chilled ethanol.
Which one of the following enzymes should be used to break the bacterial cells for isolation of DNA?