Which of the following terms are associated with a linear programming problem?
(A) Constraints
(B) Independent events
(C) Feasible region
(D) Objective function
Choose the correct answer from the options given below :

The integral \(I=\int \frac{e^x}{1-e^{2 x}} d x\) is equal to :

A random variable X has the following probability distribution where k is some number :

X	0	1	2
P(X)	k	2k	3k

Find the value of k.

The corner points of feasible region determined by the following systems of linear inequalities :
\(2 x+y \leq 10, x+3 y \leq 15, x \geq 0, y 0\) are \((0,0),(5,0),(3,4)\) and \((0,5)\)
Let \(z=p x+q y\), when \(p, q>0\) then the relation between \(p\) and \(q\), so that maximum of \(z\) occurs at both points \((3,4)\) and \((0,5)\) is:

The intervals in which the function \(f(x) = 2x^3 + 9x^2 + 12x + 20\) is increasing is given by:

The feasible region of the constraints \(x \geq 0, x+y \leq 1\) and \(x-y \leq 1\) is situated in:

Match List-I with List-II

Llst-I	Llst-(II)
(A) \(f(x)=[x]\)	(I) Continuous everywhere but not differentiable at \(x=-1\)
(B) \(f(x)=|x-1|\)	(II) Continuous everywhere except at all integral values
(C) \(f(x)=e^{|x|}\)	(III) Continuous everywhere but not differentiable at \(x=1\)
(D) \(f(x)=|x+1|\)	(IV) Continuous everywhere but not differentiable at \(x=0\)

Choose the correct answer from the options given below:

Value of \(\int \frac{2}{(x-3) \sqrt{x+1}} d x\) is : (Here \(C\) is an arbitrary constant)

Arrange the following substances in the increasing order of magnetic susceptibility.
(A) Paramagnetic substances
(B) Superconductors
(C) Diamagnetic substances
(D) Ferromagnetic substances
Choose the correct answer from the options given below :

Read the following passage carefully and answer the given questions.
Bioactive molecule, cyclosporin A, that is used as an immunosuppressive agent in organ-transplant patients, is produced by the fungus Trichoderma polysporum. Statins produced by the yeast Monascus purpureus have been commercialised as blood-cholesterol lowering agents. It acts by competitively inhibiting the enzyme responsible for synthesis of cholesterol. Baculoviruses are pathogens that attack insects and other arthropods. The majority of baculoviruses used as biological control agents are in the genus Nucleopolyhedrovirus. These viruses are excellent candidates for species-specific, narrow spectrum insecticidal applications. They have been shown to have no negative impacts on plants, mammals, birds, fish or even on non-target insects. BOD refers to the amount of the oxygen that would be consumed if all the organic matter in one liter of water were oxidised by bacteria. The sewage water is treated till the BOD is reduced. Cheese, is one of the oldest food items in which microbes were used. Different varieties of cheese are known by their characteristic texture, flavour and taste. For example, the large holes in 'Swiss cheese' are due to production of a large amount of CO₂ by a bacterium named Propionibacterium sharmanii.
Statins are produced by:

ABO blood groups are controlled by the -

Match List-I with List-II

List-I (Integral)	List-II (Value)
(A) \(\int_{-1}^1(|x|+1) d x\)	(I) 0
(B) \(\int_{-2}^2|x+1| d x\)	(II) 2
(C) \(\int_{-1}^1 3|x|^2 d x\)	(III) 5
(D) \(\int_{-1}^1 x|x| d x\)	(IV) 3

Choose the Correct answer from the options given below :

The equation of the normal to the curve \(y=4 x^3+2 \sin x\) at \((0,3)\) is :