Consider the linear programming problem (LPP):
Minimize \(Z=x+y\)
subject to
\(x+2 y \leq 4,\)
\(3 x+y \geq 3\),
\(4 x+3 y \geq 6\),
\(x, y \geq 0\).
Which of the following is correct for the above linear programming problem (LPP):
(A) The LPP has a bounded feasible region.
(B) The LPP has a unique optimal solution.
(C) The optimal value of the LPP exists at the point (3/2,0)
(D) The corner points of the feasible region are \((3 / 2,0),(3 / 5,6 / 5),(2 / 5,6 / 5)\) and \((4,0)\)
Choose the correct answer from the options given below:

The value of \(c\) for which \(f(x)=x(x-3)^2, 0 \leq x \leq 3\), satisfies Rolle's theorem is:

The function \(f: R \rightarrow R\) defined as \(f(x)=x^2+2 x\) is

The area of the region enclosed between the parabolas \(y^2 = x + 1\) and \(y^2 = -x + 1\) is:

If \(A\) and \(B\) are independent events and \(P(A)=\frac{1}{2}, P(B)=\frac{1}{3}\) then
Match List-I with List-II

List-I	List-II
(A) \(P(A \cap B)\)	(I) \(\frac{1}{2}\)
(B) \(P(\bar{A}) P(B)+P(A) P(\bar{B})\)	(II) \(\frac{1}{3}\)
(C) \(P(A \mid B)+P(B \mid A)\)	(III) \(\frac{1}{6}\)
(D) \(P(\overline{A \cap B})\)	(IV) \(\frac{5}{6}\)

Choose the correct answer from the options given below :

The general solution of the differential equation \(x d y+\left(y-e^x\right) d x=0\) is :
where C is constant of integration

Two ropes are tied along the two diagonals of a cubic room. Ropes are inclined to each other at :

In a typical test cross:

List I contains physical quantities related to mechanical oscillations of a block of mass m attached to spring of force constant k. List II contains physical quantities related to a LC oscillator. Match List I with List II.

List I	List II
(A) Mass m	(I) Charge q
(B) Force Constant k	(II) Inductance L
(C) Displacement x	(III) Reciprocal capacitance 1/C
(D) Velocity ( \(v=\frac{d x}{d t}\) )	(IV) Current \(\left(i=\frac{d q}{d t}\right)\)

On the basis of analogies between mechanical and electrical quantities;
Choose the correct answer from the options given below:

Read the following passage carefully and answer the given questions.
Biofertilisers
Biofertilisers are organisms that enrich the nutrient quality of the soil and their main sources are bacteria, fungi and cyanobacteria. You have studied about the nodules on the roots of leguminous plants formed by the symbiotic association of Rhizobium. These bacteria can fix atmospheric nitrogen into organic forms. Other bacteria can also fix atmospheric nitrogen while free-living in the soil (examples Azospirillum and Azotobacter). Fungi are also known to form symbiotic associations with plants (mycorrhiza). Many members of the genus Glomus form mycorrhiza. Plants having such associations show other benefits also, such as resistance to root-borne pathogens, tolerance to salinity and drought, and an overall increase in plant growth and development. Cyanobacteria are autotrophic microbes widely distributed in aquatic and terrestrial environments many of which can fix atmospheric nitrogen, e.g. Anabaena, Nostoc, Oscillatoria, etc. In paddy fields, cyanobacteria serve as an important biofertiliser. Blue green algae also add organic matter in to the soil and increase its fertility. Currently, in our country, a number of biofertilisers are available commercially in the market and farmers use these regularly in their fields to replenish soil nutrients and to reduce dependence on chemical fertilisers.
Choose the free living, nitrogen-fixing bacteria from the following.

Which of the following statements are correct about alkyl halides :
A. Alkyl halides are polar so they are soluble in water
B. RX on reaction with alc.KOH give alkenes
C. RX on reaction with \(AgNO_2\) give RONO
D. \(S_N1\) proceeds with inversion in configuration
E. R-X have higher boiling point than parent alkanes
Choose the correct answer from the options given below :

A random variable \(X\) has the following probability distribution:

X	0	1	2	3	4	5	6	7
P(X)	0	k	2k	2k	3k	\(k^2\)	\(2 k^2\)	\(7 k^2+k\)

The value of \(P(4 < X < 7)\) is equal to

Let \(f: R \rightarrow R\) be a function defined as \(f(x)=x^4\). Which one of the following is true?