The maximum value of the objective function \(z=2 x+3 y\) of an L.P.P. subjected to the constraints \(x-y \leq 1, x+y \leq 3, x, y \geq 0\), is

Consider the region bounded by the lines \(y-1=x, x=-2, x=3\) and \(x\)-axis. Then,
(A) The area of the bounded region is given by \(\int_{-2}^3(x+1) d x\)
(B) The numerical value of the area is \(\frac{15}{2}\) sq. units
(C) The numerical value of the area is 8 sq . units
(D) The numerical value of the area is \(\frac{17}{2}\) sq. units
Choose the correct answer from the options given below:

The product of the order and degree of the following differential equation :
\(x \left( \frac{d^2y}{dx^2} \right)^2 + \left( \frac{dy}{dx} \right)^3 + y^2 = 0\)

A letter is known to have come from either TATAPUR or from CHAKRATA. On the envelope, only two letters "TA" are visible consecutively. The probability that the letter has come from CHAKRATA is :

Two events A and B are dependent if :

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:

The value/s of \(\lambda\) for which the given system of equations will have infinitely many solutions \(x+y-6 z=0 ; \lambda x-y+2 z=0 ;-3 x+\lambda y+2 z=0\), is/are:

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.
The greater the BOD of waste water-

What is the approximate wavelength of the radiation emitted when the electron in an hydrogen atom jumps from n = \( \infty \) to n = 3?

In a feedback amplifier, if the feedback voltage is in opposite phase, the gain is less than 1. Then it will:

Read the following passage carefully and answer the given questions.
During anaerobic sludge digestion, bacteria produce a mixture of gases such as methane, hydrogen sulphide and carbon dioxide. These gases form biogas and can be used as source of energy as it is inflammable. The bioactive molecule, cyclosporin A, that is used as an immunosuppressive agent in organ-transplant patients, is produced by the fungus Trichoderma polysporum. You must have noticed that bottled fruit juices bought from the market are clearer as compared to those made at home. This is because the bottled juices are clarified by the use of pectinases and proteases. 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. The BOD test measures the rate of uptake of oxygen by micro-organisms in a sample of water and thus, indirectly, BOD is a measure of the organic matter present in the water.
Statins, used as blood-cholesterol lowering agents are produced by -

The current flowing through the two bulbs marked as 60 W, 240 V each when connected in series with 240 V source is:

Match List-I with List-II
[.] denotes the greatest integer function.

List-I	List-II
(A) \(\int_0^3[x] d x\)	(I) \(\frac{1}{2}\)
(B) \(\int_0^1[2 x] d x\)	(II) 1
(C) \(\int_0^1[3 x] d x\)	(III) \(\frac{3}{2}\)
(D) \(\int_0^1[4 x] d x\)	(IV) 3

Choose the correct answer from the options given below :