The present value of a sequence of payments of Rs. \(2000\) made at the end of every \(6\) months and continuing forever, if money is worth \(8 \%\) per annum compounded semi-annually, is:

If \(x\) is real, the minimum value of \(x^2-8 x+20\) is :

A swimmer whose speed in still water is \(6 km / hr\), swims between two points in a river and returns to the starting point. He took 20 minutes more to cover the distance upstream than downstream. If the speed of the stream is \(3 km / hr\), then the distance between the two points is:

If \(3\left[\begin{array}{cc}x & y \\ z & w\end{array}\right]=\left[\begin{array}{cc}x & 6 \\ -1 & 2 w\end{array}\right]+\left[\begin{array}{cc}4 & x+y \\ z+w & 3\end{array}\right]\), then the values of \(x, y, z\) and \(w\) are

Which of the following statements are correct in reference to the linear programming problem (LPP):
Maximize \(Z =5 x +2 y, Z =5 x +2 y, Z =5 x +2 y\)
subject to the following constraints
\(3 x+5 y \leq 15,5 x+2 y \leq 10, x \geq 0, y \geq 0\)
(A) The LPP has a unique optimal solution at \((2,0)\) only.
(B) The feasible region is bounded with corner points \((0,0),(2,0),(20 / 19,45 / 19)\) and \((0,3)\).
(C) The optimal value is unique, but there are an infinite number of optimal solutions.
(D) The feasible region is unbounded.
Choose the correct answer from the options given below:

\(\int \frac{\left(x^4-x\right)^{1 / 4}}{x^5} d x\) is equal to

Suppose \(10 \backslash \%\) of men and \(0.5 \backslash \%\) of women have grey hair. A grey haired person is selected at random. If there are equal number of males and females, then the probability of the selected person being a female is :

In a compound microscope, the distance between objective lens and eye lens is always:

The process of retrieval of information from the carrier wave at the receiver is termed as:

Electrostatics deals with the study of forces, fields and potential arising from static charges. Which of the following statements are correct ?
(A) Electrostatic force is a conservative force
(B) Charge is quantized because only integral number of electrons can be transferred from one body to the other
(C) In a uniform electric field E , an electric dipole experiences a torque \(\tau\) and a net force F .
(D) The electric field lines start from positive charges and end at negative charges. If there is a single charge, these may start or end at infinity.
(E) If the uniform surface charge density of an infinite plane sheet is positive, the electric field is directed away from the plate.
Choose the correct answer from the options given below :

Match List-l with List-II

List-I(Sugar)	List-II(Example)
(A) Invert Sugar	(I) Lactose
(B) Milk Sugar	(II) Sucrose
(C) Dextrose	(III) Fructose
(D) Ketohexose	(IV) Glucose

Choose the correct answer from the options given below:

Read the following paragraph and answer the given questions.
Bio-fertilisers are organisms that enrich the nutrient quality of the soil like cyanobacteria, while there are organisms that help to control plant diseases and pests. The very familiar beetle, the ladybird and dragonflies are useful to get rid of aphids and mosquitoes, respectively. An example of microbial biocontrol agents that can be introduced to control butterfly caterpillars is certain bacteria, which are available in sachets in the form of dry spores. These spores are mixed with water and sprayed onto vulnerable plants. Trichoderma species that are very common in the root ecosystem, are very effective biocontrol agents of several plant pathogens.Baculoviruses attack insects and other arthropods. Usage of these viruses is desirable when an ecological sensitive area is being treated.
Trichoderma is a/an:

The father of ecology in India is: