The probability of not getting 53 Sundays in a leap year is :

\(\frac{d^2}{d x^2}\left(\operatorname{det}\left[\begin{array}{cc}x^3 & x \\ 2 & e^x\end{array}\right]\right)\) equals

The feasible region of a linear programming problem is bounded. The corresponding objective function is \(Z = 3x - 4y\). The objective function attains:

The integral \(\int \frac{2 x+x^3}{1+x^2} d x\) is equal to :

Match List I with List II with regard to degree of the Differential Equation.

List - I	List - II
(A) \(\frac{d^3 y}{d x^3}+\sin y=0\)	(I) 4
(B) \(\left(\frac{d y}{d x}\right)^4+\cos \left(\frac{d y}{d x}\right)-1=0\)	(II) 1
(C) \(\left(\frac{d^2 y}{d x^2}\right)^2+\left(\frac{d y}{d x}\right)^4=0\)	(III) Not defined
(D) \(\left(\frac{d^2 y}{d x^2}\right)^5+2\left(\frac{d^3 y}{d x^3}\right)^4+7=0\)	(IV) 2

A trust invites a deposit of a lump sum amount from an individual so that an annual scholarship of Rs. 5000 is paid. Rate of interest is 5% per annum.
If the scholarship is to start at the end of this year, then the amount needed to deposit to the trust is:

If \(A=\left[\begin{array}{ccc}3 & -2 & 3 \\ 2 & 1 & -1 \\ 4 & -3 & 2\end{array}\right]\), then the matrix \((\operatorname{adj} A) A\) is equal to

The Hardy - Weinberg Principle says:
(A) Allele frequency in a population is stable and constant.
(B) Allele frequency in a population is unstable.
(C) The gene pool remains constant. It is called genetic equilibrium.
(D) The sum total of all the Allelic frequencies is 1.
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.
Nostoc is an example of :

Exposure of chloroform in air in the presence of sunlight releases a gas which is poisonous. Identify the gas.

A person can row a boat at \(5 km / hr\) in still water. If the speed of water current in a river is \(1 km / hr\), and it takes him \(1\) hour to row to a place and come back, how far off is the place?

The electric field of a plane electromagnetic wave in a medium is given by
\( E = 100 \sin(300z - 6 \times 10^{10}t) \text{ V/m} \).
The velocity of the electromagnetic wave in this medium will be:

\(\int \frac{1}{x \sqrt{x^2-9}} d x\) is equal to (given that C is constant of integration)