If \(y=\sin ^{-1} x\), then \(\left(1-x^2\right) \frac{d^2 y}{d x^2}-x \frac{d y}{d x}\) is :

Match List - I with List - II. Given \(P(A)=\frac{1}{3}\) and \(P(B)=\frac{1}{5}\) where \(A\) and \(B\) are independent.

LIST I	LIST II
A. \(P(A \cap B)\)	I. \(\frac{7}{15}\)
B. \(P \left( A ^{\prime} \cap B ^{\prime}\right)\)	II. \(\frac{2}{5}\)
C. \(P\) (at least one of the two events takes place)	III. \(\frac{1}{15}\)
D. \(P\) (only one event takes place)	IV. \(\frac{8}{15}\)

Choose the correct answer from the options given below:

A letter is known to have come either from KOLKATA or TATANAGAR. On the envelope just two consecutive letters TA are visible. The probability that letter has come from TATANAGAR is

The simplified form of \(\tan ^{-1}\left(\frac{\cos x}{1+\sin x}\right),-\frac{\pi}{2}< x <\frac{\pi}{2}\) is

If \(f(x)=3 x^2+10 x-5\), then by using differentials the appropriate value of \(f(2.03)\) is

The general solution of the differential equation \(\frac{d y}{d x}=e^{a x+b y}\) is : (Here \(C\) is an arbitrary constant)

\(\int \frac{1}{x^2\left(x^4+1\right)^{3 / 4}} d x\) is equal to :

Arrange the following steps in sequence for Ascaris infection in humans :
(A) Development of round worms in human host
(B) Excretion of parasite eggs along with the feces of infected person resulting in soil and water contamination
(C) Infection of healthy person via contaminated water, vegetables, fruits etc.
(D) Muscular pain, fever, blockage of intestinal passage
Choose the correct answer from the options given below:

\(\int e^x\left(\frac{1-x}{1+x^2}\right)^2\) dx is equal to :

Observe the diagram and answer the following questions:

Identify the correct statements with respect to retrovirus.
A. DNA genome of the virus forms complementary RNA
B. After getting into the body of a person the virus enter into macrophages
C. The viral RNA gets incorporated into host cell DNA
D. The macrophages act like virus factory
Choose the correct answer from the options given below:

Identify the correct statement:

When sodium chromate is acidified with sulfuric acid and then further treated with potassium chloride, we obtain…

Read the given paragraph carefully and answer the following questions:
Since the DNA is enclosed within the membranes, we have to break the cell open to release DNA along with other macromolecules such as RNA, proteins, polysaccharides and also lipids. This can be achieved by treating the bacterial cells/plant or animal tissue with enzymes such as lysozyme (bacteria), cellulase (plant cells), chitinase (fungus). There are some basic steps in genetically modifying an organism (i) identification of DNA with desirable genes; (ii) introduction of the identified DNA into the host; (iii) maintenance of introduced DNA in the host and transfer of the DNA to its progeny. Agrobacterium tumifaciens, a pathogen of several dicot plants is able to deliver a piece of DNA known as 'T-DNA' to transform normal plant cells into a tumor and direct these tumor cells to produce the chemicals required by the pathogen. Other molecules can be removed by appropriate treatments and purified DNA ultimately precipitates out after the addition of chilled ethanol. Selection of recombinants due to inactivation of antibiotics is a cumbersome procedure because it requires simultaneous plating on two plates having different antibiotics. Therefore, alternative selectable markers have been developed which differentiate recombinants from non-recombinants on the basis of their ability to produce colour in the presence of a chromogenic substrate. In this, a recombinant DNA is inserted within the coding sequence of an enzyme, ß'-galactosidase. This results into inactivation of the gene for synthesis of this enzyme, which is referred to as insertional inactivation.
Agrobacterium tumefaciens, a pathogen of several dicot plants, is able to deliver a piece of DNA known as: