Two persons A and B throw a die alternately till one of them gets a six and wins the game. If A begins, then the probabilities of winning of A and B respectively are

The rate of change of volume of a sphere with respect to its surface area, when radius is 4 cm, is equal to

The value of \(\left|\begin{array}{ccc}7! & 8! & 9! \\ 8! & 9! & 10! \\ 9! & 10! & 11!\end{array}\right|\) is :

A line through the points \(A(3,4,1)\) and \(B(5,1,6)\) is drawn.
The coordinates of the points where the line through the points \(A \backslash \& B\) crosses the \(X Y\) plane is:

\(\int\left(x+\frac{1}{x}\right)^2 d x\) equals :

If A and B are two events such that \(P(A) \neq 0\) and \(P(B \mid A)=1\) then

Match List-I with List-II

List-I (Integral)	List-II (Solution: C is an arbitrary constant)
(A) \(\int \frac{d x}{x^2+25}\)	(I) \(\frac{1}{10} \log \left|\frac{5+x}{5-x}\right|+C\)
(B) \(\int \frac{d x}{x^2-25}\)	(II) \(\log \left|x+\sqrt{x^2-25}\right|+C\)
(C) \(\int \frac{d x}{25-x^2}\)	(III) \(\frac{1}{5} \tan ^{-1}\left(\frac{x}{5}\right)+C\)
(D) \(\int \frac{d x}{\sqrt{x^2-25}}\)	(IV) \(\frac{1}{10} \log \left|\frac{x-5}{x+5}\right|+C\)

Choose the correct answer from the options given below:

Two pipes can fill a tank in 6 minutes and 12 minutes respectively, and a third pipe can empty the tank at the rate of 18 liters per minute. If all the pipes working together can fill the empty tank in 5 minutes, the capacity of the tank is:

The two coherent monochromatic light beams of intensities \( I \) and \( 4I \) are superposed. The maximum and minimum possible intensities in the resulting beams are :

Read the following paragraph carefully and answer the given questions.
In the year 1963 the two enzymes responsible for restricting the growth of bacteriophage in E. coli were isolated. The first restriction endonuclease - Hind II, whose functioning depended on a specific DNA nucleotide sequence was characterized five years later. It was found that Hind II always cut DNA molecules at a particular point by recognizing a specific sequence of six base pairs. This specific base sequence is known as the recognition sequence for Hind II. Besides Hind II, today we know more than 900 restriction enzymes that have been isolated from over 230 strains of bacteria, each of which recognize different recognition sequences.
When DNA from different sources is cut by the same restriction enzymes, the resultant DNA fragments have the :

If the standard electrode potential, \(E_{\text {cell }}^{\circ}=1.10 V\), then the \(E_{\text {cell }}\) for the following reaction at 298 K is
\(Zn ( s )+ Cu ^{2+}(0.1 M ) \rightarrow Zn ^{2+}(0.01 M )+ Cu ( s )\)

The cell emf depends on:

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 -