The value of \(\cot ^{-1}\left[2 \cos \left(2 \sin ^{-1} \frac{1}{2}\right)\right]\) is

Area of the region bounded by the curve \(y=\sqrt{x}\) and lines \(x+y=2, y=0\) is

If \(y =\sin ^{-1} x\), then

If maximum value of \(f(x)=2 x^3+3 x^2-6 a x+10\) occurs at \(x=-3\), then the value of \(a\) is ___________

Consider the following data of expenses (in lakhs) of an organization year wise:

Year	2001	2002	2003	2004	2005
Expenses (Rs. lakh)	160	185	220	300	510

Then expected expenses trend for the year 2006 using method of least squares is:

If \(y=3 \cos (\log x)+4 \sin (\log x)\), then choose correct option :

If A and B are two events such that \(P(A) = 0.2, P(B) = 0.4\) and \(P(A \cup B) = 0.5\) then value of \(P(A/B)\) is:

Let \(A, B, C, D\) and \(E\) be matrices of \(2 \times n, 3 \times k, 2 \times p\),
\(2 \times p\) and \(3 \times 3\) respectively.
Choose the correct statement(s) from the following :
(A) \(E B+D B\) will be defined if \(k=3, p=n\).
(B) \(E(3 \times 3) \cdot B(3 \times k) \rightarrow(3 \times k) \rightarrow n \times 3\).
(C) If \(n=p=2\), then the order of the matrix \(5 A^2-3 C\) is \(2 \times 2\).
(D) If \(n=p\), then the order of the matrix \(5 A^2-3 C\) is \(2 \times 3\).
Choose the correct answer from the options given below :

The atomic mass unit (1u) is correctly represented by :

What is the radius of the path of an electron moving with velocity of \(3 \times 10^7\) m/s in a magnetic field of \(6 \times 10^{-4}\) T perpendicular to it?

A metal wire under constant potential difference, when heated, shows that the drift velocity of the electron :

Arrange these ions into decreasing order of their ionic radii :
(A) \(\text{Ce}^{3+}\)
(B) \(\text{Pr}^{3+}\)
(C) \(\text{La}^{3+}\)
(D) \(\text{Nd}^{3+}\)
(E) \(\text{Pm}^{3+}\)
Choose the correct answer from the options given below :

We use microbes and microbially derived products almost every day. Bacteria called lactic acid bacteria (LAB) grow in milk to convert it into curd. The dough, which is used to make bread, is fermented by yeast called Saccharomyces cerevisiae. Microbes are used to produce industrial products like lactic acid, acetic acid and alcohol, which are used in a variety of processes in the industry. However, certain bacteria, which grow anaerobically on cellulosic material, produce large amount of methane along with CO2 and H2. These bacteria are collectively called methanogens, and one such common bacterium is Methanobacterium. These bacteria are commonly found in the anaerobic sludge during sewage treatment. Microbes are also used for commercial and industrial production of certain chemicals like organic acids, alcohols and enzymes.
Examples of acid producers are Aspergillus niger (a fungus) of citric acid. You are familiar with the commonly used antibiotic Penicillin. Do you know that Penicillin was the first antibiotic to be discovered, and it was a chance discovery? 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 discovery of penicillin was a chance discovery by Alexander Fleming while working on: