If A is a square matrix such that \(A^2=A\) and I is the identity matrix of same order as A, then the value of
\((A-2 I)^2-(2 A+I)^2+11 A\) is :

For \(\ x \in R -\{-1,0,1\} \), \(\int \frac{1}{x-x^5} d x\) is equal to:

Magnetic moment of a thin bar magnet is ' M '. If it is bent into a semicircular form, its new magnetic moment will be

Match List-I with List-II

List-I (Conditions)	List-II (Type of matrix)
(A) Square matrix \(A\)	(I) \(A=\left[a_{i j}\right]_{m \times m}\), where \(\left\{\begin{array}{ll}a_{i j}=0 & , i \neq j \\ a_{i j}=k & , i=j\end{array}\right.\) where \(k \neq 0\) is constant
(B) Scalar Matrix \(A\)	(II) \(A=\left[a_{i j}\right]_{m \times m}\)
(C) Diagonal matrix \(A\)	(III) \(A=\left[a_{i j}\right]_{m \times m}\), where \(\left\{\begin{array}{ll}a_{i j}=0 & , i \neq j \\ a_{i j}=1 & , i=j\end{array}\right.\)
(D) Identity matrix \(A\)	(IV) \(A=\left[a_{i j}\right]_{m \times m}\), where \(a_{i j}=0, i \neq j\)

Choose the correct answer from the options given below:

The integral \(I=\int e^x\left(\frac{x-1}{3 x^2}\right) d x\) is equal to:

For \(x \neq-1\), if \(\int \frac{x e^x d x}{(1+x)^2}=\frac{a e^x}{(1+x)^b}+c\), where \(a, b\) are fixed numbers and \(c\) is the integration constant, then \(a+b\) is equal to

The point on the curve \(y=\cos x-1, x \in\left[\frac{\pi}{2}, \frac{3 \pi}{2}\right]\) at which tangent is parallel to the x-axis is:

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.
Which one of the following statements is not true with respect to microbes?

Answer the question on the basis of passage given below :
One of the notable features of the transition element is the great variety of oxidation states it may show in its compounds. The elements which give the greatest number of oxidation states occur in or near the middle of the series. Manganese, for example exhibits all the oxidation states from +2 to +7. The lesser number of oxidation states at the extreme ends is either due to losing or sharing of few electrons (Sc, Ti) or too many d electrons of higher valence. Early in the first series scandium (II) is virtually unknown and titanium (IV) which is more stable than Ti (III) or Ti (II). On the other end, the only oxidation state of zinc is \(+ 2\).
Oxidation state of scandium is

If the elementary column transformation \(C_1 \rightarrow 2 C_1\) is carried out on the following matrix equation.
\(\left[\begin{array}{ll}1 & 3 \\ 2 & 4\end{array}\right]\left[\begin{array}{cc}-1 & 2 \\ 5 & 0\end{array}\right]=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]\) Then the transformed equation will be:

Baculoviruses are excellent candidates for species specific__________ application.

Let \(P\) and \(Q\) be any two invertible matrices of the same order. Then Match List-I with List-II

List - I (Matrix)	List - II (Equivalent matrix)
(A) \((P Q)^{-1}\)	(I) \(Q^{-1} P\)
(B) \(\left(P^{-1} Q\right)^{-1}\)	(II) \(Q P^{-1}\)
(C) \(\left(P Q^{-1}\right)^{-1}\)	(III) \(Q^{-1} P^{-1}\)
(D) \(\left(P^{-1} Q^{-1}\right)^{-1}\)	(IV) \(Q P\)

Choose the Correct answer from the options given below :

Consider \(f: R ^{+} \rightarrow[4, \infty)\) given by \(f(x)=x^2+4\)
(where \(R ^{+}\)is set of non negative real numbers) then \(f^{-1}(x)\) is :