Given that the value of a determinant of order three is zero; which one of the following is not correct?

\(\cos \left(\frac{\pi}{6}-\cos ^{-1}\left(-\frac{\sqrt{3}}{2}\right)\right)\) is equal to:

The value of derivative of the function \(\cot ^{-1}\left\{(\cos 2 x)^{1 / 2}\right\}\) at \(x=\frac{\pi}{6}\) is

For what value of \(\alpha\), the function \(f\) defined by \(f(x)=\left\{\begin{array}{ll}\alpha\left(x^2-2 x+1\right), & \text { if } x \leq 0 \\ 2 x+1, & \text { if } x>0\end{array}\right.\) is continuous at \(x=0\) ?

If \(A\) and \(B\) are square matrices of order 3 such that \(|A|=-1\) and \(|B|=5\), then the value of \(|3 A B|\) is

If \(A=\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1\end{array}\right]\), then \(A^{-1}\) is equal to :

Match List-I with List-II
If \(P(A)=\frac{3}{7}, P(B)=\frac{4}{7}\) and \(P(A \cup B)=\frac{5}{7}\)

List-I	List-II
(A) \(P(A \cap B)\)	(I) \(\frac{2}{3}\)
(B) \(P(A \mid B)\)	(II) \(\frac{5}{7}\)
(C) \(P(B \mid A)\)	(III) \(\frac{1}{2}\)
(D) \(P\left(A^{\prime} \cup B^{\prime}\right)\)	(IV) \(\frac{2}{7}\)

Choose the correct answer from the options given below :

Vapour pressure of \(\text{CHCl}_3\) and \(\text{CH}_2\text{Cl}_2\) at 298 K are 200 mmHg and 415 mmHg respectively.
Calculate the vapour pressure of the solution prepared by mixing 25.5 g of \(\text{CHCl}_3\) and 40 g of \(\text{CH}_2\text{Cl}_2\) at 298 K.

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?

If \(\vec{a}+\vec{b}+\vec{c}=\overrightarrow{0}\) and \(|\vec{a}|=3,|\vec{b}|=5,|\vec{c}|=7\), then the angle between \(\vec{a}\) and \(\vec{b}\) is:

Observe the given diagram representing global biodiversity and answer the question.

Global Bio-diversity
Identify the statements which hold true for biodiversity in India?
(A) India is amongst one of the twelve megadiversity countries of the world.
(B) India has only 8.4 percent of the world's land area.
(C) India has 8.1 percent of the global species diversity.
(D) Almost 45,000 species of plants have been recorded in India.
Choose the correct answer from the options given below :

Let \(A=\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{array}\right]\), then \(\left(A^{-1}\right)^T\) equals

Common oxidation state of lanthanoids is: