If \(|\vec{a}|=1,|\vec{b}|=2,|2 \vec{a}+\vec{b}|=2 \sqrt{3}\) then \(|\vec{a}-\vec{b}|\) is :

The derivative of \((\log x)^x\) with respect to log \(x\) is:

The function \(f ( x )=\log (\sin x )\) is ___________

The area (in sq. units) of the region \(\left\{(x, y): 3 x^2 \leq y \leq|x|\right\}\) is equal to :

Let functions \(f:\{1,3,4\} \rightarrow\{1,2,5\}\) and \(g:\{1,2,5\} \rightarrow\{1,3\}\) be defined as \(f=\{(1,2),(3,5),(4,1)\}\) and \(g=\{(1,3),(2,3),(5,1)\}\) respectively. Then, which one of the following is correct statement for \(g \circ f\)?

For the system of equations \(A X=B\), which of the following is correct?

Let \(y(x)=a(x+1) \log (x+1)+b x+5\) be the solution of the differential equation \(e^{\frac{d y}{d x}}=x+1, y(0)=5\), then the value of \((a+b)\) is :

Read the passage carefully and answer the Questions
In the formation of coordination complexes, if the inner \(d\) orbital \((n-1) d\) is used in hybridisation, the complex, is called an inner orbital or low spin or spin paired complex. And if it uses outer \(d\) orbital ( \(n d\) ) in hybridisation (like \(s p^3 d^2\) ), it is called outer orbital or high spin or spin free complex. The degeneracy of the \(d\) orbitals has been removed due to ligand electron-metal electron repulsions in the complex. This splitting of the degenerate levels due to the presence of ligands in a definite geometry is termed as crystal field splitting. For coordination complexes, the magnetic moment is determined by the number of unpaired electrons and is calculated by using the 'spin-only' formula, \(\mu=\sqrt{n(n+2)}\) where \(n\) is the number of unpaired electrons and \(\mu\) is the magnetic moment in units of Bohr magneton (BM). The coordination compounds are of great importance. These compounds are widely present in the mineral, plant and animal worlds and are known to play many important functions in the area of analytical chemistry, metallurgy, biological systems, industry and medicine.
Which of the following complex has the highest magnetic moment?

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?

Match List - I with List - II.

List - I (Magnetic Property)	List - II (Substance)
(A) Ferrimagnetic	(I) MnO
(B) Antiferromagnetic	(II) \(H _2 O\)
(C) Ferromagnetic	(III) \(ZnFe _2 O _4\)
(D) Diamagnetic	(IV) \(CrO _2\)

Choose the correct answer from the options given below:

Mercury is not considered as transition metal however it has tendency to react with chlorine to form chlorides \(X\) and \(Y\).
\(X\) reacts with \(K I\) to give red ppt while \(Y\) doesn't react. On the contrary, \(Y\) reacts with \(NH _4 OH\) to give a black ppt. Predict the formula of \(X\) and \(Y\) :

Standard reduction potentials (\(\text{E}^{\circ}\)) of \(\text{Pb}^{2+}, \text{Ag}^+, \text{Cu}^{2+}\) and \(\text{Fe}^{2+}\) are \(-0.13 \text{ V}, +0.80 \text{ V}, +0.31 \text{ V},\) and \(-0.44 \text{ V}\) respectively. Which is the strongest reducing agent?

Isomerism shown by the complex pair [Co(NH\(_3\))\(_5\)(NO\(_3\))]SO\(_4\) and [Co(NH\(_3\))\(_5\)(SO\(_4\))]NO\(_3\) is: