The mean of the number of heads in a simultaneous toss of three coins is

When two independent small samples of sizes \(n_1\) and \(n_2\) with means \(\bar{x}_1\) and \(\bar{x}_2\) respectively are drawn from populations with identical population variances, the test-statistic is computed as :

The value of \(\int_{0}^{\frac{\pi}{2}} \log \left( \frac{5 + 4 \sin x}{5 + 4 \cos x} \right) dx\) is:

A die is thrown 4 times and getting 3 is considered a success. The probability of 2 successes is :

The derivative of \(\cos ^{-1}\left(\frac{\sin x+\cos x}{\sqrt{2}}\right),-\frac{\pi}{4} < x < \frac{\pi}{4}\) with respect to \(x\) is :

If \(y=500 e^{7 x}+600 e^{-7 x}\), then \(\frac{d^2 y}{d x^2}\) is equal to :

For the function \(f(x)=e^{-2 x}(2-x)^2\), the point of local maxima is:

Read the passage carefully and answer:
Organic compounds derived from NH3 by replacing one or more H-atoms with alkyl or aryl groups will form primary, secondary and tertiary amines. The common methods which include their preparations are reduction of nitro group, amide group and nitrile group, Gabriel phthalimide synthesis and ammonolysis of alkyl halides. Amines are basic in nature due to the presence of a lone pair of electrons on the nitrogen atom. Important chemical properties of amines include acylation, alkylation and their interaction with nitrous acid. There are various tests which can distinguish different types of amines from each other, for e.g. carbylamine test, Hinsberg's test, coupling reaction etc. Another important class of compounds which can be derived from primary amines is azo compounds. These compounds are intensely colored and widely used as dyes in textile, leather and food industries.
Conversion of nitro group into primary amine is preferred using

Decomposition of \(H _2 O _2\) follows a first order reaction.
In 50 min , the concentration of \(H _2 O _2\) decreases from 0.5 to 0.125 M.
When the concentration of \(H _2 O _2\) reaches 0.05 M.
the rate of formation of \(O _2\) will be (Given \(\log 2=0.3010\) ) :

Match List-I with List-II

List-I (Process/terms related to ecosystem)	List-II (Explanations)
(A) Decomposition	(I) Highly resistant to microbial actions
(B) Humus	(II) Oxygen dependent process
(C) Stratification	(III) Break down of detritus
(D) Fragmentation	(IV) Distribution of species

Choose the correct answer from the options given below:

Answer the question on basis of passage given below :
A ketone \([A]\) which undergoes haloform reaction gives compound \([B]\) on reduction. Compound
\([B]\) on heating with sulphuric acid gives compound \([C]\) which forms monozonide \([D]\). The compound [D] on hydrolysis in the presence of zinc dust gives compound [E]. 
Compound [E] on oxidation gives \(C _2 H _4 O _2\) [F]
The compound \([ B ]\) formed can be prepared from reaction of aq. KOH . with :

Let \(e^y(x+1)=1\). Then which of the following are TRUE
(A) \(\frac{d^2 y}{d x^2}=-\frac{1}{(x+1)^2}\)
(B) \(\frac{d^2 y}{d x^2}=\left(\frac{d y}{d x}\right)^2\)
(C) \(\left.\frac{d^2 y}{d x^2}\right|_{x=0}=-1\)
(D) \(\left.\frac{d^2 y}{d x^2}\right|_{x=0}=1\)
(E) \(\left.\frac{d^2 y}{d x^2}\right|_{x=1}=\frac{1}{4}\)
Choose the correct answer from the options given below :

The power factor of an ac circuit having resistance \( R \) and inductance \( L \) connected in series is :