\(\int \frac{\log _e x}{\left(1+\log _e x\right)^2} d x\)2 is equal to

A random variable X has the following probability distribution

X	2	3	4	5
P(X)	5/k	7/k	9/k	11/k

Then the value of \(\frac{k}{4}\) is

Given a linear programming problem max \(z=22 x+18 y\) subject to \(x+y \leq 20\), \(360 x+240 y \leq 5760\) and \(x \geq 0, y \geq 0\). Its corner points are:

Match List - I with List - II

List - I (Matrix/Equations)	List - II (Values)
(A) \(\left[\begin{array}{cc}2 x+1 & 3 y \\ 0 & y^2-5 y\end{array}\right]=\left[\begin{array}{cc}x+3 & y^2+2 \\ 0 & -6\end{array}\right]\)	(I) \(x = 2, y = - 1\)
(B) \(\left[\begin{array}{ccc}1 & 2 & -1 \\ x & 0 & 3 \\ y & 3 & 4\end{array}\right]\) is symmetric	(II) \(x = 2, y = 2\)
(C) \(\left[\begin{array}{ll}x & 1\end{array}\right]\left[\begin{array}{cc}1 & 0 \\ -2 & -3\end{array}\right]\left[\begin{array}{ll}5 & 2 \\ 0 & y\end{array}\right]=0\)	(III) \(x = - 2, y = 2\)
(D) \(\left[\begin{array}{ll}x & 0 \\ 1 & 1\end{array}\right]\left[\begin{array}{ll}x & 0 \\ 1 & 1\end{array}\right]=\left[\begin{array}{cc}4 & 0 \\ -1 & y / 2\end{array}\right]\)	(IV) \(x = 2, y = 0\)

Choose the correct answer from the options given below:

The general solution of \(\frac{d y}{d x}=1+x^2+y^2+x^2 y^2\) is:
(given that \(C\) is the constant of integration)

\(\int \frac{\sqrt{1-x}}{\sqrt{1+x}} d x=\alpha \sqrt{1-x^2}+\beta \sin ^{-1} x+C\), where \(C\) is an arbitrary constant, then which of the following are TRUE?
(A) \(\alpha=1\)
(B) \(\alpha=-1\)
(C) \(\beta=1\)
(D) \(\beta=-1\)
Choose the correct answer from the options given below :

\(\tan ^{-1} 3+\tan ^{-1} \lambda=\tan ^{-1}\left(\frac{3+\lambda}{1-3 \lambda}\right)\) is valid for which values of \(\lambda\) ?

Removal of vas deferens surgically is called:

\(2 N_2 O _5 \rightarrow 4 NO _2+ O _2\)
At a given instant of time, the rate and rate constant are 1.02 \(\times\) 10\(^-\)\(\mathrm{4}\) mol L\(^-\)\(\mathrm{1}\) s\(^-\)\(\mathrm{1}\) and 3.4 \(\times\) 10\(^-\)\(\mathrm{5}\) s\(^-\)\(\mathrm{1}\), respectively.
The concentration of N\(_2\)\(\mathrm{O}\)_5 at that time will be:

Select the correct option for the statements given below.
(A) The diffraction pattern is a superposition of waves originating from each point on a single slit.
(B) The interference pattern is the superposition of two waves originating from two narrow slits.
(C) The diffraction pattern in single slit has a central bright maximum brighter than other maxima.
(D) The interference in double slit has a number of equally spaced bright and dark bands.
Choose the correct answer from the options given below :

Read the passage carefully and answer the Questions
Carbonyl compounds such as aldehydes, ketones, and carboxylic acids play an important role in the biochemical process of plants and animal kingdom.
Many aldehydes and ketones find applications in dye, perfumery and other industries.
The carbonyl carbon atom is \(s p^2\)-hybridized forming three sigma and one pi-bond with oxygen.
C-O bonds are polar bond due to high electronegativity of oxygen than carbon.
Hence the carbonyl carbon behaves as an electrophilic center and oxygen as a nucleophilic centre.
Primary and secondary alcohols yield aldehydes and ketones respectively upon oxidation with variety of oxidizing agents.
The boiling points of aldehydes and ketones differ from hydrocarbons and ethers.
Aldehydes and ketones undergo nucleophilic addition reaction due to presence of electrophilic carbonyl carbon.
Nucleophile and a proton add across the carbon oxygen double bond.
The alcohol which cannot be used to prepare carbonyl compound through oxidation is

Which of the following components were taken by S.L. Miller in a closed flask for his experiment on the origin of life?
(A) CH₄
(B) NH₃
(C) CO₂
(D) Water vapours
Choose the correct answer from the options given below:

The order of basic strength of amines in aqueous solution is :