\(\int \frac{d x}{e^x+e^{-x}}\) is equal to

If under pure competition demand and supply functions are given by \(p=\sqrt{10-x}\) and \(p=\frac{1}{2}(x-2)\) respectively, where \(p\) is price per unit and \(x\) is quantity, then the consumer surplus is :

Match List I with List II

LIST- I	LIST II
A. \(l x+m y+n z=d\) is	I. Equation of plane passing through a given point and normal to given vector
B. \(\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1\) is	II. Equation of plane in normal form
C. \((\vec{r}-\vec{a}) \cdot \vec{n}=0\)	III. Plane passing through the intersection of two planes
\(D .\left(a_1 x+b_1 y+c_1 z+d_1\right)+\lambda\left(a_2 x+b_2 y+c_2 z+d_2\right)=0\)	IV. Intercept form of plane

Choose the correct answer from the options given below :

For \(x>0\), the minimum value of \(\frac{x}{\log _e x}\) is

If \(x\) is real, the minimum value of \(x^2-8 x+20\) is :

The value of a determinant is unaltered if:

If \(f(x)=\left\{\begin{array}{ll}\frac{\tan \left(\frac{\pi}{4}-x\right)}{\cot 2 x}, & x \neq \frac{\pi}{4} \\ k, & x=\frac{\pi}{4}\end{array}\right.\)
is continuous at \(x=\frac{\pi}{4}\), then the value of \(k\) is:

The cell reaction occurring at anode in the electrolysis of aqueous \(\text{NaCl}\) solution is :

Point mutations means:

Read the passage carefully and answer the Questions
Alcohols and phenols are obtained when a hydrogen atom in aliphatic and aromatic hydrocarbons respectively, is replaced by \(OH\) group. These compounds find wide applications in industry as well as in our daily life. You may notice that ordinary spirit is used for polishing wooden furniture which is chiefly a compound containing hydroxyl group. An alcohol contains one or more hydroxyl \((OH)\) group(s) directly attached to carbon atom(s), of an aliphatic system \((CH_3OH)\) while a phenol contains \(OH\) group(s) directly attached to carbon atom(s) of an aromatic system \((C_6H_5OH)\). Alcohols are prepared by the hydration of alkenes, fermentation of glucose, and reduction of carbonyl compounds. Alcohols produce alkenes upon dehydration whereas alkenes on acid catalyzed hydration convert into alcohols.
An alkyl halide can be converted into alcohol through:

Two positive numbers whose sum is 60 and the product of the square of one number and other number is maximum are:

At equilibrium condition, the rate of generation of electron-hole pairs in Ge crystal

If the system of equations
\( \begin{array}{l} x+2 y+3 z=10 \\ -x+y+\lambda z=20 \\ 2 x+3 y+\lambda z=0 \end{array} \)
does not possess a unique solution, then \(\lambda\) is equal to