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 :

Let \(f: R \rightarrow R\) defined by \(f(x)=2 x^3-7\) for \(x \in R\). Then :
(A) \(f\) is one-one function
(B) \(f\) is many to one function
(C) \(f\) is bijective function
(D) \(f\) is into function
Choose the correct answer from the options given below:

If \(f(x) = \begin{cases} \frac{\sqrt{4+x}-2}{x}, & \text{if } x \neq 0 \\ k, & \text{if } x = 0 \end{cases}\) is continuous at \(x = 0\), then the value of k is:

If \(\left|\begin{array}{ccc}1 & -2 & 5 \\ 2 & a & -1 \\ 0 & 4 & 2a\end{array}\right|=86\), then product of all values of \(a\) is :

In a linear programming problem, the linear constraints are as follows :
\(x-3 y \geq 0, y \geq 0,0 \leq x \leq 6 \)
where \(z=x+2 y\)
Which of the following statement/s is/are correct ?
(A) The feasible region is not the first quadrant
(B) The feasible region is unbounded in the first quadrant
(C) The feasible region is bounded in the first quadrant
(D) The feasible region is a triangular region
(E) The maximum value of \(z\) is 10
Choose the correct answer from the options given below :

The particular solution of the differential equation \(\frac{d y}{d x}=8 y x\) when \(y=1\) at \(x=0\) is:

An edge of a variable cube is increasing at the rate of 3 cm/s. How fast is the volume of the cube increasing when the edge is 10 cm long?

Read the passage carefully and answer the questions.
Amines are considered to be derivatives of ammonia obtained by replacement of hydrogen atoms with alkyl or aryl groups. They are usually prepared from nitro compounds, halides, amides, imides etc. The presence of hydrogen bonding influences its physical properties. In alkylamines, a combination of electron releasing steric and hydrogen bonding factors influences the stability of substituted ammonium cations in protic polar solvents and thus affects the basic nature of amines. Alkyl amines are found to be stronger bases than ammonia which is more basic than water. Primary and secondary amines are engaged in intermolecular association due to hydrogen bonding. This intermolecular association is more in primary amines than in secondary amines. Tertiary amines do not have intermoleulcar association due to the absenceof hydrogen atom available for hydrogen bonding. Thus, the boiling point of amines depends on hydrogen bonding. In aromatic amines, electron releasing and withdrawing groups increase and decrease their basic character. Aryl diazonium salts usually obtained from arylamine undergo replacement of the diazonium group with a variety of nucleophile. Their coupling reaction of aryl diazonium salts with phenol or arylamine results in formation of dyes.
The most basic species in the following is

Which one of the following is most reactive towards substitution nucleophilic unimolecular (\(\text{S}_1\)) reactions?

Gallium and Indium can be prepared by:

Read the passage carefully and answer the questions
The speed at which a chemical reaction takes place is called the rate of reaction. The rate of reaction depends on various factors like concentration of the reactants, temperature, etc. The relation between the rate of reaction and the concentration of reacting species is represented by the equation r = k[A]^x[B]^y, where x and y are the order of the reaction with respect to the reactants A and B, respectively. The overall order of the reaction is x + y. The rate of reaction can also be increased by the use of a catalyst which provides an alternate pathway of lower activation energy. It increases the rate of forward and backward reaction to an equal extent. It does not alter the Gibbs energy of the reaction.
The rate of a gaseous reaction is given by r = k[A] [B]. If the volume of the reaction vessel is suddenly reduced to (1/4)th of its initial value, the reaction rate relating to the initial rate will become

Trisomy of Chromosome 21 in humans leads to:

___________ solution of phenol in water acts as a disinfectant.