For finding absolute maximum and minimum values of a function \(f\) given by
\(f(x)=2 x^3-15 x^2+36 x+1 \text { on }[1,5]\)
A. the critical points in \((1,5)\) are 2 and 3 .
B. the absolute maximum value of \(f\) on \([1,5]\) is 29 .
C. the absolute minimum value of \(f\) on \([1,5]\) is 24 .
D. the absolute minimum value of \(f\) on \([1,5]\) is 20 .
E. we evaluate the value of \(f\) at critical points and at the end points of the interval \([1,5]\).
Choose the correct answer from the options given below:

If \(A\) and \(B\) are two invertible matrices, then which of the following statements are correct?
(A) \(\left|A^{-1}\right|=|A|^{-1}\)
(B) \(\operatorname{adj} A=|A| A^{-1}\)
(C) \((A B)^{-1}=A^{-1} B^{-1}\)
(D) \((A+B)^{-1}=A^{-1}+B^{-1}\)
Choose the correct answer from the options given below:

The shortest distance between the lines\(\frac{x-1}{2}=\frac{y-2}{3}=\frac{z+4}{6}\)and\(\frac{x-3}{4}=\frac{y-3}{6}=\frac{z+5}{12}\) is :

The number of equivalence relation on the set {1, 2, 3} containing (1, 2) and (2, 1) is

The declared rate of return compounded semi annually equivalent to the effective rate of return \(10.25\%\) per annum is :

\(\text { If } y=\left(\log \left(x+\sqrt{x^2+a^2}\right)\right)^2 \text { and } x \neq \frac{1-a^2}{2}\)then \(\left(x^2+a^2\right) \frac{d^2 y}{d x^2}+x \frac{d y}{d x}\) is equal to :

The differential equation representing the family of curves \(y=a \sin (x+b)\), where \(a\) and \(b\) are arbitrary constants is

The function \(f(x)=2 x^3-3 x^2-12 x+4\) has:
(A) two points of local maxima.
(B) two points of local minima.
(C) one local maxima and one local minima.
(D) no local maxima or local minima.

A molecule X associates in a given solvent as per the following equation: \(X \rightleftharpoons(X) n\). For a given concentration of \(X\), the van't Hoff factor was found to be \(0 . 8 0\), and the fraction of associated molecules was \(0 . 3\). The correct value of ' \(n\) ' is:

Read the passage carefully and answer the Questions
\(KMnO _4\) is prepared by the fusion of \(MnO _2\) with an alkali metal hydroxide and an oxidizing agent like \(KNO _3\) to give a dark-green manganate ion which disproportionate to give permanganate as follows.
\(\begin{array}{l}2 MnO _2+4 KOH + O _2 \rightarrow 2 K_2 MnO _4+2 H _2 O \\ 3 K_2 MnO _4+4 H ^{+} \rightarrow 2 KMnO _4+ MnO _2+2 H _2 O \end{array}\)
On heating \(K _2 MnO _4\) decomposes at 513 K to give \(K _2 MnO _4\). Permanganate ion is tetrahedral and diamagnetic
Acidified \(KMnO _4\) acts a strong oxidizing agent which oxidizes oxalic acid, ferrous ions, nitrite ion and iodides.
Reaction of iodide ion \(\left( I ^{-}\right)\)with \(KMnO _4\) will.....

Taq polymerase is isolated from:

The probability of a shooter hitting a target is \(\frac{2}{3}\)
Minimum how many number of times must the shooter fire so that the probability of hitting the target at least once is more than 0.98.

Which one of the following is not an example of terrestrial ecosystem?