If \(\vec{a}=2 \hat{i}-3 \hat{j}+\hat{k}\) and \(\vec{b}=2 \hat{i}+\hat{j}-\hat{k}\), then which of the following statements is/are correct?
(A) \(\vec{a}\) and \(\vec{b}\) are collinear
(B) \(\vec{a}\) and \(\vec{b}\) are perpendicular
(C) Angle between \(\vec{a}\) and \(\vec{b}\) is \(\frac{\pi}{4}\)
(D) \(|\vec{a}+\vec{b}|=2 \sqrt{5}\)
Choose the correct answer from the options given below:

Match List I with List II

LIST I	LIST II
A. If \(4 \sin ^{-1} x+\cos ^{-1} x=\pi\), then \(x\) equal to	I. \(\frac{\pi}{2}\)
B. The value of \(\frac{1-\tan ^2 15^{\circ}}{1+\tan ^2 15^{\circ}}\) is	II. \(\frac{1}{2}\)
C. If \(x+\frac{1}{x}=2\), then principal value of \(\sin ^{-1} x\) is	III. \(\frac{3 \pi}{4}\)
D. Two angles of a triangle are \(\cot ^{-1} 2\) and \(\cot ^{-1} 3\), then third angle is	IV. \(\frac{\sqrt{3}}{2}\)

The semi vertical angle of a right circular cone of maximum volume of a given slant height is

If \(r_{e f f}=\) effective rate of interest, \(r=\) nominal rate of interest and \(m=\) number of conversion periods per year, the relationship between nominal rate and effective rate of interest is :

The area of the region bounded by \(y^2=9 x, x=2, x=4\) and the \(x\)-axis in the first quadrant, is

Match List - I with List - II

List - I	List - II
(A) \(\int \frac{d x}{a^2-x^2}\)	(I) \(\frac{1}{2 a} \log \left|\frac{x-a}{x+a}\right|+C\)
(B) \(\int \frac{d x}{x^2-a^2}\)	(II) \(\frac{1}{2 a} \log \left|\frac{a+x}{a-x}\right|+C\)
(C) \(\int \frac{d x}{x^2+a^2}\)	(III) \(\sin ^{-1}\left(\frac{x}{a}\right)+C\)
(D) \(\int \frac{d x}{\sqrt{a^2-x^2}}\)	(IV) \(\frac{1}{a} \tan ^{-1} \frac{x}{a}+C\)

Choose the correct answer from the options given below:

If \(a>b\) and \(c<0\), then which of the following is NOT correct?
\(a c < b c\)
\(a+c<b+c\)
\(a-c < b-c\)
\(a c>b c\)
Choose the correct answer from the options given below :

Which one of the following is not correct.

Identify the statements which are incorrect with respect to medical termination of pregnancy:
(A) MTP is relatively safe during the first trimester of pregnancy
(B) Govt. of India legalised MTP in 1981
(C) MTP is misused to get rid of desired sex determined through amniocentesis
(D) MTP is essential in cases where foetus is found to carrying harmful genes
Choose the correct answer from the options given below:

\(\int x^2 \tan ^{-1} x d x\)=

The least non-negative remainder when \(2^{101}\) is divided by \(5\) is:

The general solution of the differential equation \(\frac{d y}{d x}+\sqrt{\frac{1-y^2}{1-x^2}}=0\) is :

Read the passage carefully and answer the questions
The coordination compounds are of great importance. These compounds are widely present in the mineral, plants and animal world and are known to play many important functions in the area of analytical chemistry, metallurgy, biological systems, industry and medicine. For example, coordination compounds find use in many qualitative and quantitative chemical analysis. Some important metal extraction and purification make use of complex formation. Hardness of water can be estimated by stable complex formation of Ca\(^{2+}\) and Mg\(^{2+}\)
From which coordination complex, gold is recovered by reducing this complex with Zn ?