The value of \(\tan ^{-1}\left[2 \cos \left(2 \sin ^{-1} \frac{1}{2}\right)\right]\) is :

Match List I with List II

List - I	List - II
(A) \(|\vec{a} \times \hat{i}|^2+|\vec{a} \times \hat{j}|^2+|\vec{a} \times \hat{k}|^2\)=	(I) \(|\vec{a}|^2\)
(B) \(\frac{(\vec{a} \cdot \vec{b})^2+|\vec{a} \times \vec{b}|^2}{|\vec{b}|^2}=\)	(II) \(\sqrt{3}|\vec{a}|\)
(C) \(\vec{a}, \vec{b}\) and \(\vec{c}\) are three mutually perpendicular vectors of equal magnitude then \(|\vec{a}+\vec{b}+\vec{c}|=\)	(III) \(\frac{1}{2}|\vec{a} \times \vec{b}|\)
(D) Area of parallelogram with diagonals \(\vec{a}\) and \(\vec{b}\)	(IV) \(2|\vec{a}|^2\)

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If \(A=\left[\begin{array}{lll}1 & 1 & -2 \\ 2 & 1 & -3 \\ 5 & 4 & -9\end{array}\right]\), then \(|A|\) is equal to:

The value of the integral \(I=\int_{-1}^1\left(x+x^3+x^5\right) d x\) is:

If \(\left[\begin{array}{cc}2 x-1 & 4 x \\ 0 & y^2-1\end{array}\right]=\left[\begin{array}{cc}x+1 & 8 \\ 0 & 24\end{array}\right]\), then the value of \((x+y)\) is :

If \(\sin ^{-1} x+\sin ^{-1} y=\frac{2 \pi}{3}\), then the value of \(\cos ^{-1} x+\cos ^{-1} y\) is:

The value of \(\tan ^{-1}(1)+\cos ^{-1}\left(-\frac{\sqrt{3}}{2}\right)+\sin ^{-1}\left(\frac{1}{2}\right)\) is:

The Abingdon tortoise became extinct in Galapagos islands due to introduction of which of the following animal?

The derivative of \((\log x)^x\) with respect to log \(x\) is:

A \( 25 \, \mu\text{F} \) capacitor, a 0.10 H inductor and a \( 25 \, \Omega \) resistor are connected in series with an ac source of emf \( \epsilon = 310 \sin 314t \). What is the frequency of AC source ?

Answer the question on the basis of passage given below:
Adsorption is the phenomenon of attracting and retaining the molecules of a substance on the surface of a solid resulting into a higher concentration on the surface than in the bulk. The substance adsorbed is known as adsorbate and the substance on which adsorption takes place is called adsorbent. In physisorption adsorbate is held to the adsorbent by weak van der Waals forces and in chemisorption by strong chemical bond.
In coagulation of a positive sol, the flocculating power will be in the order:
A. \(PO _4^{3-}\)
B. \(Cl ^{-}\)
C. \([ Fe ( CN )]^{3-}\)
D. \(SO _4^{2-}\)
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Fossils are generally found in:

Match List-I with List-II

List-I	List-II
(A) \(f(x)=|x|+1\)	(I) Not differentiable at \(x=3\) only
(B) \(f(x)=|x-3|\)	(II) Not differentiable at \(x=-3\) only
(C) \(f(x)=|x+3|\)	(III) Not differentiable at \(x=3\), \(-3\) only
(D) \(f(x)=\left|x^2-9\right|\)	(IV) Not differentiable at \(x=0\) only

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