The position vector of a point which divides the line joining the points with position vectors \((\vec{a}-2 \vec{b})\) and \((2 \vec{a}+\vec{b})\) externally in the ratio \(2 : 1\), is

If \(\vec{a}=2 \hat{i}-\hat{j}+3 \hat{k}\) and \(\vec{b}=2 \hat{i}+2 \hat{j}+\hat{k}\) then
Match List-I with List-II

List-I	List-II
(A) Projection of \(\vec{a}\) on \(\vec{b}\) is	(I) \(-7 \hat{i}+4 \hat{j}+6 \hat{k}\)
(B) \(\vec{a} \times \vec{b}\) is	(II) \(\frac{1}{\sqrt{101}}(-7 \hat{i}+4 \hat{j}+6 \hat{k})\)
(C) Unit vector along \(\vec{a}+\vec{b}\) is	(III) \(\frac{5}{3}\)
(D) Unit vector perpendicular to both \(\vec{a}\) and \(\vec{b}\) is	(IV) \(\frac{1}{\sqrt{33}}(4 \hat{i}+\hat{j}+4 \hat{k})\)

Choose the correct answer from the options given below :

If \(A=\left[\begin{array}{ll}2 & 3 \\ 1 & 4\end{array}\right]\) and \(B=\left[\begin{array}{cc}1 & -2 \\ 1 & 3\end{array}\right]\), then which of the following are incorrect?
(A) \(|A|=5\)
(B) \(\operatorname{Adj} A=\left[\begin{array}{ll}4 & 3 \\ 1 & 2\end{array}\right]\)
(C) \(A(\operatorname{Adj} A)=25 I\)
(D) \(|3 B|=45\)
(E) \((A B)^{-1}=B^{-1} A^{-1}\)
Choose the correct answer from the options given below:

\(\int \frac{d x}{x^3 \sqrt{\left(1+x^4\right)}}=\)

The maximum value of the objective function \(z=2 x+3 y\) of an L.P.P. subjected to the constraints \(x-y \leq 1, x+y \leq 3, x, y \geq 0\), is

The plane \(2 x-(1+\lambda) y+3 \lambda z=0, \lambda \in R\) passes through the intersection of planes :

Pipe A can fill a tank in 24 hours and pipe B in 32 hours. If both the pipes are opened in an empty tank at the same time, then the time taken to fill the tank is:

The area (in sq. units) bounded by the parabola \(y^2=4 a x\), its latus rectum and the \(x\)-axis in the first quadrant is:

Werner was the first to describe the bonding features in coordination compounds. But his theory could not answer basic questions like :
(i) Why only certain elements possess the remarkable property of forming coordination compounds ?
(ii) Why the bonds in coordination compounds have directional properties ?
(iii) Why coordination compounds have characteristics magnetic and optical properties ?Many approaches have been put forth to explain the nature of bonding in coordination compounds viz. Valence Bond Theory (VBT), Crystal Field Theory (CFT), Ligand Field Theory (LFT) and Molecular Orbital Theory (MOT).
We shall focus our attention on elementary treatment of the application of VBT and CFT to coordination compounds.
According to this theory, the metal atom or ion under the influence of ligands can use its ( \(n -1\) )d, ns np or ns, np, nd orbitals for hybridisation to yield a set of equivalent orbitals of definite geometry such as octahedral, tetrahedral, square planar and so on.
These hybridised orbitals are allowed to overlap with ligand orbitals that can donate electron pairs for bonding.
Predict the shape of \(\left[ Ni ( CO )_4\right]\) on the basis of VBT :

Silicon can be doped to get an n-type semiconductor by using :
(A) Arsenic (As),
(B) Phosphorus (P),
(C) Boron (B),
(D) Antimony (Sb).

Total internal reflection can take place only, if

Identify the water pollinated plants from the following:
(A) Water hyacinth and Zostera
(B) Zostera and Hydrilla
(C) Water lily
(D) Vallisneria
Choose the correct answer from the options given below:

The value of \(\int_{-1}^1\left|x^3-x\right| d x\) is