Let L be the set of all lines in a plane and R be the relation on set L defined by \(R=\left\{\left(L_1, L_2\right) : L_1 \perp L_2\right\}\). Then R is
(A) an equivalence Relation
(B) a symmetric Relation
(C) not a transitive Relation
(D) a reflexive Relation
Choose the correct answer from the options given below :

The level of production where the revenue from sales is equal to the cost of production and marketing is known as

Match List-I with List-II

List-I (Matrix)	List-II (Inverse of the Matrix)
(A) \(\left[\begin{array}{cc}1 & 7 \\ 4 & -2\end{array}\right]\)	(I) \(\left[\begin{array}{cc}2 / 15 & 1 / 10 \\ -1 / 15 & 1 / 5\end{array}\right]\)
(B) \(\left[\begin{array}{cc}6 & -3 \\ 2 & 4\end{array}\right]\)	(II) \(\left[\begin{array}{cc}1 / 5 & -2 / 15 \\ -1 / 10 & 7 / 30\end{array}\right]\)
(C) \(\left[\begin{array}{cc}5 & 2 \\ -5 & 4\end{array}\right]\)	(III) \(\left[\begin{array}{cc}\frac{1}{15} & \frac{7}{30} \\ 2 / 15 & -1 / 30\end{array}\right]\)
(D) \(\left[\begin{array}{ll}7 & 4 \\ 3 & 6\end{array}\right]\)	IV) \(\left[\begin{array}{cc}2 / 15 & -1 / 15 \\ 1 / 6 & 1 / 6\end{array}\right]\)

Choose the correct answer from the options given below:

Let \(R\) be a relation on the set of integers \(Z\) such that \(R=\left\{(a, b), a=2^k b, a, b, k \in Z\right\}\), then \(R\) is :

The general solution of differential equation \(\frac{dy}{dx} - xy = e^{\frac{x^2}{2}}\) is:

If \(a, b, c\) are positive real numbers, then the least value of \((a+b+c)(a b+b c+c a)\) is :

For the system of linear equations
x + y + z = 5000
6x + 7y + 8z = 35800
6x + 7y - 8z = 7000
the values of x, y and z are:

Among the following, which alkyl halide has the highest boiling point?

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.
Which of the following pair of ions is colourless according to CFT ?

Which one of the following is not the sacred groves?

Which of the following reactions show that glucose contains five 'OH' groups?

The inverse of the matrix \(\left[\begin{array}{cc}2 & -1 \\ 1 & 0\end{array}\right]\) is:

The rate of change of volume of a sphere with respect to its surface area, when the radius is 6 cm is: