Two runners A and B complete a 150 meters race in 24 seconds and 36 seconds respectively. By how many meters will A defeat B?

The integral \(\int e^x\left(\frac{x-1}{2 x^2}\right)\)dx is equal to

The slope of the normal to the curve \(y=x^3-4 \sin x\) at \(x=0\) is:

If \(x=a \sin ^3 t\) and \(y=a \cos ^3 t\), then \(\frac{d y}{d x}\) is equal to :

The interval on which the function \(f(x)=x^3+2 x^2-1\) is decreasing, is

Area of the triangle ABC with vertices \(\left(a, a^2\right),\left(b, b^2\right)\) and \(\left(c, c^2\right)\) is equal to :

If \(\left|\begin{array}{ccc}1 & -2 & 5 \\ 2 & a & -1 \\ 0 & 4 & 2a\end{array}\right|=86\), then product of all values of \(a\) is :

Read the passage carefully and answer the Questions:
According to Valence Bond theory, the metal atom or ion under the influence of ligands can use its ( \(n-1\) )d, \(n s, n p\) or \(n s, n p, n d\) 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. While the VB theory, to a larger extent, explains the formation, structures and magnetic behaviour of coordination compounds, it suffers from the many shortcomings. The Crystal field theory (CFT) is an electrostatic model which considers the metal-ligand bond to be ionic arising purely from electrostatic interactions between the metal ion and the ligand. The five \(d\) orbitals in an isolated gaseous metal atom/ion have same energy, i.e., they are degenerate. This degeneracy is maintained if a spherically symmetrical field of negative charges surrounds the metal atom/ion. However, when this negative field is due to ligands in a complex, it becomes asymmetrical and the degeneracy of the \(d\) orbitals is lifted. It results in splitting of the \(d\) orbitals. The pattern of splitting depends upon the nature of the crystal field.
Select the incorrect statement about the Crystal field splitting in octahedral complexes.

The chromosome number per cell is 46 in:

An excess of potassium ions makes KCl crystal appear violet or liliac in colour since:
(A) Some of the anionic sites are occupied by an unpaired electron
(B)Some of the anionic sites are occupied by a pair of electrons
(C) There are vacancies at some anionic sites
(D)F-ntres are created which impart colour to the crystals.
Choose the correct answer from the options given below :

The electric potential due to an electric dipole depends on

Match List-I with List-II

List-I (Differential Equations)	List-II (Order and degree)
(A) \(\frac{d y}{d x}+e^y=0\)	(I) order 2, degree not defined
(B) \(\frac{d^2 y}{d x^2}=\left[1+\left(\frac{d y}{d x}\right)^2\right]^{3 / 2]}\)	(II) order 2, degree 1
(C) \(\left(\frac{d^2 y}{d x^2}\right)^2+e^{\left(\frac{\text { 临 })}{}\right.}=0\)	(III) order 1, degree 1
(D) \(\frac{d^2 y}{d x^2}+x \frac{d y}{d x}-2 y=\log x ; x>0\)	(IV) order 2, degree 2

Choose the correct answer from the options given below:

A machine costing Rs. \(2,00,000\) has a useful life of 5 years. The estimated scrap value is Rs. 20,000 . By using straight line method, the annual depreciation is