\(\int \tan ^{-1} \sqrt{x} d x\) equals to : (Here \(C\) is an arbitrary constant)

The equation of motion of a missile is x = 3t, y = - 4t, z = t where time 't' is in seconds and the distance is measured in kilometres.
At what distance will the rocket be from the point (0, 0, 0) in 5 seconds?

Match List I with List II

LIST I	LIST II
A. \(\left( \frac{dy}{dx} \right)^2 + \frac{dy}{dx} + 5 = 0\)	I. order 3, degree 1
B. \(xy \frac{d^2y}{dx^2} + x \left( \frac{dy}{dx} \right)^2 - y \frac{dy}{dx} = 0\)	II. order 2, degree 2
C. \(\left( \frac{d^2y}{dx^2} \right)^2 + \frac{dy}{dx} = 0\)	III. order 2, degree 1
D. \(\frac{d^3y}{dx^3} + 2\frac{d^2y}{dx^2} + \frac{dy}{dx} = 0\)	IV. order 1, degree 2

Choose the correct answer from the options given below:

The corner points of the bounded feasible region for a linear programming problem (LPP) are \(\left(0, \frac{3}{2}\right),(1,2)\) and \((4,0)\).
If the objective function is \(Z=a x+b y\), where \(a\) and \(b\) are positive,
then the condition on \(a\) and \(b\) so that the maximum of \(Z\) occurs at \((1,2)\) and \((4,0)\) is :

Which of the following functions from \(Z\) to \(Z\) is a bijective function? (where \(Z\) is set of integers)

Which of the following statements are true?
(A) Degree of the differential equation \(\frac{d^2 y}{d x^2}+y=1\) is 2.
(B) Number of arbitrary constants in the particular solution the differential equation of order 4 are 4.
(C) \(y^2=a\left(b^2-x^2\right)\) will be the general solution of a differential equation of order 2; where a and b are two arbitrary constants.
(D) \(f(x, y)=\frac{2 x-y}{x+3 y}\) is a homogeneous function of degree 1.
(E) Integrating factor of \(x \frac{d y}{d x}-y=2 x^2\) is \(\frac{1}{x}\).
Choose the correct answer from the options given below:

The inequality \(\frac{5 x-2}{3}-\frac{7 x-3}{5}>\frac{x}{4}\) holds when

In Young's double slit experiment, the width of one of the two slits is four times the width of the other slit. The ratio of the maximum to the minimum intensity in the interference pattern will be

If \(y=5 \cos x-3 \sin x\), then \(\frac{d^2 y}{d x^2}\) is equal to :

Read the passage carefully and answer the Questions
In the formation of coordination complexes, if the inner \(d\) orbital \((n - 1)d\) is used in hybridisation, the complex, is called an inner orbital or low spin or spin paired complex. And if it uses outer \(d\) orbital \((nd)\) in hybridisation (like \(sp^3d^2\)), it is called outer orbital or high spin or spin free complex. The degeneracy of the \(d\) orbitals has been removed due to ligand electron-metal electron repulsions in the complex. This splitting of the degenerate levels due to the presence of ligands in a definite geometry is termed as crystal field splitting. For coordination complexes, the magnetic moment is determined by the number of unpaired electrons and is calculated by using the 'spin-only' formula, \(\mu = \sqrt{n(n+2)}\) where \(n\) is the number of unpaired electrons and \(\mu\) is the magnetic moment in units of Bohr magneton \((BM)\). The coordination compounds are of great importance. These compounds are widely present in the mineral, plant and animal worlds and are known to play many important functions in the area of analytical chemistry, metallurgy, biological systems, industry and medicine.
Fe(II) octahedral complexes can show different 'spin only' magnetic moments for arrangements in the presence of weak and strong ligands. The 'spin only' magnetic moments of these complexes with weak and strong ligands, respectively, are

When dilute hydrochloric acid is added to sodium chromate solution the colour changes to :

If the relation \(R: A \rightarrow B\), where \(A=\{2,3,4\}\) and \(B=\{3,4,5\}\) is defined by \(R=\{(x, y): x < y, x \in A, y \in B\}\), then

Consider the lines \(l_1: \frac{x-1}{0}=\frac{y-1}{1}=\frac{2-z}{1}\) and \(l_2: \frac{x}{2}=\frac{y}{0}=\frac{2 z-1}{4}\), then which of the following are correct?
(A) Direction Ratios of \(l_1=\langle 0,1,1\rangle\)
(B) Direction Ratios of \(l_2=\langle 2,0,2\rangle\)
(C) Angle between \(l_1\) and \(l_2=\frac{\pi}{3}\)
(D) Angle between \(l_1\) and \(l_2=\frac{2 \pi}{3}\)
Choose the correct answer from the options given below :