Evaluate \(\int_0^{\frac{\pi}{4}} \frac{\sin 2 x}{\cos ^4 x+\sin ^4 x} d x=\)

Which of the following statements is/are true?
(A) \(\left(\tan ^{-1} y-x\right) d y=\left(1+y^2\right)\) dx is a differential equation where variables are separable.
(B) (1+ \(x^2\))dy + 2xydx = cot xdx(x ≠ 0) is a first order linear differential equation.
(C) (4x + 6y + 5)dy - (3y + 2x + 4)dx = 0 is not a homogeneous differential equation.
(D) (xy)dx - (x + \(\left.y^2\right)\)dy = 0 is a homogeneous differential equation.
Choose the correct answert from the option given below :

All positive integral powers of a symmetric matrix are:

If \(f(x)=\left\{\begin{array}{ll}\frac{x^2-9}{x-3}, & x \neq 3 \\ 5, & x=3\end{array}\right.\) then \(f(x):\)

The break-even point is the level of production where

Match List-I with List-II

List - I	List - II
(A) Point of minima of \(f(x)=|x+1|\)	(I) 1
(B) Minimum value of \(f(x)=|x|\)	(II) -1
(C) Maximum value of \(f(x)=1-x^2\)	(III) 2
(D) Minimum value of \(f(x)=2+\sin ^2 x\)	(IV) 0

Choose the correct answer from the options given below :

Match List I with List II and choose the correct answer from the options given below:

List - I	List - II
(A) \(\sin ^{-1} x+\cos ^{-1} x, x \in[-1,1]\)	(I) \(-\frac{\pi}{2}\)
(B) \(\tan ^{-1} \sqrt{3}-\cot ^{-1}(-\sqrt{3})\)	(II) \(-\frac{\pi}{6}\)
(C) \(\cos ^{-1}\left(\cos \frac{13 \pi}{6}\right)\)	(III) \(\frac{\pi}{2}\)
(D) \(\sin ^{-1}\left(-\frac{1}{2}\right)\)	(IV) \(\frac{\pi}{6}\)

Arrange the following em waves in the order of their increasing wavelength:
A. Visible light
B. Radio waves
C. Gamma rays
D. Ultraviolet rays
Choose the correct answer from the options given below:

What is the correct order of following steps in the preparation of potassiu Solutionm permanganate?
(A) disproportionation in a neutral or acidic solution
(B) production of dark green \(K _2 MnO _4\)
(C) \(MnO _2\)
(D) fusion with alkali metal hydroxide and \(KNO _3\)
Choose the correct answer from the options given below:

The charge flowing in a conductor varies with time as \(q(t) = (8t^2 - 4t + 5)C\). Then the current changes at the rate of :

The spin-only magnetic moment of \(\left[ MBr _4\right]^{2-}\) is 5.9 BM.
The metal M is:

Read the passage carefully and answer
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. Ligands are treated as point charges in case of anions or dipoles in case of neutral molecules. 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 (either anions or the negative ends of dipolar molecules like NH3 and H2O) in a complex, it becomes asymmetrical and the degeneracy of the dorbitals is lifted. It results in splitting of the d orbitals. The pattern of splitting depends upon the nature of the crystal field.
The colour of coordination compounds is attributed to

The energy required to remove an electron from the inner most orbit of the hydrogen atom to a distance far away from its nucleus is