Differentiation of \(\frac{x^3}{1-x^3}\) with respect to \(x^3\) is equal to:

Match List-I with List-II

List-(I) Function	Lis-(II) Derivative
(A) \(y=\sin ^{-1} x+\sin ^{-1} \sqrt{1-x^2},|x|<1\)	(I) \(\frac{d y}{d x}=\frac{1}{2 y-1}\)
(B) \(y=\sqrt{x+y}, x+y>0, y \neq \frac{1}{2}\)	(II) \(\frac{d y}{d x}=10^x \log _e 10\)
(C) \(y=\log _{10} x, x>0\)	(III) \(\frac{d y}{d x}=0\)
(D) \(y=10^*\)	(IV) \(\frac{d y}{d x}=\frac{1}{x \log _e 10}\)

Choose the correct answer from the options given below:

The value of \(\int_0^1 x e^x d x\) is :

The equation of tangent line to \(y=2 x^2+7\) which is parallel to the line \(4 x-y+3=0\) is :

The projection of \(\hat{i}-\hat{j}\) on the vector \(\hat{i}+\hat{j}\) is :

If \(A=(2,3), B=(-1,0), C=(4,6)\) then the area of the parallelogram \(A B C D\) is:

One person speaks truth in \(60 \%\) of the cases and another person in \(80 \%\) of the cases. They are likely to agree in stating the same fact in:

Question based on following passage :
Adsorption is the process of attracting and retaining the adsorbate molecules on the surface of adsorbent. It is of two types i.e. Physisrption and chemisorption. To increase the rate of reaction, the substance called 'catalyst' is added. Promotors enhance the activity of catalyst while poison decrease the catalyst activity. Solutions may be true solution, suspension and colloids. Colloids are heterogenous but quite stable. On the basis of nature of interaction, colloids may be Lyophilic or Lyophobic. On the basis of type of dispersed phase particles, there may be multimolecular colloids, macromlecular colloids and associated colloids. The congulation of colloids may be done by boiling, dialysis or by mixing oppositely charged sols.
The best coagulating agent for \(Fe(OH)_3\) solution is :

A proton, an electron, a neutron and an alpha particle have the same energy. Their de Broglie's wavelengths are given below:
(A) \( \lambda_{proton} \)
(B) \( \lambda_{electron} \)
(C) \( \lambda_{neutron} \)
(D) \( \lambda_{alpha} \)
Arrange these wavelengths in decreasing order. Choose the correct answer from the options given below:

Which enzyme modified by genetic engineering is used as a 'clot buster'?

Let A = {1,2,3}.
The number of relations on A containing (1, 2) and (2, 3),
which is reflexive and transitive but not symmetric, is:

Which of the following statements are correct?
(A) \(\tan ^{-1} x+\tan ^{-1} y=\tan ^{-1}\left(\frac{x+y}{1-x y}\right), x y<1\)
(B) \(\tan ^{-1} x+\tan ^{-1} y=\pi+\tan ^{-1}\left(\frac{x+y}{1-x y}\right), x y>1\)
(C) \(\sin ^{-1} x+\cos ^{-1} x=\frac{\pi}{2}, x \in(-1,1)\)
(D) \(\tan ^{-1} x+\cot ^{-1} x=\frac{\pi}{2}, x \in R\)
(E) \(\cos ^{-1}(-x)=-\cos ^{-1} x, x \in[-1,1]\)
Choose the correct answer from the options given below:

The density of population is decreased by the processes
(A) Natality
(B) Emigration
(C) Mortality
(D) Immigration
Choose the most appropriate combination from the options given below.