For the function \(f(x)=x^{1 / x}, x>0\), which of the following are correct?
(A) \(x=0\) is the only point where extremum may occur.
(B) The given function is maximum at \(x=e\).
(C) The function has no extreme value for \(x>0\).
(D) The maximum value of the function \(f(x)\) is \(e^{1 / e}\)
Choose the correct answer from the options given below :

Consider the following statements :
(A) \(\int_{-a}^a f(x) d x=2 \int_0^a f(x) d x\) if \(f(-x)=-f(x)\)
(B) \(\int_a^b f(x) d x=\int_a^b f(t) d t\)
(C) \(\int_0^a f(x) d x=\int_0^a f(x-a) d x\)
(D) \(\int_a^b f(x) d x=\int_a^b f(a+b-x) d x\)
Choose the correct answer from the options given below :

\(\frac{d}{d x}\left(e^{2 \log _e x^3}\right)\) equals

If \(A\) and \(B\) are independent events, then
A. not \(A\) and not \(B\) are independent
B. \(A\) and not \(B\) are independent
C. not \(A\) and \(B\) are not independent
D. \(P(A \cup B)=P(A)+P(B)\)
E. \(P(A \cap B)=P(A) P(B)\)
Choose the correct answer from the options given below:

The corner points of the bounded feasible region determined by the system of linear inequalities are \((0,0),(2,4),(0,5)\) and \((4,0)\). If the maximum value of \(Z=a x+b y\), where \(a, b>0\) occurs at both \((2,4)\) and \((4,0)\), then

The angle between 2 vectors \(\vec{a}\) and \(\vec{b}\) with magnitude 1 and 2 respectively when \(\vec{a} \cdot \vec{b}=1\) is:

For \(x \in(0,1)\), the derivative of \(\sin ^{-1} x\) with respect to \(\cos ^{-1} \sqrt{1-x^2}\) is:

The programming problem
Max Z = 2x + 3y subject to the conditions \(0 \leq x \leq 3,0 \leq y \leq 4\) is:

Answer the question on basis of passage given below :
The noble gases have closed shell electronic configuration and are monatomic gases under normal conditions. The low boiling points of lighter noble gases are due to the weak dispersion forces between the atoms and the absence of other inter atomic interaction. The direct reaction of xenon with fluorine leads to
of compounds with oxidation numbers +2, +4 and +6.
XeF4 reacts violently with water to give XeO3. The compound of Xenon exhibit stereoisomerism and their geometries can be deduced considering the total number of electrons in the valence shell.
Argon is used in arc welding because it has

Two batteries of emf 5 V with internal resistance 2 \( \Omega \) and 6 V with internal resistance 3 \( \Omega \) are connected in series with an external resistor. If the current in the circuit is 0.5 A, then the resistance of the resistor is

A balloon which always remains spherical, has a variable diameter, \(\frac{3}{2}(5 x+7)\). Then the rate of change of its volume with respect to x is

You must have seen the weed Calotropis growing in abandoned fields. The plant produces highly poisonous _________ and that is why you never see any cattle or goats browsing on this plant.

\(V_2 O _6\) catalyst is used in the manufacturing of