\(\int_{\sqrt{\log _e 2}}^{\sqrt{\log _e 4}} x e^{x^2} d x\) is equal to

\(\int \frac{\sin 2 x d x}{\sqrt{9-\cos ^4 x}}\) equals

The maximum slope of the curve \(y=-x^3+3 x^2+9 x-27\) is:

If \(\sin ^{-1} x+\sin ^{-1} y=\frac{2 \pi}{3}\), then the value of \(\cos ^{-1} x+\cos ^{-1} y\) is:

Two points A and B in a plane are related if OA = OB, where O is a fixed point. This relation is:

A man rows downstream 30 km and upstream 20 km. If he takes 5 hours to cover each distance, then the speed of stream is

If the following data is obtained from a simple random sample : 6, 7, 9, 10, 11, 17
Then the point estimate of population standard deviation is :

Identify the correct statement(s) among the following:
(A) If three electrical cells each of emf, E and internal resistance r are connected in series to a resistance R, the current drawn from them is 3E/(R+3r).
(B) If these three cells are connected in parallel combination, across resistance R then the current drawn will be E/(3R+r).
(C) If the two cells are connected in series and then one in parallel which are then connected across resistance R, then the current drawn will be 2E/(2R+2r).
(D) In a mixed grouping of cells, current will be maximum when total external resistance is equal to total equivalent internal resistance.
Choose the correct answer from the options given below :

Match List-I with List-II

List-I (Population interactions)	List-II (Species A / Species B)
(A) Competition	(I) - / 0
(B) Amensalism	(II) + / 0
(C) Commensalism	(III) + / -
(D) Parasitism	(IV) - / -

'* + 'sign for beneficial interaction, '-'for detrimental and '0 *' for neutral interaction
Choose the correct answer from the options given below:

A monochromatic source emitting light of wavelength 600 nm has a power output of 66 W. The number of photons emitted by the source in 2 minutes is: (Given: \( h = 6.6 \times 10^{-34} \text{ Js} \))

For a maximizing LPP, the constraints are:

If \(\hat{i}, \hat{j}\) and \(\hat{k}\) are unit vectors along the co-ordinate axes OX, OY and OZ respectively, then
(A) \(\hat{i} \times \hat{j}=\hat{k}\)
(B) \(\hat{k} \times \hat{i}=-\hat{j}\)
(C) \(\hat{j} \cdot \hat{j}=1\)
(D) \(\hat{j} \cdot \hat{k}=0\)
Choose the \{correct} answer from the options given below:

Arrange the following steps of PCR in correct sequence.
(A) Temperature induced denaturation of double stranded DNA.
(B) Isolation of gene of interest
(C) Amplification of gene by using Taq polymerase
(D) Primer annealing on the separated templates
Choose the correct answer from the options given below: