The value of \(\lambda\), for which the projection of \(\vec{a} = \lambda \hat{i} + \hat{j} + 4\hat{k}\) on \(\vec{b} = 2\hat{i} + 6\hat{j} + 3\hat{k}\) is 4 units

If A is a skew-symmetric matrix of order n, then

\(\int \frac{e^{2 x}-1}{e^{2 x}+1} d x\) =

Match List-I with List-II

List-I (Integral)	List-II (Solution: C is an arbitrary constant)
(A) \(\int \frac{d x}{x^2+25}\)	(I) \(\frac{1}{10} \log \left|\frac{5+x}{5-x}\right|+C\)
(B) \(\int \frac{d x}{x^2-25}\)	(II) \(\log \left|x+\sqrt{x^2-25}\right|+C\)
(C) \(\int \frac{d x}{25-x^2}\)	(III) \(\frac{1}{5} \tan ^{-1}\left(\frac{x}{5}\right)+C\)
(D) \(\int \frac{d x}{\sqrt{x^2-25}}\)	(IV) \(\frac{1}{10} \log \left|\frac{x-5}{x+5}\right|+C\)

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If \(2 P(A)=P(B)=\frac{5}{13}\) and \(P(A \mid B)=\frac{2}{5}\), then the value of \(P(A \cup B)\) is :

Which of the following statements are true?
(A) The vector equation of the line through the point (5, 2, -4) and parallel to the vector \(3 \hat{i}+2 \hat{j}-8 \hat{k}\) is \(\vec{r}=(5 \hat{i}+2 \hat{j}-4 \hat{k})+\lambda(3 \hat{i}+2 \hat{j}-8 \hat{k})\)
(B) Vector form of the equation of line \(\frac{x-5}{3}=\frac{y+4}{7}=\frac{z-6}{2}\) is \(\vec{r}=(5 \hat{i}-4 \hat{j}+6 \hat{k})+\lambda(3 \hat{i}+7 \hat{j}+2 \hat{k})\)
(C) The direction cosines of z-axis are (1, 1, 0).
(D) If a line has direction ratios 2, -1, -2, then its direction cosines are \(-2 / 3,-1 / 3,-2 / 3\).
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Two cards are drawn simultaneously at random from a well shuffled pack of 52 Cards.
Let X be the random variable which denotes number of kings in the draw. Then the probability distribution of X is

If the sequence of one strand of DNA is written as: 5'-ATCGTGCTATGAGCATGC-3'.
The sequence of its complementary strands will be?

Two similar magnetic poles having pole strengths in the ratio 1 : 4 are placed 6 cm apart. The distance from a smaller pole in strength where net magnetic field is zero will be :

The function \(f(x)=|x-1|\) is strictly increasing in the interval.

How many \(Cr-O\) bonds in dichromate ions are of the same bond length and are in resonance?

Which vector is used for introducing nematode specific genes into the tobacco host plant.

Match List-I with List-II

List-I (Diseases)	List-II (Causative organisms)
(A) Pneumonia	(I) Microsporum
(B) Ringworms	(II) Wuchereria bancrofti
(C) Elephantiasis	(III) Entamoeba histolytica
(D) Amoebiasis	(IV) Haemophilus influenzae

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