For independent events \( A_{1} , A_{2}, A_{3} , \ldots, A_{n}\) , if \(P(A_{i}\) \(= \frac{1}{i + 1}, \; i = 1, 2, 3 \ldots, n\), then the probability that none of the events occur is :

Choose the correct answer from the options given below:
A. Equation of the line passing through the point (1, 2, 3) and parallel to the vector \(3 \hat{i}+2 \hat{j}-2 \hat{k}\) is \(\frac{x-1}{3}=\frac{y-2}{2}=\frac{y-3}{-2}\)
B. Equation of line passing through (1, 2, 3) and parallel to the line given by \(\frac{x+3}{3}=\frac{4-y}{5}=\frac{z+8}{6}\) is \(\frac{x-1}{3}=\frac{y-2}{5}=\frac{z+3}{6}\).
C. Equation of line passing through the origin and (5, -2,3) is \(\frac{x}{5}=\frac{y}{-2}=\frac{z}{3}\).
D. Equation of plane passing through the point (1, 2, 3) and perpendicular to the line with direction ratio's 2, 3, -1 is 2(x - 1) + 3(y - 2) - (z - 3) = 0.
E. Equation of plane with intercepts 2, 3 and 4 on x, y and z-axis respectively is 2x + 3y + 4z = 1

If \(\vec{a}+\lambda \vec{b}\) is perpendicular to \(\vec{c}\), where
\(\vec{a}=-\hat{i}+2 \hat{j}+\hat{k}, \vec{b}=3 \hat{i}+2 \hat{j}+\hat{k}\), and \(\vec{c}=\hat{i}-\hat{j}\),
then :

The simplest form of \(\tan ^{-1}\left(\frac{\sqrt{1+x^2}-1}{x}\right), x \neq 0\) is:

Which of the following statements are correct?
(A) If \(f: R \rightarrow R\) then \(f(x)=|x|\) is continuous everywhere.
(B) If \(f: R \rightarrow R\) then \(f(x)=|x|\) is continuous everywhere but not differentiable at \(x=0\).
(C) Let \(f: R -\{0\} \rightarrow R\) then \(f(x)=1(\) or \(1 / x)\) is continuous everywhere.
(D) Let \(f: R \rightarrow R\) then \(f(x)=|x-1|+|x-2|\) is continuous everywhere but not differentiable at exactly 2 points.
(E) If \(f: R \rightarrow R\) then \(f(x)=\cot (x)\) is continuous everywhere.
Choose the correct answer from the options given below :

The curve \(y = f ( x )\) is normal probability curve, then which of the following statements are correct?
(A) mean, median and mode of the distribution coincide.
(B) the area bounded by the curve \(y = f ( x )\) and \(x\)-axis is one unit.
(C) The curve is symmetrical about the line \(x =\mu\), where \(\mu\) is the mean.
(D) y -axis is an asymptote to the curve.
Choose the correct answer from the options given below :

If \(A=\left[\begin{array}{ccc}0 & l & -3 \\ -2 & 0 & 1 \\ m & -1 & 0\end{array}\right]\) is a skew symmetric matrix, then

GnRH is secreted by:

The half life of a radioactive element is 140 days. After 560 days, one gram of the element will be reduced to:

In aldehydes, the carbonyl group is bonded to a carbon and hydrogen, while in ketones it is bonded to two carbon atoms. The carbonyl compounds in which carbonyl group is bonded to oxygen are known as carboxylic acid and their derivatives.
1,3-Diphenylprop-2-en-1-one is produced by the following.

The function \(f(x)=x+\cot ^{-1} x\) is increasing in the interval:

Answer the question on the basis of passage given below :
Nitrogen differs from rest of the members of group 15 due to its small size, high electronegativity, high ionization enthalpy and non-availability of d orbitals. Nitrogen has unique ability to form pa - pa multiple bonds with itself and with other elements having small size and high electronegativity. Heavier elements of this group do not form pa - pr bonds as their atomic orbitals are so large and diffuse that they cannot have effective overlapping.|
Match List I with List II.

LIST I (Nitrogen oxides)	LIST II (Oxidation state)
A. \(N _2 O\)	\(I. +3\)
B. \(N _2 O _3\)	\(II. +5\)
C. \(N _2 O _4\)	\(III. +1\)
D. \(N _2 O _5\)	\(IV. +4\)

Choose the correct answer from the options given below:

For the objective function \(Z = 4x + 6y\) subject to the constraints
\(3x + 2y ≥ 5, 7x + 2y ≤ 9, x ≥ 0, y ≥ 0\),
the maximum value of Z occurs at (a, b)
and the minimum value of Z occurs at (p, q)
then the value of \(\frac{a}{p}+\frac{b}{q}\) is: