For three events A, B and C of a sample space, P(exactly one of A or B occurs)= \(\mathrm{P}(\) exactly one of B or C occurs \()=\mathrm{P}(\) exactly one of C or A occurs \()=\frac{1}{4}\). If probability of all the three events occuring simultaneously is \(\frac{1}{16}\), then the probability that atleast one of the events occur is

A bag contains 2 white, 3 green and 5 red balls. If three balls are drawn one after the other without replacement, then the probability that the last ball drawn was red is

Let \(\overline{\mathrm{i}}-2 \overline{\mathrm{j}}+\overline{\mathrm{k}}, \overline{\mathrm{i}}+\overline{\mathrm{j}}-2 \overline{\mathrm{k}}, 2 \overline{\mathrm{i}}-\overline{\mathrm{j}}-\overline{\mathrm{k}}\) and \(\overline{\mathrm{i}}+\overline{\mathrm{j}}+\overline{\mathrm{k}}\) be the position vectors of four points \(\mathrm{A}, \mathrm{B}, \mathrm{C}\) and D respectively. If a point P divides AB in the ratio \(2: 1\) internally and a point \(Q\) divides \(C D\) in the ratio \(1: 2\) externally, then the ratio in which the point with position vector \(5 \overline{\mathrm{i}}-6 \overline{\mathrm{j}}-5 \overline{\mathrm{k}}\) divides PQ is

If each of the coefficients \(a, b, c\) in the equation \(a x^2+b x\) \(+c=0\) is determined by throwing a die, then the probability that the equation will have equal roots, is

The value of \(\lambda\) with \(|\lambda| < 16\) such that \(2 x^2-10 x y+12 y^2+5 x+\lambda y-3=0\) represents a pair of straight lines, is

\(\int_0^\pi x \sin ^4 x \cos ^6 x d x=\)

If \(x\) is complex, the expression \(\frac{x^2+34 x-71}{x^2+2 x-7}\) takes all values which lie in the interval \((a, b)\), find the value of \(a\) and \(b\).

\(A B C D E F\) is a regular hexagon. The sum of the vectors \(\mathbf{B E}, \mathbf{B C}, \mathbf{E F}, \mathbf{B A}, \mathbf{C F}, \mathbf{A F}\)

A clock pendulum made of invar has a period of \(0.5 \mathrm{~s}\), at \(20^{\circ} \mathrm{C}\). If the clock is used in a climate where the temperature averages to \(30^{\circ} \mathrm{C}\), how much time does the clock lose in each oscillation? (For invar, \(\alpha=9 \times 10^{-7} /{ }^{\circ} \mathrm{C}\), \(g=\) constant \()\)

If \(\overline{\mathbf{P O}}+\overline{\mathbf{O Q}}=\overline{\mathbf{Q O}}+\overline{\mathbf{O R}}\), then

A wire of length  $L$ $\mathrm{m}$ carrying a current $I \mathrm{A}$ is bent in the form of a circle. Magnitude of its magnetic moment is

If the equation \(x^4+7 x^3+18 x^2+20 x+8=0\) has a repeated root, then that repeated root is

If \(g(x)=\frac{x}{[x]}\) for \(\mathrm{x}>2\), then \(\lim _{x \rightarrow 2} \frac{g(x)-g(2)}{x-2}\) is equal to