Astudent has to write the words ABILITY, PROBABILITY, FACILITY, MOBILITY. He wrote one word and erased all the letters in it except two consecutive letters. If ' LI ' is left after erasing then the probability that the boy wrote the word PROBABLLITY is

If \(\mathbf{a}=2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+\hat{\mathbf{k}}, \mathbf{b}=\hat{\mathbf{i}}-3 \hat{\mathbf{j}}-5 \hat{\mathbf{k}}\) and \(\mathbf{c}=3 \hat{\mathbf{i}}-4 \hat{\mathbf{k}}\), then match the items of List-I with those of List-II.

\(f(x)\) is a twrice differentiable function such that \(f^{\prime \prime}(x)=-f(x)\) and \(f^{\prime}(x)=g(x)\). If \(\left.h(x)=(f(x))^2+g(x)\right)^2\) and \(h(l)=2\), then \(h(2)=\)

If \(C_1\) and \(C_2\) are the centres of similitude with respect to the circles \(x^2+y^2+6 x+8 y+24=0\) and \(x^2+y^2-6 x-8 y\) \(+9=0\) then \(C_1 C_2=\)

In triangle $ABC$, if $A$ is acute, $C$ is obtuse, $\sin A=\frac{3\sqrt{3}}{14},a=3$ and $b=5$, then $c=$

The equation of the pair of straight lines passing through the point \((2,3)\) and perpendicular to the pair of lines \(3 x^2-4 x y+5 y^2=0\) is \(a x^2+2 h x y+b y^2+2 g x+2 f y+c\) \(=0\) then \(a+b+c+f+g+h=\)

If the curves \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) and \(\frac{x^2}{25}+\frac{y^2}{16}=1\) cut each other orthogonally, then \(a^2-b^2\) equals to

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

Statement (I) : The slope of kinetic energy-displacement curve of a body in motion will be directly proportional to its acceleration. Statement (II) : From a height of \(15 \mathrm{~m}\) a ball is projected vertically upwards with a velocity of \(30 \mathrm{~m} / \mathrm{s}\). If the ball rises to the same height after hitting the ground, the loss of its energy on hitting the ground is \(30 \%\). Statement (III) : The velocity acquired by a body of mass ' \(m\) ' after travelling a fixed distance from rest under the action of a constant force is directly proportional to mass ' \(\mathrm{m}\) '. Which of the following is correct?

Match the following table. \begin{array}{|c|c|c|c|} \hline & List-I & & List-II \ \hline A. & \begin{array}{l}\text { Michelson-Morley } \ \text { experiment }\end{array} & I. & \begin{array}{l}\text { The existence of } \ \text { anti-matter }\end{array} \ \hline B. & \begin{array}{l}\text { Stern-Gerlach } \ \text { experiment }\end{array} & II. & \begin{array}{l}\text { The existence of } \ \text { de-Broglie matter waves }\end{array} \ \hline C. & \begin{array}{l}\text { Davisson-Germer } \ \text { experiment }\end{array} & III. & Electrons have spins \ \hline D. & \begin{array}{l}\text { Anderson discovery } \ \text { of positron }\end{array} & \mathrm{IV}. & \begin{array}{l}\text { The non-existence of } \ \text { ether }\end{array} \ \hline \end{array} \(\begin{array}{llll}A & B & C & D\end{array}\)

The general solution of the differential equation \((y \sin x+y) \frac{d y}{d x}-\cos ^2 x=0\)

The relation between the current \(i\) (in ampere) in a conductor and the time \(t\) (in second) is \(i=12 i+9 t^2\). The charge passing through the conductor between the times \(t=2 \mathrm{~s}\) and \(t=10 \mathrm{~s}\) is

When the origin is shifted to the point \((h, k)\) by translati:g the coordinate axes, the equation \(S=2 x^2-x y+y^2+2 x\) \(+3 y+1=0\) is changed to \(S^{\prime}=a x^2+2 h x y+b y^2-3\) \(=0\). Again by rotating the coordinate axes about the new origin through the angle \(\theta\) in the positive direction, \(\mathrm{S}^{\prime}=0\) is changed to \(\mathrm{Ax}^2+\mathrm{By}^2+\mathrm{C}=0\). Then \(\mathrm{h}+\mathrm{k}+\tan 2 \theta=\)