Line \(L_1\) of slope 2 and line \(L_2\) of slope \(\frac{1}{2}\) intersect at the origin O . In the first quadrant, \(\mathrm{P}_1, \mathrm{P}_2, \ldots . \mathrm{P}_{12}\) are 12 points on line \(L_1\) and \(Q_1, Q_2, \ldots . . Q_9\) are 9 points on line \(L_2\). Then the total number of triangles, that can be formed having vertices at three of the 22 points \(\mathrm{O}, \mathrm{P}_1, \mathrm{P}_2, \ldots \mathrm{P}_{12}\), \(\mathrm{Q}_1, \mathrm{Q}_2, \ldots . \mathrm{Q}_9\), is:

The number of \(4\) letter words (with or without meaning) that can be formed from the eleven letters of the word \('EXAMINATION'\) is

The number of real roots of the equation \(e^{6 x}-e^{4 x}-2 e^{3 x}-12 e^{2 x}+e^{x}+1=0\) is:

A fair die is tossed repeatedly until a six is obtained. Let \(\mathrm{X}\) denote the number of tosses required and let \(\mathrm{a}=\mathrm{P}(\mathrm{X}=3), \mathrm{b}=\mathrm{P}(\mathrm{X} \geq 3)\) and \(\mathrm{c}=\) \(\mathrm{P}(\mathrm{X} \geq 6 \mid \mathrm{X}>3)\). Then \(\frac{\mathrm{b}+\mathrm{c}}{\mathrm{a}}\) is equal to

\(\mathop \smallint \limits_{ - \pi /2}^{\pi /2} \frac{{{{\sin }^2}x}}{{1 + {2^x}}}dx =\) . ..  . 

Let \(x, y>0\). If \(x^{3} y^{2}=2^{15}\), then the least value of \(3 x +2 y\) is

Consider three vectors \(\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}\). Let \(|\overrightarrow{\mathrm{a}}|=2,|\overrightarrow{\mathrm{b}}|=3\) and \(\overrightarrow{\mathrm{a}}=\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}\). If \(\alpha \in\left[0, \frac{\pi}{3}\right]\) is the angle between the vectors \(\vec{b}\) and \(\vec{c}\), then the minimum value of \(27|\overrightarrow{c}-\overrightarrow{a}|^2\) is equal to :

Let \(\alpha, \beta\) be roots of \(x^2+\sqrt{2} x-8=0\). If \(\mathrm{U}_{\mathrm{n}}=\alpha^{\mathrm{n}}+\beta^{\mathrm{n}}\), then \(\frac{\mathrm{U}_{10}+\sqrt{2} \mathrm{U}_9}{2 \mathrm{U}_8}\) is equal to ............

A candidate has to go to the examination centre to appear in an examination. The candidate uses only one means of transportation for the entire distance out of bus, scooter and car. The probabilities of the candidate going by bus, scooter and car, respectively, are \(\dfrac{2}{5}\), \(\dfrac{1}{5}\) and \(\dfrac{2}{5}\). The probabilities that the candidate reaches late at the examination centre are \(\dfrac{1}{5}\), \(\dfrac{1}{3}\) and \(\dfrac{1}{4}\) if the candidate uses bus, scooter and car, respectively. Given that the candidate reached late at the examination centre, the probability that the candidate travelled by bus is:

Let \(z_1=2+3 i\) and \(z_2=3+4 i\). The set \(S =\left\{ z \in C :\left| z - z _1\right|^2-\left|z-z_2\right|^2=\left|z_1-z_2\right|^2\right\}\) represents a

Slope of a line passing through \(P(2, 3)\) and intersecting the line, \(x + y = 7\) at a distance of \(4\) units from \(P,\) is

Let \(f (x) = a^x (a > 0)\) be written as \(f( x) = f_1( x) + f_2( x)\) , where \(f_1( x)\) is an even function and \(f_2( x)\) is an odd function. Then \(f_1( x + y) + f_1( x - y )\) equals

Tangent and normal are drawn at \(P(16, 16)\) on the parabola \({y^2} = 16x\), which intersect the axis of the parabola at \(A\) and \(B\), respectively. If \(C\) is the centre of the circle through the points  \(P, A\) and \(B\) and \(\angle CPB = \theta \) , then a value of \(\tan \theta \;\)is :