Maths

Let the tangents at the points P and Q on the ellipse $\frac{x^2}{2}+\frac{y^2}{4}=1$ meet at the point $R\left(\sqrt[]{2},2\sqrt[]{2}-2\right).$ If S is the focus of the ellipse on its negative major axis, then SP² + SQ² is equal to………

A class contains b boys and g girls. If the number of ways of selecting 3 boys and 2 girls from the class is 168, then b + 3g is equal to………….

Let the coefficients of the middle terms in the expansion of ${\left(\frac{1}{\sqrt[]{6}}+\beta x\right)}^4,{\left(1-3\beta x\right)}^2$ and ${\left(1-\frac{\beta }{2}x\right)}^6,\beta >0,$ respectively form the first three terms of an A.P. If d is the common difference of this A.P., then 50 $-\frac{2d}{{\beta }^2}$ is equal to………….

Let

p : Ramesh listens to music.

p : Ramesh is out of his village.

r : It is Sunday

s : It is Saturday

Then the statement “Ramesh listens to music only if he is in his village and it is Sunday or Saturday” can be expressed as

Let A and B be two events such that $P\left(A|B\right)=\frac{2}{5},P\left(A|B\right)=\frac{1}{7}andP\left(A\cap B\right)=\frac{1}{9}.$ Consider $\left(S1\right)P\left(A\text{'}\cup B\right)=\frac{5}{6},$

$\left(S2\right)P\left(A\text{'}\cap B\text{'}\right)=\frac{1}{18}$

Then

A horizontal park is in the shape of a triangle OAB with AB = 16. A vertical lamp post OP is erected at the point O such that $\angle PAO=\angle PBO=15°and\angle PCO=45°,$ where C is the midpoint of AB. Then (OP)² is equal to

Let S be the set of all a $\in$ R for which the angel between the vectors $\overrightarrow{u}=a\left(lo{g}_{e}b\right)\widehat{i}-6\widehat{j}+3\widehat{k}$ and $\overrightarrow{v}=\left(lo{g}_{e}b\right)\widehat{i}+2\widehat{j}+2a\left(lo{g}_{e}b\right)\widehat{k},\left(b>1\right)$ is acute. Then S is equal to:

A plane P is parallel to two lines whose direction ratios are 2, 1 3 and 1, 2, 2 and it contains the point (2, 2, 2). Let P intersect the co-ordinate axes at the points A, B, C making the intercepts $\alpha ,\beta ,\gamma .$ If V is the volume of the tetrahedron OABC, where O is the origin, and p = $\mathrm{\alpha +}$
$\beta$+ $\gamma$ , then the ordered pair (V, p) is equal to:

Let the lines $\frac{x-1}{\lambda }=\frac{y-2}{1}=\frac{z-3}{2}and\frac{x+26}{-2}=\frac{y+18}{3}=\frac{z+28}{\lambda }$ be coplanar and P be the plane containing these two lines. Then which of the following points does NOT lie on P?

Let the hyperbola H : $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ pass through the point $\left(2\sqrt[]{2},-2\sqrt[]{2}\right).$ A parabola is drawn whose focus is same as the focus of H with positive abscissa and the directrix of the parabola passes through the other focus of H. If the length of the latus rectum of the parabola is e times the length of the latus rectum of H, where e is the eccentricity of H, then which of the following points lies on the parabola?