Maths

Let z = a ib, b $\ne$ 0 be complex numbers satisfying z² = $\overline{z}.2^{1-\left|z\right|}.$ Then the least value of n $\in$ N, such that zn = ${\left(z+1\right)}^n,$ is equal to………

Let $S=[-\pi ,\frac{\pi }{2})-\left\{-\frac{\pi }{2},-\frac{\pi }{4},-\frac{3\pi }{4},\frac{\pi }{4}\right\}.$ Then the number of elements in the set $|A=\left\{\theta \in S:tan\theta \left(1+\sqrt[]{5}tan\left(2\theta \right)\right)=\sqrt[]{5}-tan\left(2\theta \right)\right\}$ is……….

If $\frac{6}{3^{12}}+\frac{10}{3^{11}}+\frac{20}{3^{10}}+\frac{40}{3^9}+...+\frac{10240}{3}=2^n.$ m, where m is odd, then m. n is equal to……….

Two tangent lines ${\mathcal{l}}_{1}and{\mathcal{l}}_2$ are drawn from the point (2, 0) to the parabola 2y² = x. If the lines ${\mathcal{l}}_{1}and{\mathcal{l}}_2$ are also tangent to the circle (x – 5)² + y²= r, then 17r is equal to……….

If  is equal to 2ⁿ. m, where m is odd, then n + m is equal to………

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