If $\alpha ,\beta $are the roots of ${x^2} - ax + b = 0$ and if ${\alpha ^n} + {\beta ^n} = {V_n}$, then
${V_{n + 1}} = a{V_n} + b{V_{n - 1}}$
${V_{n + 1}} = a{V_n} + a{V_{n - 1}}$
${V_{n + 1}} = a{V_n} - b{V_{n - 1}}$
${V_{n + 1}} = a{V_{n - 1}} - b{V_n}$
Suppose $a$ is a positive real number such that $a^5-a^3+a=2$. Then,
The integer $'k'$, for which the inequality $x^{2}-2(3 k-1) x+8 k^{2}-7>0$ is valid for every $x$ in $R ,$ is
The number of real values of $x$ for which the equality $\left| {\,3{x^2} + 12x + 6\,} \right| = 5x + 16$ holds good is
Let $S=\left\{ x : x \in R \text { and }(\sqrt{3}+\sqrt{2})^{ x ^2-4}+(\sqrt{3}-\sqrt{2})^{ x ^2-4}=10\right\} \text {. }$ Then $n ( S )$ is equal to
If $\alpha ,\beta ,\gamma$ are the roots of $x^3 - x - 2 = 0$, then the value of $\alpha^5 + \beta^5 + \gamma^5$ is-