If $\alpha,\beta,\gamma, \delta$ are the roots of $x^4-100x^3+2x^2+4x+10 = 0$ then $\frac{1}{\alpha}+\frac{1}{\beta}+\frac{1}{\gamma}+\frac{1}{\delta}$ is equal to :-

Let $a$ ,$b$, $c$ , $d$ , $e$ be five numbers satisfying the system of equations

                            $2a + b + c + d + e = 6$
                            $a + 2b + c + d + e = 12$
                            $a + b + 2c + d + e = 24$
                            $a + b + c + 2d + e = 48$
                            $a + b + c + d + 2e = 96$ ,

then $|c|$ is equal to 

The value of $x$ in the given equation ${4^x} - {3^{x\,\; - \;\frac{1}{2}}} = {3^{x + \frac{1}{2}}} - {2^{2x - 1}}$is

If the sum of all the roots of the equation $e^{2 x}-11 e^{x}-45 e^{-x}+\frac{81}{2}=0$ is $\log _{ e } P$, then $p$ is equal to

The number of integers $k$ for which the equation $x^3-27 x+k=0$ has at least two distinct integer roots is

Let $\alpha$ and $\beta$ be the roots of the equation $\mathrm{x}^{2}-\mathrm{x}-1=0 .$ If $\mathrm{p}_{\mathrm{k}}=(\alpha)^{\mathrm{k}}+(\beta)^{\mathrm{k}}, \mathrm{k} \geq 1,$ then which one of the following statements is not true?