In a cubic equation coefficient of $x^2$ is zero and remaining coefficient are real has one root $\alpha = 3 + 4\, i$ and remaining roots are $\beta$ and $\gamma$ then $\alpha \beta \gamma$ is :-
$150$
$-150$
$25$
None of these
If $a$ and $b$ are the roots of equation $x^2-7 x-1=0$, then the value of $\frac{a^{21}+b^{21}+a^{17}+b^{17}}{a^{19}+b^{19}}$ is equal to $........$.
The number of solutions, of the equation $\mathrm{e}^{\sin x}-2 e^{-\sin x}=2$ is
Let $p, q$ be integers and let $\alpha, \beta$ be the roots of the equation, $x^2-x-1=0$, where $\alpha \neq \beta$. For $n=0,1,2, \ldots$, let $a_n=$ $p \alpha^n+q \beta^n$.
$FACT$ : If $a$ and $b$ are rational numbers and $a+b \sqrt{5}=0$, then $a=0=b$.
($1$) $a_{12}=$
$[A]$ $a_{11}-a_{10}$ $[B]$ $a_{11}+a_{10}$ $[C]$ $2 a_{11}+a_{10}$ $[D]$ $a_{11}+2 a_{10}$
($2$) If $a_4=28$, then $p+2 q=$
$[A] 21$ $[B] 14$ $[C] 7$ $[D] 12$
answer the quetion ($1$) and ($2$)
The equation${e^x} - x - 1 = 0$ has
Let $a, b, c, d$ be real numbers between $-5$ and $5$ such that $|a|=\sqrt{4-\sqrt{5-a}},|b|=\sqrt{4+\sqrt{5-b}},|c|=\sqrt{4-\sqrt{5+c}}$ $|d|=\sqrt{4+\sqrt{5+d}}$ Then, the product $a b c d$ is