The number of real roots of the equation, $\mathrm{e}^{4 \mathrm{x}}+\mathrm{e}^{3 \mathrm{x}}-4 \mathrm{e}^{2 \mathrm{x}}+\mathrm{e}^{\mathrm{x}}+1=0$ is
$4$
$2$
$3$
$1$
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 ($2$)
Let $\alpha ,\beta $ be the roots of ${x^2} + (3 - \lambda )x - \lambda = 0.$ The value of $\lambda $ for which ${\alpha ^2} + {\beta ^2}$ is minimum, is
Let $y = \sqrt {\frac{{(x + 1)(x - 3)}}{{(x - 2)}}} $, then all real values of $x$ for which $y$ takes real values, are
The number of real solutions of the equation $x\left(x^2+3|x|+5|x-1|+6|x-2|\right)=0$ is
Let $a$ , $b$ , $c$ are roots of equation $x^3 + 8x + 1 = 0$ ,then the value of
$\frac{{bc}}{{(8b + 1)(8c + 1)}} + \frac{{ac}}{{(8a + 1)(8c + 1)}} + \frac{{ab}}{{(8a + 1)(8b + 1)}}$ is equal to