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

The equation $e^{4 x}+8 e^{3 x}+13 e^{2 x}-8 e^x+1=0, x \in R$ has:

The population of cattle in a farm increases so that the difference between the population in year $n+2$ and that in year $n$ is proportional to the population in year $n+1$. If the populations in years $2010, 2011$ and $2013$ were $39,60$ and $123$,respectively, then the population in $2012$ was

Let $x_1,x_2,x_3 \in R-\{0\} $ ,$x_1 + x_2 + x_3\neq 0$ and $\frac{1}{x_1}+\frac{1}{x_2}+\frac{1}{x_3}=\frac{1}{x_1+x_2+x_3}$, then  $\frac{1}{{x^n}_1+{x^n}_2+{x^n}_3} =\frac{1}{{x^n}_1}+\frac{1}{{x^n}_2}+\frac{1}{{x^n}_3}$ holds good for

The number of real solutions of the equation $x\left(x^2+3|x|+5|x-1|+6|x-2|\right)=0$ is

If $(x + 1)$ is a factor of ${x^4} - (p - 3){x^3} - (3p - 5){x^2}$ $ + (2p - 7)x + 6$, then $p = $