Embibe Experts Solutions for Chapter: Theory of Equation, Exercise 1: Exercise 1

How many real numbers $x$ satisfy the equation $3^{2x+2}-3^{x+3}-3^x+3=0?$

Evaluate $X$ in its simplest form then find sum of all digits of $X.$ Where $X$ is given as $X=\sqrt{2008+2007\sqrt{2008+2007\sqrt{2008+2007\sqrt{\dots .}}}}$

If $x, y$ are positive real numbers satisfying the system of equations $x^2+y\sqrt{xy}=336, y^2+x\sqrt{xy}=112,$ then $x+y$ equals

In a class, the total numbers of boys and girls are in the ratio $4:3$. On one day it was found that $8$ boys and $14$ girls were absent from the class and that the number of boys was the square of the number of girls. What is the total number of students in the class?

If the roots of the equation $\frac{1}{x+p}+\frac{1}{x+q}=\frac{1}{r}$ are equal in magnitude and opposite is sign, then the product of roots is

Sum of the non-real roots of $\left(x^2+x-2\right)\left(x^2+x-3\right)=12,$ is

If ${\lambda }_1$ and ${\lambda }_2$ are the two values of $\lambda$, such that the roots $\alpha$ and $\beta$ of the quadratic equation, $\lambda \left(x^2-x\right)+x+5=0$ satisfy $\frac{\alpha }{\beta }+\frac{\beta }{\alpha }+\frac{4}{5}=0,$ then $\frac{{\lambda }_1}{{\lambda }_2^2}+\frac{{\lambda }_2}{{\lambda }_1^2}$ is equal to

The number of real solutions of $x-\frac{1}{x^2-4}=2-\frac{1}{x^2-4}$ is