cat 2019 Question 33

Question

The number of solutions to the equation $\mid x \mid (6x^2 + 1) = 5x^2$ is

Accepted Answer

For $x < 0$,

$-x(6x^{2} + 1) = 5x^{2}$

$\Rightarrow (6x^{2} + 1) = -5x$

$\Rightarrow (6x^{2} + 5x + 1) = 0$

$\Rightarrow (6x^{2} + 3x + 2x + 1) = 0$

$\Rightarrow (3x+1)(2x+1)=0 \;\Rightarrow\; x = -\frac{1}{3} \; ext{or}\; x = -\frac{1}{2}$

For $x = 0$, LHS = RHS = 0 (Hence, 1 solution)

For $x > 0$,

$x(6x^{2} + 1) = 5x^{2}$

$\Rightarrow (6x^{2} - 5x + 1) = 0$

$\Rightarrow (3x - 1)(2x - 1) = 0 \;\Rightarrow\; x = \frac{1}{3} \; ext{or}\; x = \frac{1}{2}$

Hence, the total number of solutions = 5

Question 33.