Consider the equation 4√2x³ − 3√2x −1 = 0 .

Statement 1: Solution of this equation is cos(π/12) .

Statement 2: This equation has only one real solution.

To analyze the given statements, we start with the cubic equation:

$4\sqrt{2}{x}^3-3\sqrt{2}x-1=0$

First, we can divide the entire equation by $\sqrt{2}$ (since $\sqrt{2}\ne 0$):

$4x^3-3x-\frac{1}{\sqrt{2}}=0$

This can be rewritten as:

$4x^3-3x=\frac{1}{\sqrt{2}}$

Recall the trigonometric triple-angle identity for cosine:

$cos\left(3\theta \right)=4{cos}^3\left(\theta \right)-3cos\left(\theta \right)$

Let us substitute $x=cos\left(\theta \right)$ into our equation, where we restrict $\theta \in [0,\pi ]$ to find unique real roots:

$cos\left(3\theta \right)=\frac{1}{\sqrt{2}}$

Since $\theta \in [0,\pi ]$, the angle $3\theta$ lies in the interval $[0,3\pi ]$. Within this interval, we solve for $3\theta$ where the cosine value is $\frac{1}{\sqrt{2}}$:

1) $3\theta =\frac{\pi }{4}\Rightarrow \theta =\frac{\pi }{12}$
2) $3\theta =2\pi -\frac{\pi }{4}=\frac{7\pi }{4}\Rightarrow \theta =\frac{7\pi }{12}$
3) $3\theta =2\pi +\frac{\pi }{4}=\frac{9\pi }{4}\Rightarrow \theta =\frac{9\pi }{12}=\frac{3\pi }{4}$

Each of these three values of $\theta$ lies in the range $[0,\pi ]$ and yields a distinct value for $x=cos\left(\theta \right)$:

$x_1=cos\left(\frac{\pi }{12}\right)$
$x_2=cos\left(\frac{7\pi }{12}\right)$
$x_3=cos\left(\frac{3\pi }{4}\right)$

Thus, we have three distinct real solutions. Let us evaluate the statements:

Statement 1: "Solution of this equation is cos(π/12)". Since $x_1=cos(\pi /12)$ is indeed a root of the equation, Statement 1 is true.

Statement 2: "This equation has only one real solution". Since we found three distinct real solutions, Statement 2 is false.

Therefore, Statement 1 is true but Statement 2 is false.