Question 39

If $\begin{bmatrix} \tan x & 1 \\ 1 & \tan x \end{bmatrix} \begin{bmatrix} \cos^2 x \\ \sin x \cos x \end{bmatrix} = \begin{bmatrix} a \\ b \end{bmatrix} \frac{1}{2}$ where $b = 2a$ , then $0 \leq x \leq \pi$ that satisfies is....

$\frac{\pi}{6}$
$\frac{\pi}{12}$
$\frac{5\pi}{6}$
$\frac{5\pi}{12}$

ONLY $(1), (2), (3)$ are correct

ONLY $(1)$ and $(3)$ are correct

ONLY $(2)$ and $(4)$ are correct

ONLY $(4)$ is correct

ALL statements are correct

Explanation

Given the matrices

\begin{bmatrix} \tan x & 1 \\ 1 & \tan x \end{bmatrix} \begin{bmatrix} \cos^2 x \\ \sin x \cos x \end{bmatrix} = \begin{bmatrix} a \\ b \end{bmatrix}

Multiply the matrices

This is matrix multiplication, so

\begin{bmatrix} a \\ b \end{bmatrix} = \begin{bmatrix} \tan x \cdot \cos^2 x + \sin x \cdot \cos x \\ \cos^2 x + \tan x \cdot \sin x \cos x \end{bmatrix}

Change $\tan x = \frac{\sin x}{\cos x}$ , to

\begin{bmatrix} a \\ b \end{bmatrix} = \begin{bmatrix} \left(\frac{\sin x}{\cos x}\right) \cdot \cos^2 x + \sin x \cdot \cos x \\ \cos^2 x + \left(\frac{\sin x}{\cos x}\right) \cdot \sin x \cos x \end{bmatrix}

Simplify to

\begin{bmatrix} a \\ b \end{bmatrix} = \begin{bmatrix} \sin x \cdot \cos x + \sin x \cdot \cos x \\ \cos^2 x + \sin^2 x \end{bmatrix}

\begin{bmatrix} a \\ b \end{bmatrix} = \begin{bmatrix} 2 \sin x \cdot \cos x \\ \sin^2 x + \cos^2 x \end{bmatrix}

We know that the value $2 \sin x \cdot \cos x = \sin 2x$ and the value $\sin^2 x + \cos^2 x = 1$ , then

\begin{bmatrix} a \\ b \end{bmatrix} = \begin{bmatrix} \sin 2x \\ 1 \end{bmatrix}

Therefore, the value $a = \sin 2x$ and $b = 1$ .

Substitute into the given condition

Substitute $b = 1$ into $b = 2a$ to get $a = \frac{1}{2}$ .

Then, substitute $a = \frac{1}{2}$ into $a = \sin 2x$ to get $\sin 2x = \frac{1}{2}$ .

The values of $2x$ that satisfy are

2x_1 = \frac{\pi}{6} + k \cdot 2\pi \quad \text{or} \quad 2x_2 = \frac{5\pi}{6} + k \cdot 2\pi

x_1 = \frac{\pi}{12} + k\pi \quad \text{or} \quad x_2 = \frac{5\pi}{12} + k\pi

The values of $x$ that satisfy are $\frac{\pi}{12}$ or $\frac{5\pi}{12}$ .

Therefore, the correct statements are the second and fourth.

Command Palette

Number 39

Explanation

Multiply the matrices

Substitute into the given condition