Q194 / 269 · NDA 2025 (Phase 1) · Mathematics
Consider the following for the two (02) items that follow:
Let $2\sin\alpha + \cos\alpha = 2$, where $0 < \alpha < 90°$.
What is $2\sin 2\alpha + \cos 2\alpha$ equal to?
Let $2\sin\alpha + \cos\alpha = 2$, where $0 < \alpha < 90°$.
What is $2\sin 2\alpha + \cos 2\alpha$ equal to?
- A$\dfrac{11}{10}$
- Correct answerB$\dfrac{11}{5}$
- C$\dfrac{12}{5}$
- D$\dfrac{13}{5}$
Solution
Given $2\sin\alpha + \cos\alpha = 2$
$\cos\alpha = 2(1 - \sin\alpha)$
Squaring both sides:
$\cos^2\alpha = 4(1 - \sin\alpha)^2$
$1 - \sin^2\alpha = 4(1 - 2\sin\alpha + \sin^2\alpha)$
$5\sin^2\alpha - 8\sin\alpha + 3 = 0$
$(5\sin\alpha - 3)(\sin\alpha - 1) = 0$
Since $0 < \alpha < 90^\circ$, $\sin\alpha \neq 1$. Thus, $\sin\alpha = \dfrac{3}{5}$.
This gives $\cos\alpha = \sqrt{1 - \left(\dfrac{3}{5}\right)^2} = \dfrac{4}{5}$.
Now, $\sin 2\alpha = 2\sin\alpha\cos\alpha = 2\left(\dfrac{3}{5}\right)\left(\dfrac{4}{5}\right) = \dfrac{24}{25}$
$\cos 2\alpha = \cos^2\alpha - \sin^2\alpha = \left(\dfrac{4}{5}\right)^2 - \left(\dfrac{3}{5}\right)^2 = \dfrac{7}{25}$
Therefore, $2\sin 2\alpha + \cos 2\alpha = 2\left(\dfrac{24}{25}\right) + \dfrac{7}{25} = \dfrac{48 + 7}{25} = \dfrac{55}{25} = \dfrac{11}{5}$.
Answer: $\dfrac{11}{5}$
$\cos\alpha = 2(1 - \sin\alpha)$
Squaring both sides:
$\cos^2\alpha = 4(1 - \sin\alpha)^2$
$1 - \sin^2\alpha = 4(1 - 2\sin\alpha + \sin^2\alpha)$
$5\sin^2\alpha - 8\sin\alpha + 3 = 0$
$(5\sin\alpha - 3)(\sin\alpha - 1) = 0$
Since $0 < \alpha < 90^\circ$, $\sin\alpha \neq 1$. Thus, $\sin\alpha = \dfrac{3}{5}$.
This gives $\cos\alpha = \sqrt{1 - \left(\dfrac{3}{5}\right)^2} = \dfrac{4}{5}$.
Now, $\sin 2\alpha = 2\sin\alpha\cos\alpha = 2\left(\dfrac{3}{5}\right)\left(\dfrac{4}{5}\right) = \dfrac{24}{25}$
$\cos 2\alpha = \cos^2\alpha - \sin^2\alpha = \left(\dfrac{4}{5}\right)^2 - \left(\dfrac{3}{5}\right)^2 = \dfrac{7}{25}$
Therefore, $2\sin 2\alpha + \cos 2\alpha = 2\left(\dfrac{24}{25}\right) + \dfrac{7}{25} = \dfrac{48 + 7}{25} = \dfrac{55}{25} = \dfrac{11}{5}$.
Answer: $\dfrac{11}{5}$
Hence, the correct answer is option: 'B'.