Question : If $\sec \theta-2 \cos \theta=\frac{7}{2}$, where $\theta$ is a positive acute angle, then the value of $\sec \theta$ is:
Option 1: 6
Option 2: 8
Option 3: 5
Option 4: 4
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Correct Answer: 4
Solution :
Given, $\sec \theta-2 \cos \theta=\frac{7}{2}$
We know, $\cos\theta = \frac{1}{\sec\theta}$
⇒ $\sec \theta-\frac{2}{\sec \theta}=\frac{7}{2}$
⇒ $(\sec^2\theta - 2)2=7\sec\theta$
⇒ $2\sec^2\theta - 7\sec\theta -4 = 0$
Solving it as a quadratic equation, using middle-term splitting,
⇒ $2\sec^2\theta - 8\sec\theta + \sec\theta - 4 = 0$
⇒ $2\sec\theta (\sec\theta-4)+1(\sec\theta - 4) = 0$
⇒ $(2\sec\theta + 1)(\sec\theta-4)=0$
⇒ $\sec\theta = -\frac{1}{2}$ and $\sec \theta = 4$
It is given that $\theta$ is an acute angle which means the value $\sec\theta$ should be positive.
$\therefore\sec\theta = 4$
Hence, the correct answer is 4.
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