Assertion (A): If 4 cos θ = 11 sin θ, the value of $\dfrac{11\text{cosθ} - 7 \text{sinθ}}{11 \text{cosθ} + 7 \text{sinθ}}=\dfrac{93}{149}$.

Reason (R): tan θ = $\dfrac{\text{sin θ}}{\text{cos θ}}$

1. A is true, R is false.
2. A is false, R is true.
3. Both A and R are true.
4. Both A and R are false.

Both A and R are true.

Explanation

Given,

4 cos θ = 11 sin θ

⇒ sin θ = $\dfrac{4}{11}$ cos θ

⇒ $\dfrac{sin θ}{cos θ} = \dfrac{4}{11}$

Let sin θ = 4a and cos θ = 11a.

$\dfrac{11\text{cosθ} - 7 \text{sinθ}}{11 \text{cosθ} + 7 \text{sinθ}}\\[1em] = \dfrac{11 \times 11a - 7 \times 4a}{11 \times 11a + 7 \times 4a}\\[1em] = \dfrac{121a - 28a}{121a + 28a}\\[1em] = \dfrac{93a}{149a}\\[1em] = \dfrac{93}{149}$

∴ Assertion (A) is true.

sin θ = $\dfrac{Perpendicular}{Hypotenuse}$

cos θ = $\dfrac{Base}{Hypotenuse}$

tan θ = $\dfrac{Perpendicular}{Base}$

= $\dfrac{\dfrac{Perpendicular}{Hypotenuse}}{\dfrac{Base}{Hypotenuse}}$

= $\dfrac{\text{sin θ}}{\text{cos θ}}$

∴ Reason (R) is true.

Hence, both Assertion (A) and Reason (R) are true.