NDA 2026 — Mathematics PYQ

1. Determine $\cot \theta$ and the signs of trigonometric functions:

Given the equation:

$$3\cot \theta +4=0$$

$$3\cot \theta =-4$$

$$\cot \theta =-\frac{4}{3}$$

Since $\theta$ lies in the fourth quadrant, $\cos \theta$ is positive, while $\sin \theta$ and $\tan \theta$ are negative.

2. Find $\sin \theta$ and $\cos \theta$:

We know that $\cot \theta =\frac{\cos \theta }{\sin \theta }=-\frac{4}{3}$. Let $\cos \theta =4k$ and $\sin \theta =-3k$.

Using the identity ${\sin }^2\theta +{\cos }^2\theta =1$:

$$(-3k{)}^2+(4k{)}^2=1$$

$$9k^2+16k^2=1$$

$$25k^2=1⟹k^2=\frac{1}{25}⟹k=\frac{1}{5}$$

Thus:

$$\sin \theta =-3\left(\frac{1}{5}\right)=-\frac{3}{5}$$

$$\cos \theta =4\left(\frac{1}{5}\right)=\frac{4}{5}$$

3. Evaluate $\sin 2\theta$ and $\cos 2\theta$:

Using double angle formulas:

$$\sin 2\theta =2\sin \theta \cos \theta =2\left(-\frac{3}{5}\right)\left(\frac{4}{5}\right)=-\frac{24}{25}$$

$$\cos 2\theta ={\cos }^2\theta -{\sin }^2\theta ={\left(\frac{4}{5}\right)}^2-{\left(-\frac{3}{5}\right)}^2=\frac{16}{25}-\frac{9}{25}=\frac{7}{25}$$

4. Final Calculation:

$$\sin 2\theta +\cos 2\theta =-\frac{24}{25}+\frac{7}{25}=-\frac{17}{25}$$

Correct Option: (b) $-\frac{17}{25}$