CBSE Class 10 Mathematics Standard sample paper 2023-24: question 30

Hints

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Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

1. Given $1+{\sin }^2\theta =3\sin \theta \cos \theta$.

   Here $\cos \theta \ne 0$: if $\cos \theta =0$, the left side would be 2 and the right side 0. So we may divide both sides by ${\cos }^2\theta$:
   $$\frac{1}{{\cos }^2\theta }+{\tan }^2\theta =3\tan \theta$$

   ½ mark

2. $\Rightarrow {\sec }^2\theta +{\tan }^2\theta =3\tan \theta$

   ½ mark

3. Use ${\sec }^2\theta =1+{\tan }^2\theta$: $1+{\tan }^2\theta +{\tan }^2\theta =3\tan \theta$

   ½ mark

4. $\Rightarrow 2{\tan }^2\theta -3\tan \theta +1=0$

   ½ mark

5. Put $\tan \theta =x$: $2x^2-3x+1=0$, so $(x-1)(2x-1)=0$ and $x=1$ or $\frac{1}{2}$.

   Hence $\tan \theta =1$ or $\frac{1}{2}$. Proved.

   1 mark

Answer: $\tan \theta =1$ or $\frac{1}{2}$ (proved)

Revise the chapter

• Chapter: Introduction to Trigonometry (notes, formulas, where marks are lost and every past question)
• Topic: Proving Simple Trigonometric Identities