Evaluate (5² 60°+ 4² 30°- ² 45°)/(² 30°+ ² 30°)

MathsCBSE Class 10 · Standard · Board exam 2023, set 30/4/1 · 2 marks · Very short answer

CBSE Class 10 Mathematics Standard 2023: question 25(A)

Set in the CBSE Class 10 board exam of 2023, paper 30/4/1, section B, question 25(A), for 2 marks. Try it first; the hints and our step-by-step solution are below.

Q25(A) · 2 marks · Very short answerIntroduction to Trigonometry

Evaluate $\frac{5 cos ^{2} 6 0 ^{\circ} + 4 sec ^{2} 3 0 ^{\circ} - tan ^{2} 4 5 ^{\circ}}{sin ^{2} 3 0 ^{\circ} + cos ^{2} 3 0 ^{\circ}}$

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 30/4/1.

Hints

Open one hint at a time and try again before the next.

Hint 1

The denominator is a well-known identity. What is $sin^{2} θ + cos^{2} θ$ ?

Hint 2

Use $cos 6 0^{\circ} = \frac{1}{2}$ , $sec 3 0^{\circ} = \frac{2}{3}$ and $tan 4 5^{\circ} = 1$ , then add the fractions.

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.

Use the standard values: $cos 6 0^{\circ} = \frac{1}{2}$ , $sec 3 0^{\circ} = \frac{2}{3}$ , $tan 4 5^{\circ} = 1$ .
In the denominator, $sin^{2} 3 0^{\circ} + cos^{2} 3 0^{\circ} = 1$ by the identity $sin^{2} θ + cos^{2} θ = 1$ .
Substituting these values:
$\frac{5 ( \frac{1}{2} ) ^{2} + 4 ( \frac{2}{3} ) ^{2} - ( 1 ) ^{2}}{1} = \frac{5}{4} + \frac{16}{3} - 1$
1½ marks
Take the LCM 12:
$\frac{15}{12} + \frac{64}{12} - \frac{12}{12} = \frac{67}{12}$
½ mark

Answer: $\frac{67}{12}$

Where the marks are

Substitutes the correct values of $cos 6 0^{\circ}$ , $sec 3 0^{\circ}$ , $tan 4 5^{\circ}$ and gets denominator $= 1$ (1½ marks)
Simplifies to $\frac{67}{12}$ (½ mark)

Revise the chapter

Chapter: Introduction to Trigonometry (notes, formulas, where marks are lost and every past question)

Practise this chapterBoard-style questions from Introduction to Trigonometry, one at a time, marked against the answer key and marking points.Back to the full paper2023 board paper, set 30/4/1 (Standard): every question, by section.

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CBSE Class 10 Mathematics Standard 2023: question 25(A)

Hints

Step-by-step solution

Revise the chapter

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