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

MathsCBSE Class 10 · Standard · Board exam 2024, set 30/4/1 · 2 marks · Very short answer

CBSE Class 10 Mathematics Standard 2024: question 23(a)

Set in the CBSE Class 10 board exam of 2024, paper 30/4/1, section B, question 23(a), for 2 marks. Try it first; the hints and our step-by-step solution are below.

Q23(a) · 2 marks · Very short answerIntroduction to Trigonometry

Evaluate : $\frac{5 c o s ^{2} 6 0 ^{\circ} + 4 s e c ^{2} 3 0 ^{\circ} - t a n ^{2} 4 5 ^{\circ}}{s i n ^{2} 3 0 ^{\circ} + s i n ^{2} 6 0 ^{\circ}}$

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 board paper, set 30/4/1.

Hints

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

Hint 1

Write down the standard values of cos 60°, sec 30°, tan 45°, sin 30° and sin 60°.

Hint 2

Square each value carefully; notice the denominator becomes $\frac{1}{4} + \frac{3}{4}$ .

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.

Substitute $cos 6 0^{\circ} = \frac{1}{2}$ , $sec 3 0^{\circ} = \frac{2}{3}$ , $tan 4 5^{\circ} = 1$ , $sin 3 0^{\circ} = \frac{1}{2}$ , $sin 6 0^{\circ} = \frac{3}{2}$ :
$\frac{5 ( \frac{1}{2} ) ^{2} + 4 ( \frac{2}{3} ) ^{2} - ( 1 ) ^{2}}{( \frac{1}{2} ) ^{2} + ( \frac{3}{2} ) ^{2}} = \frac{\frac{5}{4} + \frac{16}{3} - 1}{\frac{1}{4} + \frac{3}{4}}$
1½ marks
$= \frac{\frac{15 + 64 - 12}{12}}{1} = \frac{67}{12}$
½ mark

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

Where the marks are

Correct substitution of all trigonometric values (1½ marks)
Simplifies to 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 paper2024 board paper, set 30/4/1 (Standard): every question, by section.

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CBSE Class 10 Mathematics Standard 2024: question 23(a)

Hints

Step-by-step solution

Revise the chapter

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