Ex. 35. Statement I: Common value(s) of x satisfying the equations log sin x ( sec x + 8 ) > 0 and log sin x cos x + log cos x sin x = 2 in ( 0 , 4 π ) does not exist. Statement II: On solving above trigonometric equations we have to take intersection of trigonometric chains given by sec x > 1 and x = n π + π 4 , n ∈ I .

Solution: log sin x cos x + log cos x sin x = 2 only when sin x = cos x ⇒ x = n π + π 4 Also, log sin x ( sec x + 8 ) > 0 ⇒ sec x + 8 < 1 ⇒ sec x < − 7 Clearly, x = π 4 + n π satisfy the first equation. Statement I is true and Statement II is false. ∴ Answer is (c).

Trigonometry & Inverse Trigonometry

Trigonometric Equations with Logarithms

Grade 11

Question:

Ex. 35. Statement I: Common value(s) of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> satisfying the equations <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mo>log</mo><munder><mi>sin</mi><mi>x</mi></munder></mrow><mo>(</mo><mi>sec</mi><mi>x</mi><mo>+</mo><mn>8</mn><mo>)</mo><mo>></mo><mn>0</mn></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mo>log</mo><munder><mi>sin</mi><mi>x</mi></munder></mrow><mi>cos</mi><mi>x</mi><mo>+</mo><mrow><mo>log</mo><munder><mi>cos</mi><mi>x</mi></munder></mrow><mi>sin</mi><mi>x</mi><mo>=</mo><mn>2</mn></mrow></math> in <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mn>4</mn><mi>π</mi><mo>)</mo></math> does not exist.Statement II: On solving above trigonometric equations we have to take intersection of trigonometric chains given by <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>sec</mi><mi>x</mi><mo>></mo><mn>1</mn></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>x</mi><mo>=</mo><mi>n</mi><mi>π</mi><mo>+</mo><mfrac><mi>π</mi><mn>4</mn></mfrac></mrow></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><mi mathvariant="double-struck">I</mi></math>.

(a) Statement I is true, Statement II is true; Statement II is a correct explanation for Statement I.

(b) Statement I is true, Statement II is true; Statement II is not a correct explanation for Statement I.

(c) Statement I is true, Statement II is false.

(d) Statement I is false, Statement II is true.

Step-by-Step Solution

Key Concept: Solve both logarithmic-trigonometric inequalities and equations separately, then find their intersection.

Solution:<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mo>log</mo><munder><mi>sin</mi><mi>x</mi></munder></mrow><mi>cos</mi><mi>x</mi><mo>+</mo><mrow><mo>log</mo><munder><mi>cos</mi><mi>x</mi></munder></mrow><mi>sin</mi><mi>x</mi><mo>=</mo><mn>2</mn></mrow></math> only when <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>sin</mi><mi>x</mi><mo>=</mo><mi>cos</mi><mi>x</mi></mrow></math>⇒ <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>x</mi><mo>=</mo><mi>n</mi><mi>π</mi><mo>+</mo><mfrac><mi>π</mi><mn>4</mn></mfrac></mrow></math>Also, <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mo>log</mo><munder><mi>sin</mi><mi>x</mi></munder></mrow><mo>(</mo><mi>sec</mi><mi>x</mi><mo>+</mo><mn>8</mn><mo>)</mo><mo>></mo><mn>0</mn></mrow></math>⇒ <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>sec</mi><mi>x</mi><mo>+</mo><mn>8</mn><mo><</mo><mn>1</mn></mrow></math> ⇒ <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>sec</mi><mi>x</mi><mo><</mo><mo>−</mo><mn>7</mn></mrow></math>Clearly, <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>x</mi><mo>=</mo><mfrac><mi>π</mi><mn>4</mn></mfrac><mo>+</mo><mi>n</mi><mi>π</mi></mrow></math> satisfy the first equation.Statement I is true and Statement II is false.∴ Answer is (c).

Correct Answer: C

Question:

Step-by-Step Solution

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