SATHEE: Straight Line and Pair of Straight Lines 1 Question 2

Straight Line and Pair of Straight Lines 1 Question 2

2. The equation $y = \sin x \sin (x + 2) - \sin^{2} (x + 1)$ represents a straight line lying in

(a) second and third quadrants only

(b) first, second and fourth quadrants

(d) third and fourth quadrants only

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Solution:

Key Idea Use formulae :

$2 \sin A \sin B = \cos (A - B) - \cos (A + B)$ and $\cos 2 θ = 1 - 2 \sin^{2} θ$

Given equation is $y = \sin x \sin (x + 2) - \sin^{2} (x + 1)$

$= \frac{1}{2} [\cos 2 - \cos (2 x + 2)] - \frac{1}{2} [1 - \cos (2 x + 2)]$

$[∵ 2 \sin A \sin B = \cos (A - B) - \cos (A + B)$ and $\cos 2 θ = 1 - 2 \sin^{2} θ \Rightarrow 2 \sin^{2} θ = 1 - \cos 2 θ]$

$= \frac{1}{2} \cos 2 - \frac{1}{2} \cos (2 x + 2) - \frac{1}{2} + \frac{1}{2} \cos (2 x + 2)$

$= \frac{1}{2} (\cos (2) - 1) = - \frac{1}{2} (2 \sin^{2} (1))$

$= - \sin^{2} (1) < 0 \Rightarrow y < 0$

and as we know that $y < 0$ , is in third and fourth quadrants only.

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Straight Line and Pair of Straight Lines 1 Question 2

2. The equation y=sin⁡xsin⁡(x+2)−sin2⁡(x+1) represents a straight line lying in

Solution:

Key Idea Use formulae :

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2. The equation $y = \sin x \sin (x + 2) - \sin^{2} (x + 1)$ represents a straight line lying in