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SSC CGL 2023 PYQ — Question 51

SSC CGL - 25 July 2023 - Shift 3 • Quantitative Aptitude • Q51

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If r = 15(sinθ\theta + cosθ\theta) and s =16(sinθ\theta - cosθ\theta), then the value of r2152+s2162\frac{r^2}{15^2}+\frac{s^2}{16^2}

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Correct answer: D

Explanation

The correct answer is option 4 i.e. 2.

 Given, r = 15(sin⁡θ\sin\theta + cosθ\theta) 

 ⇒ r15\frac{r}{15} = (sin⁡θ\sin\theta + cosθ\theta)

⇒ r2152\frac{r^2}{15^2} = (sin⁡θ\sin\theta + cosθ\theta)2                         [squaring both sides]

⇒ r2152\frac{r^2}{15^2} = sin⁡2θ\sin^2\theta + cos⁡2θ\cos^2\theta + 2sinθ\thetacosθ\theta              ....................1

And s = 16(sinθ−cosθsin\theta-cos\theta)

⇒ s16\frac{s}{16} = (sin⁡θ\sin\theta - cosθ\theta)

⇒ s2162\frac{s^2}{16^2} = (sin⁡θ\sin\theta - cosθ\theta)2                       [squaring both sides]

⇒ s2162\frac{s^2}{16^2} = sin⁡2θ\sin^2\theta + cos⁡2θ\cos^2\theta - 2sinθ\thetacosθ\theta              ....................2

Adding equation 1 and 2,

⇒ r2152+s2162\frac{r^2}{15^2}+\frac{s^2}{16^2} = 2( sin⁡2θ\sin^2\theta+cos⁡2θ\cos^2\theta) = 2(1)                      [sin⁡2θ\sin^2\theta + cos⁡2θ\cos^2\theta = 1]

Hence,  r2152+s2162\frac{r^2}{15^2}+\frac{s^2}{16^2} = 2