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NTA Abhyas 2020
Exam
NTA Abhyas
Year
2020
Subject
Mathematics
Chapter
Trigonometric Equations

NTA Abhyas 2020 · Mathematics · Trigonometric Equations

The sum of all the solutions in [0,100] for the equation sin + cos...

The sum of all the solutions in [0,100]\left[0,100\right] for the equation sin\pix+cos\pix=0sin \pix + cos \pix = 0 is

Options

  1. A25502550
  2. B50255025
  3. C25252525
  4. D50505050

Correct answer

B. 50255025

Step-by-step solution

sin\pix+cos\pix=0sin \pix + cos \pix = 0
tan\pix=−1=tan3π4tan \pix = - 1 = tan\dfrac{3\pi}{4}
\pix=nπ+3π4⇒x=n+34\pix = n\pi + \dfrac{3\pi}{4} ⇒ x = n + \dfrac{3}{4}
Given, x∈[0,100],x \in \left[0,100\right], hence nn will range from 00 to 99.99 .
xx will be 34,74,114,.....,3994\dfrac{3}{4}, \dfrac{7}{4}, \dfrac{11}{4}, . . . . . , \dfrac{399}{4}
{S}_{n} = \dfrac{n}{2} \left[2a + \left( n - 1\right)d\right]<br />=50\left[\dfrac{6}{4} +99 \times1\right] = 5025$$
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