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Solve this Ques of jee main exam February session 2021​

Answer» <html><body><h3><u>EXPLANATION</u>.</h3><p>⇒ sin⁻¹[x² + 1/3] + cos⁻¹[x² - 2/3] = x².</p><p>As we know that,</p><p></p><p>Range of sinθ = [-1,1].</p><p></p><p>There are three digit exists between [-1,1].</p><p>⇒ - 1, <a href="https://interviewquestions.tuteehub.com/tag/0-251616" style="font-weight:bold;" target="_blank" title="Click to know more about 0">0</a>, 1.</p><p></p><p>⇒ [x² + 1/3].</p><p>in this particular equation, we get.</p><p>Only two <a href="https://interviewquestions.tuteehub.com/tag/possibility-1160394" style="font-weight:bold;" target="_blank" title="Click to know more about POSSIBILITY">POSSIBILITY</a> exists = 0,1.</p><p></p><p>⇒ [x² - 2/3].</p><p>In this particular case, we get.</p><p>Three possibility exists = -1, 0, 1.</p><p></p><p>Total we can <a href="https://interviewquestions.tuteehub.com/tag/say-1195451" style="font-weight:bold;" target="_blank" title="Click to know more about SAY">SAY</a> that,</p><p>There are six [6] possibility exists.</p><p></p><p>⇒ [x² + 1/3]     [x² - 2/3].</p><p>⇒ 0                  -1.</p><p>⇒ 0                   0.</p><p>⇒ 0                    1.</p><p>⇒ 1                     -1.</p><p>⇒ 1                     0.</p><p>⇒ 1                     1.</p><p></p><p>Now, we can equation as,</p><p>⇒ sin⁻¹[x² + 1/3] + cos⁻¹[x² - 2/3] = x².</p><p></p><p><a href="https://interviewquestions.tuteehub.com/tag/put-11868" style="font-weight:bold;" target="_blank" title="Click to know more about PUT">PUT</a> the value (0, -1) in the equation, we get.</p><p>⇒ sin⁻¹[(0)² + 1/3] + cos⁻¹[(-1)² - 2/3] = x².</p><p>⇒ sin⁻¹(0) + cos⁻¹(1) = x².</p><p>⇒ 0 + π = x². [L.H.S.]</p><p></p><p>Now again put the value of x² = π in the equation, we get.</p><p>⇒ sin⁻¹[π + 1/3] + cos⁻¹[π - 2/3] = π.</p><p>⇒ sin⁻¹[3.14 + 0.33] + cos⁻¹[3.14 - 0.6667] = π. [R.H.S.]</p><p></p><p>As we can see that,</p><p>⇒ L.H.S ≠ R.H.S.</p><p>So 1st possibility is rejected.</p><p></p><p>Put the value (0,0) in the equation, we get.</p><p>⇒ sin⁻¹[x² + 1/3] + cos⁻¹[x² - 2/3] = x².</p><p>⇒ sin⁻¹[(0)² + 1/3] + cos⁻¹[(0)² - 2/3] = x².</p><p>⇒ sin⁻¹(0) + cos⁻¹[0] = x².</p><p>⇒ 0 + π/2 = x². [L.H.S.]</p><p></p><p>Put the value of π/2 in the equation, we get.</p><p>⇒ sin⁻¹[π/2 + 1/3] + cos⁻¹[π/2 - 2/3] = π/2.</p><p>⇒ sin⁻¹[1.57 + 0.33] + cos⁻¹[1.57 - 0.667] = 1.57. [R.H.S.]</p><p></p><p>As we can see that,</p><p>⇒ L.H.S. ≠ R.H.S.</p><p>So 2nd possibility is <a href="https://interviewquestions.tuteehub.com/tag/also-373387" style="font-weight:bold;" target="_blank" title="Click to know more about ALSO">ALSO</a> rejected.</p><p></p><p>Put the value (0,-1) in the equation, we get.</p><p>⇒ sin⁻¹[x² + 1/3] + cos⁻¹[x² - 2/3] = x².</p><p>⇒ sin⁻¹[(0)² + 1/3] + cos⁻¹[(-1)² - 2/3] = x².</p><p>⇒ sin⁻¹(0) + cos⁻¹(1) = x².</p><p>⇒ 0 + 0 = x². [L.H.S.].</p><p></p><p>Put the value of x² = 0 in the equation, we get.</p><p>⇒ sin⁻¹[0 + 1/3] + cos⁻¹[0 - 2/3] = 0.</p><p>⇒ sin⁻¹(0) + cos⁻¹(-1) = 0. [R.H.S].</p><p></p><p>As we can see that,</p><p>⇒ L.H.S. ≠ R.H.S.</p><p>So 3rd possibility is also rejected.</p><p></p><p>Put the value (1, -1) in the equation, we get.</p><p>⇒ sin⁻¹[x² + 1/3] + cos⁻¹[x² - 2/3] = x².</p><p>⇒ sin⁻¹[(1)² + 1/3] + cos⁻¹[(-1)² - 2/3] = x².</p><p>⇒ sin⁻¹(1) + cos⁻¹(-1) = x².</p><p>⇒ π/2 + π = x².</p><p>⇒ 3π/2 = x². [L.H.S.].</p><p></p><p>Put the value of x² = 3π/2 in the equation, we get.</p><p>⇒ sin⁻¹[3π/2 + 1/3] + cos⁻¹[3π/2 - 2/3] = 3π/2.</p><p>⇒ sin⁻¹[9π + 2/6] + cos⁻¹[9π - 4/6] = 3π/2. [R.H.S.].</p><p></p><p>As we can see that,</p><p>⇒ L.H.S. ≠ R.H.S.</p><p>So, 4th possibility is also rejected.</p><p></p><p>Put the value (1,0) in the equation, we get.</p><p>⇒ sin⁻¹[x² + 1/3] + cos⁻¹[x² - 2/3] = x².</p><p>⇒ sin⁻¹[(1)² + 1/3] + cos⁻¹[(0)² - 2/3] = x².</p><p>⇒ sin⁻¹(1) + cos⁻¹(0) = x².</p><p>⇒ π/2 + π/2 = x².</p><p>⇒ x² = π. [L.H.S.].</p><p></p><p>Put the value of x² = π in the equation, we get.</p><p>⇒ sin⁻¹[π + 1/3] + cos⁻¹[π - 2/3] = π.</p><p>⇒ sin⁻¹[3.14 + 0.33] + cos⁻¹[3.14 - 0.667] = π. [R.H.S.].</p><p></p><p>As we can see that,</p><p>⇒ L.H.S. ≠ R.H.S.</p><p>So 5th possibility is also rejected.</p><p></p><p>Put the value (1,1) in the equation, we get.</p><p>⇒ sin⁻¹[x² + 1/3] + cos⁻¹[x² - 2/3] = x².</p><p>⇒ sin⁻¹[(1)² + 1/3] + cos⁻¹[(1)² - 2/3] = x².</p><p>⇒ sin⁻¹(1) + cos⁻¹(1) = x².</p><p>⇒ π/2 = x². [L.H.S.].</p><p></p><p>Put the value of x² = π/2 in the equation, we get.</p><p>⇒ sin⁻¹[π/2 + 1/3] + cos⁻¹[π/2 - 2/3] = π/2.</p><p>⇒ sin⁻¹[1.57 + 1.33] + cos⁻¹[1.57 - 0.667] = 1.57. [R.H.S.].</p><p></p><p>As we can see that,</p><p>⇒ L.H.S. ≠ R.H.S.</p><p>So 6th possibility is also rejected.</p><p></p><p>As we can observe all the possibility and see that no one possibility is matched.</p><p>Hence, Number of solutions = 0.</p><p><strong><u>Option [A] is correct answer</u></strong>.</p><p></p></body></html>


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