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If a is Any Real Number, the Number of Roots of \[\Cot X - \Tan X = A\] in the First Quadrant is (Are). - Mathematics

MCQ
Sum

If a is any real number, the number of roots of \[\cot x - \tan x = a\] in the first quadrant is (are).

Options

  • 2

  • 0

  • 1

  • none of these

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Solution

1
Given:
\[\cot x - \tan x = a\]
\[ \Rightarrow \frac{1}{\tan x} - \tan x = a\]
\[ \Rightarrow 1 - \tan^2 x = a \tan x\]
\[ \Rightarrow \tan^2 x + a \tan x - 1 = 0\]
It is a quadratic equation.
If tan x = z, , then the equation becomes
\[z^2 + az - 1 = 0\]

\[\Rightarrow z = \frac{- a \pm \sqrt{a^2 + 4}}{2}\]
\[ \Rightarrow \tan x = \frac{- a \pm \sqrt{a^2 + 4}}{2}\]
\[ \Rightarrow x = \tan^{- 1} \left( \frac{- a \pm \sqrt{a^2 + 4}}{2} \right)\]
There are two roots of the given equation, but we need to find the number of roots in the first quadrant.
There is exactly one root of the equation, that is,
\[x = \tan^{- 1} \left( \frac{- a + \sqrt{a^2 + 4}}{2} \right)\].
  Is there an error in this question or solution?
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APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 11 Trigonometric equations
Q 4 | Page 27
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