Question
has
- no solution
- one solution
- two solutions
- three solutions
Hint:
In this question, we have to find how many solution this equation does. The equation is
. Check for , the value of x in all those quadrant where tan is positive and tan is negative.
The correct answer is: one solution
has
Here we have to find the how many of solution this equation does.
Firstly, we have equation,
![vertical line tan space x vertical line equals tan space x plus fraction numerator 1 over denominator cos begin display style space end style x end fraction left parenthesis 0 less or equal than x less or equal than 2 pi right parenthesis](data:image/png;base64,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)
We know x ∈ 2π
Lets spilt into two parts,
x ∈ [ 0 , π/2 ] U [π , 3 π/2 ] and [π/2 , π] U [3 π/2 , 2 π ]
At 1st case ,
x ∈ [ 0 , π/2 ] U [π , 3 π/2 ]
at this region tan is always positive,
then we can write,
|tanx| = tanx
So,
![vertical line tan space x vertical line equals tan space x plus fraction numerator 1 over denominator cos begin display style space end style x end fraction](data:image/png;base64,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)
![tan space x equals tan space x plus fraction numerator 1 over denominator cos begin display style space end style x end fraction](data:image/png;base64,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)
= 0
So it has no solution because, cosx in denominator and we know cosx ≠ 0 ..
At 2nd case,
x ∈ [π/2 , π] U [3 π/2 , 2 π ] ,
at this range tan is always negative,
so | tanx| = -tanx
we have,
![vertical line tan space x vertical line equals tan space x plus fraction numerator 1 over denominator cos begin display style space end style x end fraction](data:image/png;base64,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)
-![tan space x equals tan space x plus fraction numerator 1 over denominator cos begin display style space end style x end fraction](data:image/png;base64,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)
-2 tanx = ![fraction numerator 1 over denominator cos space x end fraction](data:image/png;base64,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)
= -1/2
sinx = -1/2
at [π/2 , π] sine is positive but it not possible ,
but at [3 π/2 , 2 π ] sine is negative,
so x = 330°
Hence it has only one solution,
Therefore , the correct answer is one solution.
In this question, we have to find the number of solution. In each quadrant tan value is different. Make two different case, one is where tan is positive ,[ 0 , π/2 ] U [π , 3 π/2 ] . And second case , tan is negative at [π/2 , π] U [3 π/2 , 2 π ] .
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