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NTA Abhyas 2020

Exam: NTA Abhyas
Year: 2020
Subject: Mathematics
Chapter: Trigonometric Ratios & Identities

NTA Abhyas 2020 · Mathematics · Trigonometric Ratios & Identities

If sin A and cos A are the roots of the equation 4x^ 2 - 3x + a =...

If

sin A

and

cos A

are the roots of the equation

4x^{2} - 3x + a = 0, sin A + cos A + tan A + cot A+ sec A + cosecA = 7

and

0 < A < \dfrac{\pi}{2},

then the value of

a

must be

Options

A $\dfrac{7}{25}$
B $\dfrac{25}{7}$
C $\dfrac{28}{25}$
D $\dfrac{25}{28}$

Correct answer

C. $\dfrac{28}{25}$

Step-by-step solution

sin A and cos A

are the roots of the equation

4x^{2} - 3x + a = 0

<br/>

⇒ sin A + cos A = \dfrac{3}{4}

and

sin Acos A = \dfrac{a}{4}

<br/>Also, \left(sin A + cos A\right) + \left(tan A + cot A\right) + \left(sec A + cosecA\right) = 7

⇒ \left(sin A + cos A\right) + \left(\dfrac{1}{sin Acos A}\right) + \left(\dfrac{sin A + cos A}{sin Acos A}\right) = 7

⇒ \dfrac{3}{4} + \dfrac{1}{\dfrac{a}{4}} + \dfrac{\dfrac{3}{4}}{\dfrac{a}{4}} = 7

⇒ \dfrac{3}{4} + \dfrac{4}{a} + \dfrac{3}{a} = 7

⇒ \dfrac{7}{a} = 7 - \dfrac{3}{4} = \dfrac{25}{4}

⇒ a = \dfrac{28}{25}

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