QJEEM21Ma8b32: JEE Main 2021 Mathematics - Class 11 JEEM21Ma8b32 Solution: Let and . If is a quadratic equation whose roots are 1/5 and…

Question

Verified Solution • 2025–26 Session moderate

Let

\alpha = \mathop {\max }\limits_{x \in R} \{ {8^{2\sin 3x}}{.4^{4\cos 3x}}\}

and

\beta = \mathop {\min }\limits_{x \in R} \{ {8^{2\sin 3x}}{.4^{4\cos 3x}}\}

. If

8{x^2} + bx + c = 0

is a quadratic equation whose roots are

\alpha

^1/5 and

\beta

^1/5, then the value of c

-

b is equal to :

Accepted Answer

\alpha = \mathop {\max }\limits_{x \in R} \{ {8^{2\sin 3x}}{.4^{4\cos 3x}}\}

= \max \{ {2^{6\sin 3x}}{.2^{8\cos 3x}}\}

= max\{ {2^{6\sin 3x + 8\cos 3x}}\}

and

\beta = \min \{ {8^{2\sin 3x}}{.4^{4\cos 3x}}\} = \min \{ {2^{6\sin 3x + 8\cos 3x}}\}

Now range of

6sin3x + 8cos3x

= \left[ { - \sqrt {{6^2} + {8^2}} , + \sqrt {{6^2} + {8^2}} } \right] = [ - 10,10]

\alpha

= 2¹⁰ &

\beta

= 2^{$-$ 10}

So,

\alpha

^1/5 = 2² = 4

\Rightarrow

\beta

^1/5 = 2^{$-$ 2} = 1/4

quadratic 8x² + bx + c = 0

- {b \over 8} = {{17} \over 4}

\Rightarrow

b = -34

{c \over 8} = 1

\Rightarrow

c = 8

\therefore

c – b = 8 + 34 = 42

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