AssertionA- A- B- C are positive angles such that A-B-C-7- then maximum value of A B C-1-3-3ReasonR- A-M- G-M-

The correct option is C Both A and R are trueAssertion (A) : tan (A+B+C)=tan A +tan B+tan C tan A tan B tan C/1 tan A tan B tan A tan C tan B tan C ...

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Byju's Answer

Standard XII

Mathematics

Cos(A+B)Cos(A-B)

AssertionA: ...

Question

Assertion(A): $A, B, C$ are positive angles such that $A + B + C = 7 π$ , then maximum value of $cot A cot B cot C = \frac{1}{3 \sqrt{3}}$
Reason(R): $A . M . \geq G . M .$

A is true, R is false

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Both A and R are false

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A is false, R is true

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Both A and R are true

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Solution

The correct option is C Both A and R are true
Assertion (A) : $t a n (A + B + C) = \frac{t a n A + t a n B + t a n C - t a n A t a n B t a n C}{1 - t a n A t a n B - t a n A t a n C - t a n B t a n C}$
$t a n (7 π) = 0 \Rightarrow t a n (A + B + C) = 0 G i v e n (A + B + C = 7 π)$
$t a n A + t a n B + t a n C = t a n A t a n B t a n C$ ... (i)
Taking 3 positive numbers $t a n A, t a n B, t a n C$
$A . M . \geq G . M .$
$\Rightarrow \frac{t a n A + t a n B + t a n C}{3} \geq (t a n A t a n B t a n C)^{1 / 3}$
$\frac{t a n A t a n B t a n C}{3} \geq (t a n A t a n B t a n C) 1 / 3$ [From (i)]
$(t a n A t a n B t a n C) 2 / 3 \geq 3$
$\Rightarrow c o t A c o t B c o t C \leq \frac{1}{3 \sqrt{3}}$ --------True.
Reason (R)
A.M. of three positive numbers $\geq$ their G.M. ------------- True.
Therefore, both are True.
Hence, option 'D' is correct.

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Assertion(A): A, B, C are positive angles such that A+B+C=7π, then maximum value of cotAcotBcotC=13√3Reason(R): A.M.≥G.M.

Assertion(A): $A, B, C$ are positive angles such that $A + B + C = 7 π$ , then maximum value of $cot A cot B cot C = \frac{1}{3 \sqrt{3}}$
Reason(R): $A . M . \geq G . M .$