If the value of 3 36^ +5 18^ 5 36^ -3 18^ is a 5 -b c , where a, b, c are natural numbers and gcd(a, c)=1 , then a+b+c is equal to :

Correct answer: B. 52 aligned & 3( 5 +1) 4 +5( 5 -1 4 ) 5( 5 +1 4 )-3( 5 -1 4 ) = 8 5 -2 2 5 +8 & = 4 5 -1 5 +4 × 5 -4 5 -4 aligned aligned & = 20-16 5 - 5 +4 -11 & = 17 5 -24 11 a=17, b=27, c=11 & a+b+c=52 aligned

JEE Main 20248 AprilEvening Shift

Exam: JEE Main
Year: 2024
Date: 8 April
Shift: Evening Shift
Subject: Mathematics
Chapter: Trigonometric Ratios & Identities

JEE Main 2024 · 8 April · Evening Shift · Mathematics · Trigonometric Ratios & Identities

If the value of 3 36^ +5 18^ 5 36^ -3 18^ is a 5 -b c , where a, b,...

If the value of

\frac{3 \cos 36^{\circ}+5 \sin 18^{\circ}}{5 \cos 36^{\circ}-3 \sin 18^{\circ}}

\frac{a \sqrt{5}-b}{c}

, where

a, b, c

are natural numbers and

\operatorname{gcd}(a, c)=1

, then

a+b+c

is equal to :

Options

Correct answer

B. 52

Step-by-step solution

\begin{aligned} & \frac{\frac{3(\sqrt{5}+1)}{4}+5\left(\frac{\sqrt{5}-1}{4}\right)}{5\left(\frac{\sqrt{5}+1}{4}\right)-3\left(\frac{\sqrt{5}-1}{4}\right)}=\frac{8 \sqrt{5}-2}{2 \sqrt{5}+8} \\ & =\frac{4 \sqrt{5}-1}{\sqrt{5}+4} \times \frac{\sqrt{5}-4}{\sqrt{5}-4}\end{aligned}

\begin{aligned} & =\frac{20-16 \sqrt{5}-\sqrt{5}+4}{-11} \\ & =\frac{17 \sqrt{5}-24}{11} \Rightarrow a=17, b=27, c=11 \\ & a+b+c=52\end{aligned}

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