Math, asked by Anonymous, 4 months ago

solve it! fast ...........​

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Answered by hotcupid16
5

Given,

\longrightarrow\sf{f(x)=ax^2+bx+c\quad\quad\dots(1)}

for \sf{a,\ b,\ c\in\mathbb{Z},} such that,

\longrightarrow\sf{f\left(\cos\left(\dfrac{\pi}{7}\right)\right)=\sin\left(\dfrac{\pi}{7}\right)\sin\left(\dfrac{3\pi}{7}\right)+\sin\left(\dfrac{3\pi}{7}\right)\sin\left(\dfrac{5\pi}{7}\right)+\sin\left(\dfrac{5\pi}{7}\right)\sin\left(\dfrac{\pi}{7}\right)}

By product - to - sum rule, we have,

\sf{\sin A\sin B=\dfrac{1}{2}\big[\cos(A-B)-\cos(A+B)\big]}

Then,

\begin{aligned}\longrightarrow\sf{f\left(\cos\left(\dfrac{\pi}{7}\right)\right)}&=\sf{\dfrac{1}{2}\left[\cos\left(\dfrac{\pi}{7}-\dfrac{3\pi}{7}\right)-\cos\left(\dfrac{\pi}{7}+\dfrac{3\pi}{7}\right)\right]}\\\\&+\sf{\dfrac{1}{2}\left[\cos\left(\dfrac{3\pi}{7}-\dfrac{5\pi}{7}\right)-\cos\left(\dfrac{3\pi}{7}+\dfrac{5\pi}{7}\right)\right]}\\\\&+\sf{\dfrac{1}{2}\left[\cos\left(\dfrac{5\pi}{7}-\dfrac{\pi}{7}\right)-\cos\left(\dfrac{5\pi}{7}+\dfrac{\pi}{7}\right)\right]}\\\\&\end{aligned}

Since \sf{\cos(-x)=\cos x,}

\begin{aligned}\longrightarrow\sf{f\left(\cos\left(\dfrac{\pi}{7}\right)\right)}&=\sf{\dfrac{1}{2}\left[\cos\left(\dfrac{2\pi}{7}\right)-\cos\left(\dfrac{4\pi}{7}\right)\right]}\\\\&+\sf{\dfrac{1}{2}\left[\cos\left(\dfrac{2\pi}{7}\right)-\cos\left(\dfrac{8\pi}{7}\right)\right]}\\\\&+\sf{\dfrac{1}{2}\left[\cos\left(\dfrac{4\pi}{7}\right)-\cos\left(\dfrac{6\pi}{7}\right)\right]}\\\\&\end{aligned}

\begin{aligned}\longrightarrow\sf{f\left(\cos\left(\dfrac{\pi}{7}\right)\right)}&=\sf{\dfrac{1}{2}\left[\cos\left(\dfrac{2\pi}{7}\right)-\cos\left(\dfrac{4\pi}{7}\right)\right]}\\\\&+\sf{\dfrac{1}{2}\left[\cos\left(\dfrac{2\pi}{7}\right)-\cos\left(\pi+\dfrac{\pi}{7}\right)\right]}\\\\&+\sf{\dfrac{1}{2}\left[\cos\left(\dfrac{4\pi}{7}\right)-\cos\left(\pi-\dfrac{\pi}{7}\right)\right]}\\\\&\end{aligned}

Since  \sf{\cos(\pi-x)=\cos(\pi+x)=-\cos x,}

\begin{aligned}\longrightarrow\sf{f\left(\cos\left(\dfrac{\pi}{7}\right)\right)}&=\sf{\dfrac{1}{2}\left[\cos\left(\dfrac{2\pi}{7}\right)-\cos\left(\dfrac{4\pi}{7}\right)\right]}\\\\&+\sf{\dfrac{1}{2}\left[\cos\left(\dfrac{2\pi}{7}\right)+\cos\left(\dfrac{\pi}{7}\right)\right]}\\\\&+\sf{\dfrac{1}{2}\left[\cos\left(\dfrac{4\pi}{7}\right)+\cos\left(\dfrac{\pi}{7}\right)\right]}\\\\&\end{aligned}

\longrightarrow\sf{f\left(\cos\left(\dfrac{\pi}{7}\right)\right)=\cos\left(\dfrac{2\pi}{7}\right)+\cos\left(\dfrac{\pi}{7}\right)}

Since \sf{\cos(2x)=2\cos^2x-1,}

\longrightarrow\sf{f\left(\cos\left(\dfrac{\pi}{7}\right)\right)=2\cos^2\left(\dfrac{\pi}{7}\right)+\cos\left(\dfrac{\pi}{7}\right)-1\quad\quad\dots(2)}

Hence, on comparing (2) with (1), we get,

\longrightarrow\sf{f(x)=2x^2+x-1}

Therefore,

\longrightarrow\sf{f(2)=2(2)^2+2-1}

\longrightarrow\sf{\underline{\underline{f(2)=9}}}

Hence 9 is the answer.

Answered by Anonymous
1

Answer:

Given :

Area of rhombus = 120 cm²

Length of diagonal = 8 cm

To find :

Length of another diagonal

According to the question,

\sf{ :  \implies Area \: of \: rhombus =  \dfrac{1}{2}  \times d _{1} \times d_{2}   }

 \\

 \sf  : \implies{ {120 \: cm}^{2} =  \dfrac{1}{2}   \times 8 \: cm \times x}

 \\

 \sf :  \implies{ {120 \: cm}^{2} \times 2 = 8 \: cm \times x }

 \\

 \sf :  \implies{ {240 \: cm}^{2}  = 8 \: cm \times x}

 \\

 \sf  : \implies{ \dfrac{240}{8}  \: cm = x}

 \\

 { \underline{ \boxed{  \sf  \pink{ :  \implies{   \bm3 \bm0 \: c m =x}}}}}

{ \therefore{ \underline{\sf{So \:,the \:  length \:  of \:  other \:  diagonal  \: is \:    3 0 \: cm}}}}

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