Question

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Answer 1

Answer:

(i) Value of x - 1/x is ± 8/3.

(ii) Value of x³ - 1/x³ is ± 728/27.

Step-by-step explanation:

Given :

$\implies \dfrac{ {x}^{2} + 1 }{x} = 3 \dfrac{1}{3}$

$\implies \dfrac{ {x}^{2}}{x} + \dfrac{1}{x} = 3 \dfrac{1}{3}$

$\implies x + \dfrac{1}{x} = 3 \dfrac{1}{3}$

RHS can be written as

$\implies x + \dfrac{1}{x} = 3 + \dfrac{1}{3}$

Comparing on both sides

$\implies x = 3 \ \ OR \ \ x = \dfrac{1}{3}$

(i) Find the value of x - 1/x

When x = 3

$\implies x - \dfrac{1}{x} = 3 - \dfrac{1}{3}$

$\implies x - \dfrac{1}{x} = \dfrac{9 - 1}{3}$

$\implies x - \dfrac{1}{x} = \dfrac{8}{3}$

When x = 1/3

$\implies x - \dfrac{1}{x} = \dfrac{1}{3} - 3$

$\implies x - \dfrac{1}{x} = \dfrac{1 - 9}{3}$

$\implies x - \dfrac{1}{x} = - \dfrac{8}{3}$

Therefore the value of x - 1/x is ± 8/3.

(ii) Find the value of x³ - 1/x³

When x = 3

$\implies x^3 - \dfrac{1}{x^3 } = 3^3 - \dfrac{1}{3^3 }$

$\implies x^3 - \dfrac{1}{x^3 } = 27 - \dfrac{1}{27}$

$\implies x^3 - \dfrac{1}{x^3 } = \dfrac{729 - 1}{27}$

$\implies x^3 - \dfrac{1}{x^3 } = \dfrac{728}{27}$

When x = 1/3

$\implies x^3 - \dfrac{1}{x^3 } = \dfrac{1}{3^3 } - 3^3$

$\implies x^3 - \dfrac{1}{x^3 } = \dfrac{1}{27 } - 27$

$\implies x^3 - \dfrac{1}{x^3 } = \dfrac{1 - 729}{27}$

$\implies x^3 - \dfrac{1}{x^3 } = - \dfrac{728}{27}$

Therefore the value of x³ - 1/x³ is ± 728/27.

Answer 2

Answer:

Given: The edges of a triangluar Board are 6 cm, 8 cm and 10 cm. & The cost of painting it at the rate of Rs 5 per cm².

Need to find: The Cost of painting?

⠀⠀━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

❍ Let's say, that the sides of triangle be x, y and z are 6 cm, 8 cm & 10 cm respectively.

⌑ We'll use Heron's formula to find out the area of the given triangular Board. First we'll calculate semi – perimeter of the triangle —

» Semi - perimeter is Gievn by sum of all sides of triangle —

\begin{gathered}:\implies\sf S = \Bigg\{\dfrac{x + y + z}{2}\Bigg\}\\\\\\\end{gathered}

:⟹S={

2

x+y+z

}

\begin{gathered}:\implies\sf S = \Bigg\{\dfrac{6 + 8 + 10}{2}\Bigg\}\\\\\\\end{gathered}

:⟹S={

2

6+8+10

}

\begin{gathered}:\implies\sf S = \cancel\dfrac{24}{2}\\\\\\\end{gathered}

:⟹S=

2

24

\begin{gathered}:\implies{\boxed{\pmb{\frak{S = 12\;cm}}}}\\\\\end{gathered}

:⟹

S=12cm

S=12cm

\begin{gathered}\therefore{\underline{\textsf{Hence, Semi-perimeter of the triangle is \textbf{12 cm.}}}}\\\end{gathered}

∴

Hence, Semi-perimeter of the triangle is 12 cm.

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✇ Now, Calculating area of the triangle —

⠀

\begin{gathered}\star\;\underline{\pmb{\boxed{\sf{Area_{\:(triangle)} = \sqrt{s\Big(s - a\Big)\Big(s - b\Big)\Big(s - c\Big)}}}}}\\\\\end{gathered}

⋆

Area

(triangle)

=

s(s−a)(s−b)(s−c)

Area

(triangle)

=

s(s−a)(s−b)(s−c)

\begin{gathered}:\implies\sf Area_{\:(triangle)} = \sqrt{12\Big(12 - 6\Big)\Big(12 - 8\Big)\Big(12 - 10\Big)} \\\\\\\end{gathered}

:⟹Area

(triangle)

=

12(12−6)(12−8)(12−10)

\begin{gathered}:\implies\sf Area_{\:(triangle)} = \sqrt{12\times 6 \times 4 \times 2}\\\\\\\end{gathered}

:⟹Area

(triangle)

=

12×6×4×2

\begin{gathered}:\implies\sf Area_{\;(triangle)} = \sqrt{576}\\\\\\\end{gathered}

:⟹Area

(triangle)

=

576

\begin{gathered}:\implies\underline{\boxed{\pmb{\frak{Area_{\:(triangle)} = 24\;cm^2}}}}\;\bigstar\\\\\end{gathered}

:⟹

Area

(triangle)

=24cm

2

Area

(triangle)

=24cm

2

★

\begin{gathered}\therefore{\underline{\sf{Hence,\; Area\; of \;the\; triangle \;is \;{\pmb{\sf{24\;cm^2}}}.}}}\\\end{gathered}

∴

Hence,Areaofthetriangleis

24cm

2

24cm

2

.

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» Now, we've to find out the cost of painting the board at rupees 5 per cm². So, Let's Solve :

⠀

\begin{gathered}\longrightarrow{ \pmb{\mathbb{ COST = RATE \times AREA }}}\\\\\end{gathered}

⟶

COST=RATE×AREA

COST=RATE×AREA

\begin{gathered}\longrightarrow\sf Cost = 5 \times 24 \\\\\end{gathered}

⟶Cost=5×24

\begin{gathered}\longrightarrow\underline{\boxed{\pmb{\frak{Cost = 120}}}}\;\bigstar\\\\\end{gathered}

⟶

Cost=120

Cost=120

★

\therefore{\underline{\sf{Hence,\; the \;Cost\; of\; painting \;is~ {\pmb{\sf{Option~ c)~120~ rs}}}.}}}∴

Hence,theCostofpaintingis

Option c) 120 rs

Option c) 120 rs.