\displaystyle ( i) \ x=\sqrt{\frac{5x}{2} +\frac{1}{6}}

\displaystyle ( ii) \ x-3=2\sqrt{2}

ক) \displaystyle a+b+c=9 এবং \displaystyle ab+bc+ca=31 হলে, \displaystyle a^{2} +b^{2} +c^{2} এর মান নির্নয় কর ।

খ) \displaystyle ( i) নং সম্পর্ক হতে, \displaystyle \frac{216x^{6} -1}{27x^{3}} এর মান নির্নয় কর ।

গ) দেখাও যে, \displaystyle \left(\sqrt{x}\right)^{3} +\left(\frac{1}{\sqrt{x}}\right)^{3} =10\sqrt{2}

ক)

দেওয়া আছে,

\displaystyle a+b+c=9

এবং, \displaystyle ab+bc+ca=31

আমরা জানি,

\displaystyle a^{2} +b^{2} +c^{2} =( a+b+c)^{2} -2( ab+bc+ca)

\displaystyle =( 9)^{2} -2.31

\displaystyle =81\ -62

\displaystyle =19

খ)

দেওয়া আছে,

\displaystyle  \begin{array}{{>{\displaystyle}l}} \Longrightarrow x=\sqrt{\frac{5x}{2} +\frac{1}{6}}\ \end{array}
\displaystyle \Longrightarrow x^{2} =\left(\sqrt{\frac{5x}{2} +\frac{1}{6}}\right)^{2}

\displaystyle \Longrightarrow x^{2} =\frac{5x}{2} +\frac{1}{6}

\displaystyle \Longrightarrow x^{2} =\frac{15x+1}{6}

\displaystyle \Longrightarrow 6x^{2} =15x+1

\displaystyle \Longrightarrow 6x^{2} -1=15x

\displaystyle \Longrightarrow \frac{6x^{2}}{3x} -\frac{1}{3x} =\frac{15x}{3x}

\displaystyle \Longrightarrow 2x-\frac{1}{3x} =5 ——- (১ নং)

এখন,

\displaystyle \frac{216x^{6} -1}{27x^{3}}

\displaystyle =\frac{216x^{6}}{27x^{3}} -\frac{1}{27x^{3}}

\displaystyle =8x^{3} -\frac{1}{27x^{3}}

\displaystyle =( 2x)^{3} -\left(\frac{1}{3x}\right)^{3}

\displaystyle =\left( 2x-\frac{1}{3x}\right)^{3} +3.2x.\frac{1}{3x}\left( 2x-\frac{1}{3x}\right)

\displaystyle =( 5)^{3} +2.5 [ মান বসাই ]

\displaystyle =5.5.5+10

\displaystyle =125+10

\displaystyle =135

নির্নয় মান: \displaystyle 135

গ)

দেওয়া আছে,

\displaystyle x-3=2\sqrt{2}

\displaystyle \Longrightarrow x=3+2\sqrt{2}

\displaystyle \Longrightarrow x=2+2\sqrt{2} +1

\displaystyle \Longrightarrow x=\left(\sqrt{2}\right)^{2} +2.\sqrt{2} .1+( 1)^{2}

\displaystyle \Longrightarrow x=\left(\sqrt{2} +1\right)^{2}

\displaystyle \Longrightarrow \sqrt{x} =\sqrt{\left(\sqrt{2} +1\right)^{2}} [ বর্গমূল করে ]

\displaystyle \Longrightarrow \sqrt{x} =\sqrt{2} +1
আবার,
\displaystyle \frac{1}{\sqrt{x}} =\frac{1}{\sqrt{2} +1}

\displaystyle \Longrightarrow \frac{1}{\sqrt{x}} =\frac{1.\left(\sqrt{2} -1\right)}{\left(\sqrt{2} +1\right) .\left(\sqrt{2} -1\right)} [ উভয়পক্ষ \displaystyle \sqrt{2} -1 দ্বারা ভাগ করে পাই ]

\displaystyle \Longrightarrow \frac{1}{\sqrt{x}} =\frac{\left(\sqrt{2} -1\right)}{\left(\sqrt{2}\right)^{2} -( 1)^{2}}

\displaystyle \Longrightarrow \frac{1}{\sqrt{x}} =\frac{\left(\sqrt{2} -1\right)}{2-1}

\displaystyle \Longrightarrow \frac{1}{\sqrt{x}} =\frac{\left(\sqrt{2} -1\right)}{1}

\displaystyle \Longrightarrow \frac{1}{\sqrt{x}} =\left(\sqrt{2} -1\right)

সুতরাং, \displaystyle \sqrt{x} +\frac{1}{\sqrt{x}} =\sqrt{2} +1+\sqrt{2} -1

\displaystyle =2\sqrt{2}

এখন, \displaystyle \left(\sqrt{x}\right)^{3} +\left(\frac{1}{\sqrt{x}}\right)^{3}

\displaystyle =\left(\sqrt{x} +\frac{1}{\sqrt{x}}\right)^{3} -3.\sqrt{x} .\frac{1}{\sqrt{x}}\left(\sqrt{x} +\frac{1}{\sqrt{x}}\right)[ \displaystyle a^{3} +b^{3} =( a+b)^{3} -3ab( a+b) সূত্র প্রয়োগ করে পাই ]

\displaystyle =\left( 2\sqrt{2}\right)^{3} -3.2\sqrt{2} [ মান বসাই ]

\displaystyle =( 2)^{3} .\left(\sqrt{2}\right)^{3} -6\sqrt{2}

\displaystyle =8.\sqrt{2} .\sqrt{2} .\sqrt{2} -6\sqrt{2}

\displaystyle =8.\left(\sqrt{2}\right)^{2} .\sqrt{2} -6\sqrt{2}

\displaystyle =8.2\sqrt{2} -6\sqrt{2}

\displaystyle =16\sqrt{2} -6\sqrt{2}

\displaystyle =10\sqrt{2}

সুতরাং \displaystyle \left(\sqrt{x}\right)^{3} +\left(\frac{1}{\sqrt{x}}\right)^{3} =10\sqrt{2} ( প্রমানিত )

2) যদি \displaystyle x-y=6,xy=7,z=2\sqrt{2} +\sqrt{7} হলে,

ক) \displaystyle 4a-5b এর ঘন নির্নয় কর ।

খ) দেখাও যে, \displaystyle x^{3} -y^{3} -4( x+y)^{2} =x^{2} +y^{2} +( x-y)^{2}

গ) \displaystyle z^{5} -\frac{1}{z^{5}} এর মান নির্নয় কর ।

ক)

\displaystyle 4a-5b এর ঘন\displaystyle =( 4a-5b)^{3}

\displaystyle =( 4a)^{3} -3.( 4a)^{2} .5b+3.4a.( 5b)^{2} -( 5b)^{3}

\displaystyle =64a^{3} -3.16a^{2} .5b+3.4a.25b^{2} -125b^{3}

\displaystyle =64a^{3} -240a^{2} b+300ab^{2} -125b^{3}

খ)

দেওয়া আছে,

\displaystyle x-y=6,xy=7

বামপক্ষ\displaystyle =x^{3} -y^{3} -4( x+y)^{2}

\displaystyle =( x-y)^{3} +3xy( x-y) -4\left{( x-y)^{2} +4xy\right}

\displaystyle =( 6)^{3} +3.7.6-4\left( 6^{2} +4.7\right)

\displaystyle =6.6.6+126-4( 36+28)

\displaystyle =216+126-4.64

\displaystyle =86
ডানপক্ষ\displaystyle =x^{2} +y^{2} +( x-y)^{2}

\displaystyle =( x-y)^{2} +2xy+6^{2}

\displaystyle =( 6)^{2} +2.7+36

\displaystyle =36+14+36

\displaystyle =86
সুতরাং \displaystyle x^{3} -y^{3} -4( x+y)^{2} =x^{2} +y^{2} +( x-y)^{2} ( দেখানো হল )

Post Author: showrob

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