[গণিত] হলে, (x,y) নির্ণয় কর।
উত্তর: [গণিত] সমতা অনুযায়ী, [গণিত] (i) নং হতে, [গণিত] (iii) হতে, [গণিত] = 2 [গণিত]
HSC Higher Math 1st Paper Chapter 1: ম্যাট্রিকস ও নির্ণায়ক CQ — Prosthuti প্রশ্নব্যাংক
HSC · Higher Math 1st Paper · Chapter 1: ম্যাট্রিকস ও নির্ণায়ক · সৃজনশীল
হলে, (x,y) নির্ণয় কর।
উদ্দীপক
A=\left[\[\begin{array}{ccc}1 & 3 & 2 \\ 2 & 0 & 3 \\ 1 & -1 & 1\end{array}\]\right], D=\left|\[\begin{array}{ccc}x & y & z \x^2 & y^2 & z^2 \x^3-1 & y^3-1 & z^3-1\end{array}\]\right| এবং,
সোনার বাংলা কলেজ · 2024
উত্তর
\left[\[\begin{array}{l} 2 x+3 y \x-2 y \end{array}\]\right]=\left[\[\begin{array}{c} 13 \\ -4 \end{array}\]\right]
সমতা অনুযায়ী,
(i) নং হতে,
(iii) হতে,
= 2
উত্তর
দেওয়া আছে,
A=\left[\[\begin{array}{ccc}1 & 3 & 2 \\ 2 & 0 & 3 \\ 1 & -1 & 1\end{array}\]\right]
\therefore A^2=\left[\[\begin{array}{ccc}1 & 3 & 2 \\ 2 & 0 & 3 \\ 1 & -1 & 1\end{array}\]\right]\left[\[\begin{array}{ccc}1 & 3 & 2 \\ 2 & 0 & 3 \\ 1 & -1 & 1\end{array}\]\right]
=\left[\[\begin{array}{lll}1+6+2 & 3+0-2 & 2+9+2 \\ 2+0+3 & 6+0-3 & 4+0+3 \\ 1-2+1 & 3+0-1 & 2-3+1\end{array}\]\right]
=\left[\[\begin{array}{ccc}9 & 1 & 13 \\ 5 & 3 & 7 \\ 0 & 2 & 0\end{array}\]\right]
=\left[\[\begin{array}{ccc}9 & 1 & 13 \\ 5 & 3 & 7 \\ 0 & 2 & 0\end{array}\]\right] \cdot\left[\[\begin{array}{ccc}1 & 3 & 2 \\ 2 & 0 & 3 \\ 1 & -1 & 1\end{array}\]\right]
=\left[\[\begin{array}{lll}9+2+13 & 27+0-13 & 18+3+13 \\ 5+6+7 & 15+0-7 & 10+9+7 \\ 0+4+0 & 0+0+0 & 0+6+0\end{array}\]\right]
=\left[\[\begin{array}{ccc}24 & 14 & 34 \\ 18 & 8 & 26 \\ 4 & 0 & 6\end{array}\]\right]
দেওয়া আছে,
\begin{gathered}=\left[\begin{array}{ccc}24 & 14 & 34 \\ 18 & 8 & 26 \\ 4 & 0 & 6\end{gathered}\right]-2\left[\[\begin{array}{ccc}9 & 1 & 13 \\ 5 & 3 & 7 \\ 0 & 2 & 0\end{array}\]\right]+\left[\[\begin{array}{ccc}1 & 3 & 2 \\ 2 & 0 & 3 \\ 1 & -1 & 1\end{array}\]\right]- 2\left[\[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\]\right]\end{gathered}
\end{array}
=\left[\[\begin{array}{lll}24-18+1-2 & 14-2+3-0 & 34-26+2+0 \\ 18-10+2+0 & 8-6+0-2 & 26-14+3+0 \\ 4-0+1-0 & 0-4-1+0 & 6+0+1-2\end{array}\]\right]
=\left[\[\begin{array}{ccc}5 & 15 & 10 \\ 10 & 0 & 15 \\ 5 & -5 & 5\end{array}\]\right]\end{array}
উত্তর
দেওয়া আছে,
D=\left|\[\begin{array}{ccc}x & y & z \x^2 & y^2 & z^2 \x^2-1 & y^3-1 & z^2-1\end{array}\]\right|
=\left|\[\begin{array}{lll}x & y & z \x^2 & y^2 & z^2 \x^2 & y^3 & z^3\end{array}\]\right|-\left|\[\begin{array}{ccc}x & y & z \x^2 & y^2 & z^2 \\ 1 & 1 & 1\end{array}\]\right|
=x y z\left|\[\begin{array}{lll}1 & 1 & 1 \x^2 & y & z \x^2 & y^2 & z^2\end{array}\]\right|-\left|\[\begin{array}{lll}1 & 1 & 1 \x & y & z \x^2 & y^2 & z^2\end{array}\]\right|
=(x y z-1)\left|\[\begin{array}{lll}1 & 1 & 1 \x & y & z \x^r & y^r & z^r\end{array}\]\right|
=(x y z-1)\left|\[\begin{array}{ccc}0 & 0 & 1 \x-y & y-z & z \x^2-y^2 & y^2=z^2 & z^2\end{array}\]\right|
(proved)