Grade 11 Basic Mathematics by Sukunda Pustak Vawan Notes and Solutions | Nepal
Disclaimer:
1. If \rm \omega be a complex cube root of unity, show that
a) \rm (1 + \omega - \omega^2)^3 - ( 1 - \omega + \omega^2)^3 = 0
Solution:
Given,
\rm = (1 + \omega - \omega^2)^3 - (1 - \omega + \omega^2)^3
We know,
\rm 1 + \omega + \omega^2 = 0
\rm = (- \omega^2 - \omega^2)^3 - (1 + \omega^2 - \omega)^3
\rm = (-2 \omega ^2)^3 - (- \omega - \omega)^3
\rm = (-2 \omega^2)^3 - (- 2 \omega)^3
\rm = -8 \cdot \omega^6 + 8 \omega^3
\rm = - 8 \cdot 1 + 8 \cdot 1
\rm = 0
b) $\rm (2 + \omega + \omega^2)^3 + (1 + \omega - \omega^2)^8 - (1 - 3 \omega + \omega^2)^4 = 1$
Solution:
Given,
\rm = (2 + \omega + \omega^2)^3 + (1 + \omega - \omega^2) ^8 - (1 - 3 \omega + \omega^2)^4
We know, $\rm 1 + \omega + \omega^2 = 0$
$\rm = (2 + (-1))^3 + (- \omega^2 - \omega^2)^8 - (1 + \omega^2 - 3 \omega)^4$
$\rm = (2 -1 )^3 + (- 2\omega ^2)^8 - (-\omega - 3 \omega)^4$
$\rm = (1)^3 + (-2 \omega^2)^8 - (- 4 \omega)^4$
\rm = 1 + 2^8 (\omega)^8 - 4^4 (\omega)^4
\rm = 1 + 2^8 (1) - 4^4(1)
\rm = 1 + 4^4 - 4^4
\rm = 1 + 0
\rm = 1
2 a) If \alpha = \frac{1}{2} ( - 1 + \sqrt{-3}) and \beta = \frac{1}{2} ( - 1 - \sqrt{-3}), show that \alpha^4 + \alpha^2 \beta^2 + \beta^4 = 0.
Solution:
Given,
\alpha = \frac{1}{2} ( - 1 + \sqrt{-3})
\beta = \frac{1}{2} ( - 1 - \sqrt{-3})
Now,
\alpha^4 + \alpha^2 \beta^2 + \beta^4 = ( \alpha^2)^2 + (\beta^2)^2 + \alpha^2 \beta^2)
\rm = (\alpha^2 + \beta^2)^2 - 2 \alpha^2 \beta^2 + \alpha^2 \beta^2
\rm = (\alpha^2 + \beta^2)^2 - (\alpha^2 \beta^2)
\rm = (\alpha^2 + \beta^2)^2 - (\alpha \beta)^2
\rm = (\alpha + \beta + \alpha \beta)(\alpha + \beta - \alpha \beta) - (i)
And,
\rm \alpha \beta = \left ( \frac{1}{2} ( - 1 + \sqrt{-3}) \right ) \left ( \frac{1}{2} ( - 1 - \sqrt{-3}) \right )
\rm = \frac{-1}{4} ( - 1 + \sqrt{-3})(1 + \sqrt{-3})
\rm = \frac{-1}{4} ( - 1 - \sqrt{-3} + \sqrt{-3} + \sqrt{-3} \sqrt{-3})
\rm = \frac{-1}{4} ( - 1 + (\sqrt{3}i)(\sqrt{3}i))
\rm = \frac{-1}{4} (-1 + 3 i^2)
\rm = \frac{-1}{4} ( - 1 -3)
\rm = \frac{-4}{4}
\rm = -1
Again
\rm \alpha + \beta = \frac{1}{2} ( - 1 + \sqrt{-3}) + \frac{1}{2} ( - 1 - \sqrt{-3})
\rm = \frac{1}{2} \left ( - 1 + \sqrt{-3} - 1 - \sqrt{-3} \right )
\rm = \frac{1}{2} \left ( -2 \right )
\rm = -1
From equation (i)
\rm (\alpha + \beta + \alpha \beta)(\alpha + \beta - \alpha \beta)
\rm = (1 + (-1)) ( 1 - (-1))
\rm = 0 \times (2)
\rm = 0
i) x + y + z = 0
Solution:
Given,
\rm x + y + z = (a + b) + a \omega + b \omega^2 + a \omega^2 + b \omega
\rm = a + a \omega + a \omega^2 + b + b \omega + b\omega^2
\rm = a(1 + \omega +\omega^2) + b(1 + \omega + \omega^2)
We know, \rm (1 + \omega + \omega^2) = 0
\rm = a (0) + b(0)
\rm = 0 + 0
\rm = 0
ii) xyz = \rm a^3 + b^3
Solution:
Given,
\rm xyz = (a + b)(a \omega + b \omega^2) (a \omega^2 + b\omega)
\rm = (a +b) ( a \omega^3 + a b \omega^2 + ab \omega^4 + b \omega^3)
\rm = (a+b) ( a \cdot 1 + ab \omega^2 ( 1 + \omega^2) + b \cdot 1)
\rm = (a + b)(a + ab \omega^2 ( - \omega) + b)
\rm = (a + b) (a - ab \omega^3 + b)
\rm = (a + b)(a - ab \codt 1 + b)
\rm = (a + b)(a - ab + b)
\rm = a^3 + b^3
0 Comments
You can let us know your questions in the comments section as well.