\((\frac{(1+i}{1-i})^m=1\)
\(⇒(\frac{1+i}{1-i}×\frac{1+i}{1+i})^m=1\)
\(⇒(\frac{(1+i)^2}{1^2-1^2})^m=1\)
\(⇒(\frac{1^2+i^2+2i}{2})^m=1\)
\(⇒(\frac{1-1+2i}{2})^m=1\)
\(⇒(\frac{2i}{2})^m=1\)
\(⇒i^m=1\)
\(\text{∴\,m=4k, where k is some integer.}\)
\(\text{Therefore, the least positive integer is 1. }\)
\(\text{Thus, the least positive integral value of m is 4 (= 4 × 1). }\)
Let α,β be the roots of the equation, ax2+bx+c=0.a,b,c are real and sn=αn+βn and \(\begin{vmatrix}3 &1+s_1 &1+s_2\\1+s_1&1+s_2 &1+s_3\\1+s_2&1+s_3 &1+s_4\end{vmatrix}=\frac{k(a+b+c)^2}{a^4}\) then k=
Figures 9.20(a) and (b) refer to the steady flow of a (non-viscous) liquid. Which of the two figures is incorrect ? Why ?
A Complex Number is written in the form
a + ib
where,
The Complex Number consists of a symbol “i” which satisfies the condition i^2 = −1. Complex Numbers are mentioned as the extension of one-dimensional number lines. In a complex plane, a Complex Number indicated as a + bi is usually represented in the form of the point (a, b). We have to pay attention that a Complex Number with absolutely no real part, such as – i, -5i, etc, is called purely imaginary. Also, a Complex Number with perfectly no imaginary part is known as a real number.