117989
The value of \(1+\sum_{\mathrm{k}=0}^{14}\left\{\cos \frac{(2 \mathrm{k}+1) \pi}{15}+\mathrm{i} \sin \frac{(2 \mathrm{k}+1) \pi}{15}\right\}\) is
117990
If \(\left(\frac{3}{2}+\frac{i \sqrt{3}}{2}\right)^{50}=3^{25}(x+i y)\), where \(x\) and \(y\) are real, then the ordered pair \((x, y)\) is given by
117991
The least positive integer n. for which \(\frac{(1+i)^n}{(1-i)^{n-2}}\) is positive is
1 4
2 3
3 2
4 1
Explanation:
D \(\frac{(1+\mathrm{i})^{\mathrm{n}}}{(1-\mathrm{i})^{\mathrm{n}-2}}\) is positive for what value of \(\mathrm{n}\) \(\left(\frac{1+i}{1-i}\right)^n \times(1-i)^2=\left[\frac{(1+i)^2}{\left(1^2-i^2\right)}\right]^n \times(1-i)^2\) \(=\frac{(1-1+2 i)^n}{2^n} \times(-2 i)\) \(=-\frac{(2 i)^{n+1}}{2^n}=\frac{2^{n+1}}{2^n}\left[-1(i)^{n+1}\right]\) \(=-2(i)^{n+1}=2\) For least positive integer, \(\mathrm{n}+1=2\) \(\mathrm{n}=1\)
Karnataka CET-2010
Complex Numbers and Quadratic Equation
117992
If \(x+i y=(-1+i \sqrt{3})^{2010}\), then \(x=\)
117989
The value of \(1+\sum_{\mathrm{k}=0}^{14}\left\{\cos \frac{(2 \mathrm{k}+1) \pi}{15}+\mathrm{i} \sin \frac{(2 \mathrm{k}+1) \pi}{15}\right\}\) is
117990
If \(\left(\frac{3}{2}+\frac{i \sqrt{3}}{2}\right)^{50}=3^{25}(x+i y)\), where \(x\) and \(y\) are real, then the ordered pair \((x, y)\) is given by
117991
The least positive integer n. for which \(\frac{(1+i)^n}{(1-i)^{n-2}}\) is positive is
1 4
2 3
3 2
4 1
Explanation:
D \(\frac{(1+\mathrm{i})^{\mathrm{n}}}{(1-\mathrm{i})^{\mathrm{n}-2}}\) is positive for what value of \(\mathrm{n}\) \(\left(\frac{1+i}{1-i}\right)^n \times(1-i)^2=\left[\frac{(1+i)^2}{\left(1^2-i^2\right)}\right]^n \times(1-i)^2\) \(=\frac{(1-1+2 i)^n}{2^n} \times(-2 i)\) \(=-\frac{(2 i)^{n+1}}{2^n}=\frac{2^{n+1}}{2^n}\left[-1(i)^{n+1}\right]\) \(=-2(i)^{n+1}=2\) For least positive integer, \(\mathrm{n}+1=2\) \(\mathrm{n}=1\)
Karnataka CET-2010
Complex Numbers and Quadratic Equation
117992
If \(x+i y=(-1+i \sqrt{3})^{2010}\), then \(x=\)
117989
The value of \(1+\sum_{\mathrm{k}=0}^{14}\left\{\cos \frac{(2 \mathrm{k}+1) \pi}{15}+\mathrm{i} \sin \frac{(2 \mathrm{k}+1) \pi}{15}\right\}\) is
117990
If \(\left(\frac{3}{2}+\frac{i \sqrt{3}}{2}\right)^{50}=3^{25}(x+i y)\), where \(x\) and \(y\) are real, then the ordered pair \((x, y)\) is given by
117991
The least positive integer n. for which \(\frac{(1+i)^n}{(1-i)^{n-2}}\) is positive is
1 4
2 3
3 2
4 1
Explanation:
D \(\frac{(1+\mathrm{i})^{\mathrm{n}}}{(1-\mathrm{i})^{\mathrm{n}-2}}\) is positive for what value of \(\mathrm{n}\) \(\left(\frac{1+i}{1-i}\right)^n \times(1-i)^2=\left[\frac{(1+i)^2}{\left(1^2-i^2\right)}\right]^n \times(1-i)^2\) \(=\frac{(1-1+2 i)^n}{2^n} \times(-2 i)\) \(=-\frac{(2 i)^{n+1}}{2^n}=\frac{2^{n+1}}{2^n}\left[-1(i)^{n+1}\right]\) \(=-2(i)^{n+1}=2\) For least positive integer, \(\mathrm{n}+1=2\) \(\mathrm{n}=1\)
Karnataka CET-2010
Complex Numbers and Quadratic Equation
117992
If \(x+i y=(-1+i \sqrt{3})^{2010}\), then \(x=\)
117989
The value of \(1+\sum_{\mathrm{k}=0}^{14}\left\{\cos \frac{(2 \mathrm{k}+1) \pi}{15}+\mathrm{i} \sin \frac{(2 \mathrm{k}+1) \pi}{15}\right\}\) is
117990
If \(\left(\frac{3}{2}+\frac{i \sqrt{3}}{2}\right)^{50}=3^{25}(x+i y)\), where \(x\) and \(y\) are real, then the ordered pair \((x, y)\) is given by
117991
The least positive integer n. for which \(\frac{(1+i)^n}{(1-i)^{n-2}}\) is positive is
1 4
2 3
3 2
4 1
Explanation:
D \(\frac{(1+\mathrm{i})^{\mathrm{n}}}{(1-\mathrm{i})^{\mathrm{n}-2}}\) is positive for what value of \(\mathrm{n}\) \(\left(\frac{1+i}{1-i}\right)^n \times(1-i)^2=\left[\frac{(1+i)^2}{\left(1^2-i^2\right)}\right]^n \times(1-i)^2\) \(=\frac{(1-1+2 i)^n}{2^n} \times(-2 i)\) \(=-\frac{(2 i)^{n+1}}{2^n}=\frac{2^{n+1}}{2^n}\left[-1(i)^{n+1}\right]\) \(=-2(i)^{n+1}=2\) For least positive integer, \(\mathrm{n}+1=2\) \(\mathrm{n}=1\)
Karnataka CET-2010
Complex Numbers and Quadratic Equation
117992
If \(x+i y=(-1+i \sqrt{3})^{2010}\), then \(x=\)
117989
The value of \(1+\sum_{\mathrm{k}=0}^{14}\left\{\cos \frac{(2 \mathrm{k}+1) \pi}{15}+\mathrm{i} \sin \frac{(2 \mathrm{k}+1) \pi}{15}\right\}\) is
117990
If \(\left(\frac{3}{2}+\frac{i \sqrt{3}}{2}\right)^{50}=3^{25}(x+i y)\), where \(x\) and \(y\) are real, then the ordered pair \((x, y)\) is given by
117991
The least positive integer n. for which \(\frac{(1+i)^n}{(1-i)^{n-2}}\) is positive is
1 4
2 3
3 2
4 1
Explanation:
D \(\frac{(1+\mathrm{i})^{\mathrm{n}}}{(1-\mathrm{i})^{\mathrm{n}-2}}\) is positive for what value of \(\mathrm{n}\) \(\left(\frac{1+i}{1-i}\right)^n \times(1-i)^2=\left[\frac{(1+i)^2}{\left(1^2-i^2\right)}\right]^n \times(1-i)^2\) \(=\frac{(1-1+2 i)^n}{2^n} \times(-2 i)\) \(=-\frac{(2 i)^{n+1}}{2^n}=\frac{2^{n+1}}{2^n}\left[-1(i)^{n+1}\right]\) \(=-2(i)^{n+1}=2\) For least positive integer, \(\mathrm{n}+1=2\) \(\mathrm{n}=1\)
Karnataka CET-2010
Complex Numbers and Quadratic Equation
117992
If \(x+i y=(-1+i \sqrt{3})^{2010}\), then \(x=\)