Previous Year Questions on Complex Numbers

Complex Numbers - Previous Year Questions

Previous Year Questions on Complex Numbers

Previous year questions on Complex Numbers are essential for understanding advanced mathematical concepts. Originating from Classical Algebra, complex numbers form a bridge between algebra and geometry. These questions are categorized topicwise and yearwise, helping students excel in examinations. For additional resources, refer to Mathematics Notes and Mathematics Questions.

Vidyasagar University

2023-24 (CBCS)

Find the product of all the values of \( \left(1+i \right)^{\frac{2}{5}} \). [2]
Let \(z\) be a complex variable such that \(\left| \frac{z-i}{z+i}\right|=k \). Show that the point \(z\) lies on a circle in the complex plane if \(k\ne 1\) and \(z\) lies on a straight line if \(k= 1\). [5]
Prove that \(x^{n}=\prod_{k=0}^{\frac{n-2}{2} }\left[x^{2}-2x\cos{\frac{(2k+1)\pi}{n}}+1 \right] \), if \(n\) be an even positive integer. Deduce that \(\sin{\frac{\pi}{16}}\sin{\frac{3\pi}{16}}\sin{\frac{5\pi}{16}}\sin{\frac{7\pi}{16}}=\frac{1}{8\sqrt{2}} \) [4+3]

2021-22 (CBCS)

Find all values \(i^{\frac{1}{7}}\). [4]
Find the general values of the equation \( \left(\cos\theta+i~\sin\theta \right)\left(\cos 2\theta+i~\sin 2\theta \right)…\left(\cos n\theta+i~\sin n\theta \right)=-i\), where \(\theta\) is real. [2]

2020-21 (CBCS)

If \(z\) is a complex number satisfying the condition \(\left|z-\frac{3}{z} \right|=2\). Find the greatest and the least value of \(|z|\). [2]
If \(n\) be a positive integer and \((7+2i)^{n}=a+ib\), then prove that \(a^{2}+b^{2}=(53)^{n}\). Hence express \((53)^{2}\) as the sum of two squares. [2]
If \(i^{\alpha+i~\beta}=\alpha+i~\beta\) then prove that \(\alpha^{2}+\beta^{2}=e^{-(4n+1)\pi \beta}\). [2]
Find the roots of \(z^{n}=(z+1)^{n+1}\), where \(n\) is a positive integer, show that the points which represent them in the Argand diagram are collinear. [6]
Find the \(x^{8}+y^{8}=\prod \left[x^{2}-2xy\cos{\frac{r\pi}{8}}+y^{2} \right]\), \(r=1,3,5,7\). [6]

2019-20 (CBCS)

Find the sum of \(99^{th}\) powers of the roots of the equation \(x^{7}-1=0\). [2]
\(z\) is a variable complex number such that \(|z|=2\). Show that the point \(z+\frac{1}{z}\) lies on an ellipse. [2]
If \(x=\log \tan \left(\frac{\pi}{4}+\frac{\theta}{2} \right)\), where \(\theta\) is real, prove that \(\theta=i~\log \tan \left(\frac{\pi}{4}+i~\frac{x}{2} \right) \). [5]
If \(z_{1},z_{2}\) and \(a\) are complex numbers where \(a\ne 0\), show that \(a^{z_{1}}.a^{z_{2}\ne a^{z_{1}+z_{2}}}\) but \( \left( \text{ the p.v. of } a^{z_{1}}\right).\left(\text{ the p.v. of }a^{z_{2}}\right)\ne \text{ the p.v. of }a^{z_{1}+z_{2}} \). [4]

2018-19 (CBCS)

If the complex numbers \(z_{1}\), \(z_{2}\) and \(z_{3}\) represent the three points \(P\), \(Q\), \(R\) lie on a straight line. [2]
Prove that \(x^{n}+1=\prod_{k=0}^{\frac{n-2}{2}}\left[x^{2}-2x\cos{\frac{(2k+1)\pi}{n}}+1 \right] \). If \(n\) be an even integer then deduce the \( \sin{\frac{\pi}{16}}\sin{\frac{3\pi}{16}}\sin{\frac{5\pi}{16}}\sin{\frac{7\pi}{16}}=\frac{1}{8\sqrt{2}} \). [5]
Show that the solution of the equation \((1+x)^{n}-(1-x)^{n}=0\) are \(x=i~\tan{\frac{\pi r}{n}}\), where \(r = \begin{cases} 0,1,2,…,n-1 &\text{if n be odd} \\ 0,1,2,…,\frac{n}{2}-1,\frac{n}{2}+1,…,n-1 &\text{if n be even } \end{cases}\). [5]

2017-18 (CBCS)

If \(x+iy\) moves on the straight line \(3x+4y+5=0\), then find the minimum value of \(|x+iy|\). [2]
If \(\left(1+i~\tan{\alpha} \right)^{1+i~\tan{\beta}}\) can have real values then show that one of them is \(\left(\sec{\alpha}\right)^{\sec^{2}\beta} \). [5]
If \(x+\frac{1}{x}=2\cos{\alpha} \), \(y+\frac{1}{y}=2\cos{\beta} \), \(z+\frac{1}{z}=2\cos{\gamma} \) and \(x++y+z=0 \) then prove that \(\displaystyle\sum_{}^{}\sin{4\alpha}=2\displaystyle\sum_{}^{}\sin{\beta+\gamma} \) \(\displaystyle\sum_{}^{}\cos{4\alpha}=2\displaystyle\sum_{}^{}\cos{\beta+\gamma} \). [5]

FAQs

What are complex numbers?
Complex numbers are numbers that include a real part and an imaginary part, written as \(a + bi\), where \(i\) is the imaginary unit.
Why study previous year questions on Complex Numbers?
They help identify important patterns and frequently asked questions in exams.
How are complex numbers applied in real life?
They are used in signal processing, electrical engineering, and quantum physics.
Where can I find topicwise questions on Complex Numbers?
Visit Complex Numbers Questions for organized materials.
What is Euler’s formula in complex numbers?
Euler’s formula states that \(e^{i\theta} = \cos\theta + i\sin\theta\), linking exponential functions to trigonometry.
What is the polar form of a complex number?
The polar form represents a complex number as \(r(\cos\theta + i\sin\theta)\), where \(r\) is the modulus and \(\theta\) is the argument.
What is the importance of the Argand plane?
The Argand plane provides a graphical representation of complex numbers, aiding visualization of operations like addition and multiplication.
What are some reliable books for complex numbers?
Refer to Suggested Books for comprehensive resources.
Can complex numbers be linked to other topics?
Yes, they connect to Theory of Equations and Inequalities.
Are previous year questions solved?
Yes, solutions to previous year questions are available in books and educational resources.