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Complex Numbers and Quadratic Equations

Multiple Choice Questions

71.

The roots α and β of a quadratic equation, satisfy the relations α + β = α² + β² and αβ = α²β². What is the number of such quadratic equations ?

72.

If p², q² and r²(where p, q, r > 0) are in GP, then which of the following is/are correct ?

1) p, q and r are in GP.

2) ln(p), ln(q) and ln(r) are in AP

Select the correct answer using the code given below :

1 only
2 only
Both 1 and 2
Neither 1 nor 2

73.
What is the modulus of the complex number $\frac{\cos (θ) + isin (θ)}{\cos (θ) - isin (θ)} where, i = \sqrt{- 1} ?$

$\frac{1}{2}$

1

$\frac{3}{2}$

2

$We have, \frac{\cos (θ) + isin (θ)}{\cos (θ) - isin (θ)} \Rightarrow \frac{\cos (θ) + isin (θ)}{\cos (θ) - isin (θ)} \times \frac{\cos (θ) + isin (θ)}{\cos (θ) + isin (θ)} [By rationalization] \Rightarrow \frac{(\cos (θ) + isin (θ))}{\cos^{2} (θ) + \sin^{2} (θ)} \Rightarrow \cos^{2} (θ) - \sin^{2} (θ) + 2 isin (θ) \cos (θ) [\cos^{2} (θ) + \sin^{2} (θ) = 1] \Rightarrow \cos (2 θ) + isin (2 θ) [∵ \cos^{2} (θ) - \sin^{2} (θ) = \cos (2 θ) & 2 \sin (θ) \cos (θ) = \sin (2 θ)] So, |\cos (2 θ) + isin (2 θ)| \Rightarrow \sqrt{\cos^{2} (2 θ) + \sin^{2} (2 θ)} \Rightarrow 1$

Previous Year Papers

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Multiple Choice Questions

73. What is the modulus of the complex number cosθ + isinθcosθ - isinθ where, i = - 1 ? 12 1 32 2

73.
What is the modulus of the complex number $\frac{\cos (θ) + isin (θ)}{\cos (θ) - isin (θ)} where, i = \sqrt{- 1} ?$

$\frac{1}{2}$

1

$\frac{3}{2}$

2