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Class 11 ch 4

Complex numbers and quadratic equations: JEE Main-style questions

Original JEE Main-style questions on complex numbers and quadratic equations, with a worked solution and the usual trap for each. Tap an option, or type a numerical answer, to check it.

  • 6 questions
  • 3 free
  • 4 multiple choice, 2 numerical value
  • No calculator

Learn the chapters first: Class 11, Complex Numbers and Quadratic Equations.

Questions

Q1

·JEE Main style·Multiple choicePowers of i

If \(z = \dfrac{1 + i}{1 - i}\), then \(z^{2026}\) equals

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Answer: (a) \(-1\)

\(z = \dfrac{(1 + i)^2}{(1 - i)(1 + i)} = \dfrac{2i}{2} = i\).

\(2026 = 4 \times 506 + 2\), so \(i^{2026} = i^2 = -1\).

Q2

·JEE Main style·Multiple choiceSum and product of roots

If \(\alpha\) and \(\beta\) are the roots of \(x^2 - 4x + 1 = 0\), then \(\alpha^3 + \beta^3\) equals

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Answer: (c) \(52\)

\(\alpha + \beta = 4\), \(\alpha\beta = 1\).

\(\alpha^3 + \beta^3 = (\alpha + \beta)^3 - 3\alpha\beta(\alpha + \beta) = 64 - 12 = 52\).

Q3

·JEE Main style·Numerical valueNature of roots

Find the number of integers \(k\) for which \(x^2 - kx + 9 = 0\) has no real roots.

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Answer: 11

No real roots: discriminant \(k^2 - 36 < 0\), so \(-6 < k < 6\).

Integers \(-5, -4, \dots, 5\): 11 of them.

Trap: \(k = \pm 6\) give equal (real) roots, so they are excluded.

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Next steps

All questions are original, written by CBSE Math Revision in the style of each exam; none are taken from NTA, IIT, CBSE or NCERT papers. Every answer was checked twice: by a computer re-solve and by hand. No calculator needed. Spotted a mistake? Tell us.