NCERT Class 11 Maths: Complex Numbers Important Questions

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NCERT Class 11 Maths – Chapter: Complex Numbers / एनसीईआरटी कक्षा 11 गणित – अध्याय: सम्मिश्र संख्याएँ

20 Bilingual Questions (Practice & HOTS) / 20 द्विभाषी प्रश्न (अभ्यास और HOTS)

Argand diagram of a complex number. Image: Wikimedia Commons, CC0.

Q1. Find the multiplicative inverse of 3 + 4i
3 + 4i का गुणात्मक प्रतिलोम ज्ञात कीजिए।

Q2. If \(z_1 = 2 - i\) and \(z_2 = -2 + i\), find \(\text{Re}(z_1z_2 / \bar{z}_1)\)
यदि \(z_1 = 2 - i\) और \(z_2 = -2 + i\) है, तो \(\text{Re}(z_1z_2 / \bar{z}_1)\) ज्ञात कीजिए।

Q3. Express \((1 + i)^4\) in the form \(a + ib\)
\((1 + i)^4\) को \(a + ib\) के रूप में व्यक्त कीजिए।

Q4. Find the value of \(i^9 + i^{19}\)
\(i^9 + i^{19}\) का मान ज्ञात कीजिए।

Q5. Find the conjugate and modulus of \(z = -5 + 12i\)
\(z = -5 + 12i\) का संयुग्मी और मापांक ज्ञात कीजिए।

Q6. If \(\frac{(1+i)^2}{2-i} = x+iy\), find \(x+y\)
यदि \(\frac{(1+i)^2}{2-i} = x+iy\), तो \(x+y\) का मान ज्ञात कीजिए।

Q7. Find the square roots of \(-3 - 4i\)
\(-3 - 4i\) के वर्गमूल ज्ञात कीजिए।

Q8. Convert \(1 / (1 - i)\) into \(a + ib\) form
\(1 / (1 - i)\) को \(a + ib\) रूप में बदलिए।

Q9. Find \(x\) and \(y\) if \((x + iy)(2 - 3i) = 4 + i\)
यदि \((x + iy)(2 - 3i) = 4 + i\), तो \(x\) और \(y\) ज्ञात कीजिए।

Q10. Show that \((1 + i)^n + (1 - i)^n = 2^{n/2 + 1} \cos(n\pi/4)\)
सिद्ध कीजिए कि \((1 + i)^n + (1 - i)^n = 2^{n/2 + 1} \cos(n\pi/4)\)

Q11. If \(z = \frac{1+7i}{(2-i)^2}\), find \(|z|\)
यदि \(z = \frac{1+7i}{(2-i)^2}\), तो \(|z|\) का मान ज्ञात कीजिए।

Q12. Find the smallest positive integer \(n\) for which \(\left(\frac{1+i}{1-i}\right)^n = 1\)
\(n\) का वह न्यूनतम धनात्मक मान ज्ञात कीजिए जिसके लिए \(\left(\frac{1+i}{1-i}\right)^n = 1\) हो।

Q13. If \((x + iy)^3 = u + iv\), show that \(u/x + v/y = 4(x^2 - y^2)\)
यदि \((x + iy)^3 = u + iv\), तो सिद्ध कीजिए \(u/x + v/y = 4(x^2 - y^2)\)

Q14. If \(|z_1 + z_2| = |z_1| + |z_2|\), show that \(\text{arg}(z_1) = \text{arg}(z_2)\)
यदि \(|z_1 + z_2| = |z_1| + |z_2|\), तो सिद्ध कीजिए \(\text{arg}(z_1) = \text{arg}(z_2)\)

Q15. Find real \(a\) such that \(3i^3 - 2ai^2 + (1-a)i + 5\) is real
\(a\) का वह वास्तविक मान ज्ञात कीजिए जिसके लिए \(3i^3 - 2ai^2 + (1-a)i + 5\) वास्तविक हो।

Q16. If \(|\beta| = 1\), find \(\left|\frac{\beta - \alpha}{1 - \bar{\alpha}\beta}\right|\)
यदि \(|\beta| = 1\), तो \(\left|\frac{\beta - \alpha}{1 - \bar{\alpha}\beta}\right|\) का मान ज्ञात कीजिए।

Q17. Prove that \(\left|\frac{z - 1}{z + 1}\right| = 1\) lies on the imaginary axis
सिद्ध कीजिए कि \(\left|\frac{z - 1}{z + 1}\right| = 1\) काल्पनिक अक्ष पर स्थित है।

Q18. If \(z = x + iy\), prove that \(|z^2| = |z|^2\)
यदि \(z = x + iy\), तो सिद्ध कीजिए \(|z^2| = |z|^2\)

Q19. Find the square root of \(i\)
\(i\) का वर्गमूल ज्ञात कीजिए।

Q20. If \(1, \omega, \omega^2\) are cube roots of unity, show \((1 - \omega + \omega^2)^5 + (1 + \omega - \omega^2)^5 = 32\)
यदि \(1, \omega, \omega^2\) इकाई के घनमूल हैं, तो सिद्ध कीजिए \((1 - \omega + \omega^2)^5 + (1 + \omega - \omega^2)^5 = 32\)

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💡 Important Exam & Study Tips

• Master NCERT Fundamentals: Ensure thorough familiarity with standard textbook definitions, formulas, and in-text examples before solving advanced problems.
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• Practice Under Timed Conditions: Regular practice with previous years' question papers (PYQs) builds exam speed and accuracy.

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