JEE Main Algebra — Numbers & Matrices: Complex Numbers and Quadratic Equations – Chapter Test

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Difficulty: 10 medium · 10 hard

Complex Numbers and Quadratic Equations MCQ questions with answers

The first 3 of 20 questions. Unlock the series for the rest, each with a solution.

1. Q1. The value of (cos 20° + i sin 20°)⁹ is:

   (cos 20° + i sin 20°)⁹ का मान है:

   • A. 1
   • B. −1
   • C. i
   • D. −i
   Solution

   By De Moivre's theorem the expression equals cos 180° + i sin 180° = −1.

   डी मॉयवर प्रमेय द्वारा यह व्यंजक cos 180° + i sin 180° = −1 के बराबर है।

2. Q2. The sum of all values of k for which the equation x² + kx + (k + 3) = 0 has equal roots is:

   k के उन सभी मानों का योग जिनके लिए समीकरण x² + kx + (k + 3) = 0 के मूल समान हों, है:

   • A. 6
   • B. −4
   • C. 4
   • D. −12
   Solution

   Equal roots require the discriminant k² − 4(k + 3) = 0, i.e. k² − 4k − 12 = 0, whose roots k = 6 and k = −2 sum to 4.

   समान मूलों के लिए विविक्तकर (discriminant) k² − 4(k + 3) = 0 होना आवश्यक है, अर्थात् k² − 4k − 12 = 0, जिसके मूल k = 6 और k = −2 का योग 4 है।

3. Q3. Assertion (A): The roots of x² + x + 1 = 0 are the non-real cube roots of unity ω and ω². Reason (R): 1 + ω + ω² = 0 and ω³ = 1. Choose the correct option.

   अभिकथन (A): x² + x + 1 = 0 के मूल इकाई के अवास्तविक घनमूल ω और ω² हैं। कारण (R): 1 + ω + ω² = 0 और ω³ = 1। सही विकल्प चुनिए।

   • A. A is true but R is false / A सत्य है परन्तु R असत्य है
   • B. Both A and R are true and R is the correct explanation of A / A और R दोनों सत्य हैं और R, A की सही व्याख्या है
   • C. Both A and R are true but R is not the correct explanation of A / A और R दोनों सत्य हैं परन्तु R, A की सही व्याख्या नहीं है
   • D. A is false but R is true / A असत्य है परन्तु R सत्य है
   Solution

   Both are true, and because ω satisfies ω² + ω + 1 = 0, the relations in R are exactly why ω and ω² are the roots of x² + x + 1 = 0, so R correctly explains A.

   दोनों सत्य हैं, और चूँकि ω समीकरण ω² + ω + 1 = 0 को संतुष्ट करता है, R में दिए गए संबंध ठीक वही कारण हैं जिनसे ω और ω² समीकरण x² + x + 1 = 0 के मूल बनते हैं, अतः R, A की सही व्याख्या है।

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