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NCERT Solutions for Class 11 Math Ch-4: Complex Numbers and Quadratic Equations
NCERT Solutions for Class 11 Maths Chapter 4 Complex Numbers and Quadratic Equations introduces the concept of imaginary and complex numbers, their algebraic operations, the modulus and argument, and solving quadratic equations with complex roots.
Overview of Topics & Subtopics
Important Questions and Answers
Q1: What is a complex number, and how is it represented in standard form?
Answer: A complex number is a number of the form z = a + ib, where a and b are real numbers, and i is the imaginary unit satisfying i² = −1. Here, a is called the real part and b is called the imaginary part of the complex number.
Q2: Define the modulus and conjugate of a complex number z = a + ib.
Answer: The modulus of z = a + ib is given by |z| = √(a² + b²), representing its distance from the origin in the Argand plane. The conjugate of z, denoted z̄, is a − ib, obtained by reversing the sign of the imaginary part.
Q3: How are complex numbers represented on the Argand plane?
Answer: On the Argand plane, a complex number z = a + ib is represented as a point (a, b), where the horizontal axis represents the real part and the vertical axis represents the imaginary part of the complex number.
Q4: When does a quadratic equation ax² + bx + c = 0 have complex (non-real) roots?
Answer: A quadratic equation ax² + bx + c = 0 has complex, non-real roots when its discriminant, D = b² − 4ac, is negative; in such cases, the roots are given by x = (−b ± i√(4ac − b²))/(2a), forming a conjugate pair of complex numbers.
Q5: State the value of i^4n, where n is any positive integer.
Answer: The value of i^4n is always 1 for any positive integer n, since the powers of i repeat in a cycle of 4: i¹ = i, i² = −1, i³ = −i, i⁴ = 1, and this cycle repeats thereafter.
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