Complex Numbers and Quadratic Equations — Class 11 Mathematics NCERT Solutions (Free)
Free step-by-step NCERT solutions for Class 11 Mathematics chapter "Complex Numbers and Quadratic Equations" — 8 important questions with detailed answers for CBSE board exam preparation.
TL;DR: Free step-by-step NCERT solutions for Class 11 Mathematics chapter "Complex Numbers and Quadratic Equations" — 8 important questions with detailed ans…
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NCERT Solutions for Class 11 Mathematics — Complex Numbers and Quadratic Equations. Step-by-step answers to all 8 textbook questions from this chapter, written for CBSE board preparation and free to use.
Complex Numbers and Quadratic Equations — All 8 Questions Solved
Q1. Express (3 + 2i) + (1 - 4i) and (3 + 2i) - (1 - 4i) in the form a + bi.
Step 1: Add the complex numbers.
(3 + 2i) + (1 - 4i)
= 3 + 1 + 2i - 4i
= 4 + (2 - 4)i
= 4 - 2i
Step 2: Subtract the complex numbers.
(3 + 2i) - (1 - 4i)
= 3 - 1 + 2i - (-4i)
= 2 + 2i + 4i
= 2 + 6i
Step 3: Verify using the general rule.
For addition: (a + bi) + (c + di) = (a + c) + (b + d)i
For subtraction: (a + bi) - (c + di) = (a - c) + (b - d)i
Final Answer: (3 + 2i) + (1 - 4i) = 4 - 2i; (3 + 2i) - (1 - 4i) = 2 + 6i
Q2. Multiply (2 + 3i)(1 - i) and express the result in the form a + bi.
Step 1: Use the distributive property (FOIL).
(2 + 3i)(1 - i)
= 2(1) + 2(-i) + 3i(1) + 3i(-i)
= 2 - 2i + 3i - 3i²
Step 2: Use i² = -1.
= 2 - 2i + 3i - 3(-1)
= 2 - 2i + 3i + 3
= (2 + 3) + (-2 + 3)i
= 5 + i
Step 3: Verify by expanding carefully.
(2 + 3i)(1 - i) = 2·1 + 2·(-i) + 3i·1 + 3i·(-i)
= 2 - 2i + 3i + 3
= 5 + i ✓
Final Answer: (2 + 3i)(1 - i) = 5 + i
Q3. Find the complex conjugate and modulus of z = 3 - 4i. Also find z · z̄.
Step 1: Find the complex conjugate z̄.
If z = 3 - 4i, then z̄ = 3 + 4i
Step 2: Find the modulus |z|.
|z| = √(3² + (-4)²) = √(9 + 16) = √25 = 5
Step 3: Calculate z · z̄.
z · z̄ = (3 - 4i)(3 + 4i)
= 3² - (4i)²
= 9 - 16i²
= 9 - 16(-1)
= 9 + 16
= 25
Step 4: Verify the relationship.
Notice that z · z̄ = |z|²
25 = 5² ✓
Final Answer: z̄ = 3 + 4i, |z| = 5, z · z̄ = 25
Q4. Divide (5 + 2i) by (1 + i) and express the result in the form a + bi.
Step 1: Write the division as a fraction.
(5 + 2i) / (1 + i)
Step 2: Multiply numerator and denominator by the conjugate of the denominator.
The conjugate of (1 + i) is (1 - i).
= (5 + 2i)(1 - i) / [(1 + i)(1 - i)]
Step 3: Calculate the denominator.
(1 + i)(1 - i) = 1² - i² = 1 - (-1) = 1 + 1 = 2
Step 4: Calculate the numerator.
(5 + 2i)(1 - i)
= 5(1) + 5(-i) + 2i(1) + 2i(-i)
= 5 - 5i + 2i - 2i²
= 5 - 5i + 2i + 2
= 7 - 3i
Step 5: Divide.
(7 - 3i) / 2 = 7/2 - 3i/2 = 7/2 - (3/2)i
Final Answer: (5 + 2i) / (1 + i) = 7/2 - (3/2)i
Q5. Solve the quadratic equation x² - 4x + 5 = 0 using the quadratic formula.
Step 1: Identify coefficients in ax² + bx + c = 0.
a = 1, b = -4, c = 5
Step 2: Calculate the discriminant.
Δ = b² - 4ac = (-4)² - 4(1)(5) = 16 - 20 = -4
Step 3: Since Δ < 0, the solutions are complex.
Step 4: Apply the quadratic formula.
x = (-b ± √Δ) / (2a)
x = (-(-4) ± √(-4)) / (2·1)
x = (4 ± √(-4)) / 2
x = (4 ± 2i) / 2
x = 2 ± i
Step 5: Write the two solutions.
x₁ = 2 + i
x₂ = 2 - i
Step 6: Verify by substitution.
For x = 2 + i:
(2 + i)² - 4(2 + i) + 5
= 4 + 4i + i² - 8 - 4i + 5
= 4 + 4i - 1 - 8 - 4i + 5
= 0 ✓
Final Answer: x = 2 + i or x = 2 - i
Q6. Find i⁴⁵ and simplify your answer.
Step 1: Identify the pattern of powers of i.
i¹ = i
i² = -1
i³ = i² · i = -1 · i = -i
i⁴ = i² · i² = (-1)(-1) = 1
i⁵ = i⁴ · i = 1 · i = i
The pattern repeats every 4 powers: i, -1, -i, 1, i, -1, -i, 1, ...
Step 2: Find the remainder when 45 is divided by 4.
45 = 4 × 11 + 1
45 ≡ 1 (mod 4)
Step 3: Use the pattern.
Since the remainder is 1:
i⁴⁵ = i¹ = i
Final Answer: i⁴⁵ = i
Q7. If z = 1 + i, find z² and z³ in the form a + bi.
Step 1: Calculate z².
z² = (1 + i)²
= 1² + 2(1)(i) + i²
= 1 + 2i + (-1)
= 2i
Step 2: Calculate z³ using z³ = z² · z.
z³ = z² · z = (2i)(1 + i)
= 2i + 2i²
= 2i + 2(-1)
= 2i - 2
= -2 + 2i
Step 3: Verify z³ by direct calculation.
z³ = (1 + i)³
= (1 + i)²(1 + i)
= (2i)(1 + i)
= 2i + 2i²
= 2i - 2 = -2 + 2i ✓
Final Answer: z² = 2i, z³ = -2 + 2i
Q8. Show that if α and β are roots of ax² + bx + c = 0, then α + β = -b/a and αβ = c/a (Vieta's formulas).
Step 1: Use the quadratic formula to express the roots.
α = (-b + √Δ) / (2a)
β = (-b - √Δ) / (2a)
where Δ = b² - 4ac
Step 2: Calculate α + β.
α + β = [(-b + √Δ) / (2a)] + [(-b - √Δ) / (2a)]
= [(-b + √Δ) + (-b - √Δ)] / (2a)
= -2b / (2a)
= -b/a ✓
Step 3: Calculate αβ.
αβ = [(-b + √Δ) / (2a)] · [(-b - √Δ) / (2a)]
= [(-b)² - (√Δ)²] / (4a²)
= [b² - Δ] / (4a²)
= [b² - (b² - 4ac)] / (4a²)
= 4ac / (4a²)
= c/a ✓
Step 4: Alternative verification using factorization.
If α and β are roots, then:
ax² + bx + c = a(x - α)(x - β)
= a[x² - (α + β)x + αβ]
= ax² - a(α + β)x + aαβ
Comparing coefficients:
Coefficient of x: -a(α + β) = b ⟹ α + β = -b/a
Constant term: aαβ = c ⟹ αβ = c/a
Final Answer: Vieta's formulas α + β = -b/a and αβ = c/a are proved.
Complex Numbers and Quadratic Equations — Practice MCQs with Answers
Attempt these 20 multiple-choice questions after working through the solutions above. Each carries the correct option and the reasoning behind it, so a wrong answer tells you which idea to revisit.
MCQ 1. What is the conjugate of the complex number $3 + 4i$?
- A. $3 - 4i$
- B. $-3 + 4i$
- C. $-3 - 4i$
- D. $4 + 3i$
Answer: A. $3 - 4i$ — The conjugate of a complex number $a+bi$ is $a-bi$. You simply change the sign of the imaginary part.
MCQ 2. Simplify $i^{101}$.
- A. $i$
- B. $-i$
- C. $1$
- D. $-1$
Answer: A. $i$ — Powers of $i$ cycle with a period of 4 ($i, -1, -i, 1$). The exponent $101$ leaves a remainder of $1$ when divided by $4$.
MCQ 3. If $z = \frac{1+2i}{1-3i}$, then what is $z$ in the form $a+bi$?
- A. $-\frac{1}{2} + \frac{1}{2}i$
- B. $\frac{1}{2} + \frac{1}{2}i$
- C. $-\frac{1}{2} - \frac{1}{2}i$
- D. $\frac{1}{2} - \frac{1}{2}i$
Answer: A. $-\frac{1}{2} + \frac{1}{2}i$ — To divide complex numbers, multiply the numerator and denominator by the conjugate of the denominator to rationalize the expression.
MCQ 4. For what value of $k$ will the quadratic equation $x^2 + 6x + k = 0$ have real and equal roots?
- A. $9$
- B. $3$
- C. $6$
- D. $0$
Answer: A. $9$ — For a quadratic equation to have real and equal roots, its discriminant ($D = b^2 - 4ac$) must be equal to zero.
MCQ 5. What is the modulus of the complex number $z = (1+i)^2$?
- A. $2$
- B. $\sqrt{2}$
- C. $1$
- D. $4$
Answer: A. $2$ — First, simplify the complex number by expanding $(1+i)^2$, then calculate its modulus.
MCQ 6. Find the principal argument of $z = -1 - i$.
- A. $\frac{\pi}{4}$
- B. $-\frac{3\pi}{4}$
- C. $\frac{3\pi}{4}$
- D. $-\frac{\pi}{4}$
Answer: B. $-\frac{3\pi}{4}$ — The complex number $-1-i$ lies in the third quadrant of the Argand plane. The principal argument for a third quadrant complex number is $-\pi + \alpha$, where $\alpha$ is the reference angle.
MCQ 7. If $\alpha$ and $\beta$ are the roots of the equation $2x^2 - 5x + 7 = 0$, what is the value of $\frac{1}{\alpha} + \frac{1}{\beta}$?
- A. $\frac{5}{7}$
- B. $-\frac{5}{7}$
- C. $\frac{7}{5}$
- D. $-\frac{7}{5}$
Answer: A. $\frac{5}{7}$ — First, simplify the expression $\frac{1}{\alpha} + \frac{1}{\beta}$ to $\frac{\alpha+\beta}{\alpha\beta}$. Then, use Vieta's formulas to find the sum and product of the roots.
MCQ 8. If $|z+1| = |z-1|$, what is the locus of $z$?
- A. $x$-axis
- B. $y$-axis
- C. circle
- D. straight line not passing through origin
Answer: B. $y$-axis — Geometrically, the condition $|z+1| = |z-1|$ means that the complex number $z$ is equidistant from the points $-1$ (representing $-1+0i$) and $1$ (representing $1+0i$) on the Argand plane. The locus of such points is the perpendicular bisector of the line segment joining these two points, which is the imaginary axis ($y$-axis).
MCQ 9. If $x^2 + px + q = 0$ and $x^2 + qx + p = 0$ (where $p \neq q$) have a common root, then what is the relationship between $p$ and $q$?
- A. $p+q+1 = 0$
- B. $p-q=1$
- C. $p+q=1$
- D. $p^2-q^2=0$
Answer: A. $p+q+1 = 0$ — If a root is common to both equations, it must satisfy both. Subtracting the equations will help in finding the common root.
MCQ 10. If one root of the quadratic equation $ax^2 + bx + c = 0$ is the reciprocal of the other, then what is the relation between $a, b, c$?
- A. $a=c$
- B. $a=-c$
- C. $b=c$
- D. $a=b$
Answer: A. $a=c$ — If the roots are reciprocals of each other, their product is 1. The product of roots for a quadratic equation $ax^2+bx+c=0$ is $c/a$.
MCQ 11. Find the square root of $3+4i$.
- A. $\pm(2+i)$
- B. $\pm(1+2i)$
- C. $\pm(2-i)$
- D. $\pm(1-2i)$
Answer: A. $\pm(2+i)$ — Assume the square root is $x+iy$. Squaring this and equating real and imaginary parts with $3+4i$ will give a system of equations for $x$ and $y$. Also, equating moduli helps.
MCQ 12. If $\omega$ is a non-real cube root of unity, then $(1-\omega)(1-\omega^2)$ is equal to:
- A. $1$
- B. $0$
- C. $3$
- D. $4$
Answer: C. $3$ — Expand the product and use the fundamental properties of cube roots of unity: $1+\omega+\omega^2=0$ and $\omega^3=1$.
MCQ 13. For what value of $m$ will the quadratic equation $x^2 - (m-3)x + m = 0$ have real roots?
- A. $(-\infty, 1] \cup [9, \infty)$
- B. $[1, 9]$
- C. $(1, 9)$
- D. $(-\infty, 9]$
Answer: A. $(-\infty, 1] \cup [9, \infty)$ — For a quadratic equation to have real roots, its discriminant ($D = b^2 - 4ac$) must be greater than or equal to zero.
MCQ 14. If $z$ is a complex number such that $\text{Re}(z) > 0$ and $|z|=2$, then $z$ lies on a:
- A. Circle in the right half-plane
- B. Circle in the left half-plane
- C. Line segment
- D. Point
Answer: A. Circle in the right half-plane — The condition $|z|=2$ means $z$ is on a circle of radius 2 centered at the origin. The condition $\text{Re}(z)>0$ restricts $z$ to the right half of the complex plane.
MCQ 15. If the roots of $x^2 - 8x + k = 0$ are $\alpha$ and $\beta$ such that $\alpha^2 + \beta^2 = 34$, then what is the value of $k$?
- A. $15$
- B. $10$
- C. $18$
- D. $24$
Answer: A. $15$ — Use Vieta's formulas to express the sum and product of roots in terms of coefficients, and the algebraic identity $\alpha^2+\beta^2 = (\alpha+\beta)^2 - 2\alpha\beta$.
MCQ 16. Find the multiplicative inverse of $z = (3+4i)$.
- A. $\frac{3}{25} - \frac{4}{25}i$
- B. $\frac{3}{25} + \frac{4}{25}i$
- C. $\frac{1}{3} - \frac{1}{4}i$
- D. $-\frac{3}{25} + \frac{4}{25}i$
Answer: A. $\frac{3}{25} - \frac{4}{25}i$ — The multiplicative inverse of a complex number $z$ is $1/z$. To express this in the form $a+bi$, multiply the numerator and denominator by the conjugate of $z$.
MCQ 17. A quadratic equation whose roots are $2+i$ and $2-i$ is:
- A. $x^2 - 4x + 5 = 0$
- B. $x^2 + 4x - 5 = 0$
- C. $x^2 - 5x + 4 = 0$
- D. $x^2 + 5x - 4 = 0$
Answer: A. $x^2 - 4x + 5 = 0$ — For a quadratic equation with roots $\alpha$ and $\beta$, the equation can be formed as $x^2 - (\alpha+\beta)x + \alpha\beta = 0$.
MCQ 18. If $(x+y) + i(x-y) = 4+6i$, then what are the values of $x$ and $y$?
- A. $x=5, y=-1$
- B. $x=-1, y=5$
- C. $x=5, y=1$
- D. $x=-5, y=1$
Answer: A. $x=5, y=-1$ — For two complex numbers to be equal, their real parts must be equal, and their imaginary parts must be equal. This forms a system of linear equations.
MCQ 19. If $z$ is a complex number such that $|z|=1$, then the maximum value of $|z+2|$ is:
- A. $1$
- B. $2$
- C. $3$
- D. $4$
Answer: C. $3$ — This problem uses the triangle inequality for complex numbers, $|z_1+z_2| \le |z_1|+|z_2|$. The maximum value is achieved when the complex numbers are aligned.
MCQ 20. The roots of the equation $ax^2+bx+c=0$ are rational if the discriminant $D=b^2-4ac$ is:
- A. A perfect square
- B. A negative number
- C. Zero
- D. A positive number
Answer: A. A perfect square — For the roots of a quadratic equation to be rational, the expression under the square root in the quadratic formula (the discriminant) must be a perfect square.
How to Use These Solutions
Attempt each question yourself first and only then compare with the worked answer. Marks in the CBSE board exam are awarded for the METHOD as much as the final result, so reproduce the steps rather than memorising the last line. Where a solution states a law, a formula or a definition, learn that wording — examiners look for it.
Related practice for this chapter: chapter MCQs, previous-year questions, revision notes and sample papers.