Arpita Srivastava Content Writer
Content Writer | Updated On - Jul 25, 2025
Complex numbers and Quadratic Equations is an important chapter included in NCERT Class 11 Mathematics. It is used to combine quadratic measurements with the roots of complex numbers.
- Complex numbers and quadratic equations are used to solve equations that have imaginary and real numbers.
- The concept is commonly used to solve algebraic problems and equations.
- Complex numbers are derived from the nth root of quadratic equations.
- A quadratic equation is a two-dimensional polynomial equation that can be written as:
\(ax^2 + bx + c = 0.\)
- where x denotes the variable, a and b are the numerical coefficients, and c is the constant term
- In this, the coefficient with the highest degree can never be zero.
- The quadratic equation method can be used to solve speed and time problems.
- Complex numbers are represented in terms of real numbers and imaginary numbers, which can be expressed as follows:
\(x + iy.\)
- In this, x and y are real numbers, and “i” is called the imaginary number.
- The solution of any quadratic equation can be obtained using the formula:
\(x=\frac{(-b±√(b²-4ac))}{2a}\)
Some important formulas used in complex numbers are as follows:
- Addition of two Complex Numbers: (a + bi) +(c + di)
\((a + c) + (b + d) i.\)
- Subtraction of two Complex Numbers: (a + bi) − (c + di)
\((a − c) + (b − d) i.\)
- Multiplication of two Complex Numbers: (a + bi) × (c + di)
\( (ac − bd) + (ad + bc) i.\)
- Division of two Complex Numbers: \(\frac{(a + ib) }{ (c + id)}\)
\(\frac{(ac + bd) }{ (c_2 + d_2)} + \frac{i (bc – ad) }{ (c_2 + d_2)}\)
Complex Numbers and Quadratic Equation MCQs
Ques: Add two complex numbers: z1 = 9+4i and z2 = 8+3i?
- 17+7i
- 17
- 7i
- 17
Click here for the answer
Ans: (a) 17+7i
Explanation: Here a=9, b=4, c=8, d=3
⇒ z1+ z2 = (a+c)+(b+d)i
⇒ z1 + z2 = (9+8)+(4+3)i
∴ z1 + z2 = 17+7i
Ques: Subtract two complex numbers: z1 = 7+9i and z2 = 8+2i?
- 17+i
- 17
- -1+7i
- -2+7i
Click here for the answer
Ans: (c) -1+7i
Explanation: Here a=7, b=9, c=8, d=2
⇒ z1 - z2 = (a − c) + (b − d) i.
⇒ z1 - z2 = (7- 8)+(9-2)i
∴ z1 - z2 = -1+7i
Ques: z1 = 19+4i and z2 = 8+13i?
- 27+17i
- 17
- 7i
- 17
Click here for the answer
Ans: (a) 27+17i
Explanation: Here a=19, b=4, c=8, d=13
⇒ z1+ z2 = (a+c)+(b+d)i
⇒ z1 + z2 = (19+8)+(4+13)i
∴ z1 + z2 = 27+17i
Ques: Suppose a + ib = c + id, then determine whether
- a2 + c2 = 0
- a2 + b2 = c2 + d2
- b2 + d2 = 0
- b2 = c2 + d2
Click here for the answer
Ans: (b) a2 + b2 = c2 + d2
Explanation: It is given that: a + ib = c + id
⇒ |a + ib| = |c + id|
⇒ √(a2 + b2) = √(c2 + d2)
⇒ Squaring on both sides, we will get: a2 + b2 = c2 + d2
Ques: Determine the required value of complex equation: 1 + i2 + i4 + i6 + … + i2n
- Positive
- Negative
- Cannot be evaluated
- 0
Click here for the answer
Ans: (d) 0
Explanation: Since it is given that: 1 + i2 + i4 + i6 + … + i2n = 1 – 1 + 1 – 1 + … (–1)n
∴ This cannot be evaluated unless the value of n is known.
Ques: Determine the value of required arg (x) where x < 0
- 2
- π/2
- Π
- 1
Click here for the answer
Ans: (c) π
Explanation: Suppose z = x + 0i and x < 0
⇒ Since the point (-x, 0) lies on the negative side of the real axis,
⇒ |z| = |x + oi| = √[(-1)2 + 0)] = 1
∴ Principal argument (z) = π
Ques: Express the following complex expression in the form of a + bi: (2 – i) – (-2 + i6)
- 1
- 4 – 7i
- 4 + 7i
- 3
Click here for the answer
Ans: (b) 4 – 7i
Explanation: (2 – i) – (-2 + i6) = 2 – i + 2 – i6
∴ the result is 4 – 7i
Ques: Find the conjugate of required complex numbers: √-5 + 4i2.
- √5i + 4
- 2
- √5i - 4
- √5i
Click here for the answer
Ans: (c)√5i - 4
Explanation: Simplify the expression -
∴ the result is √-5 + 4i2 = √5i - 4
Ques: Determine the value of √(-25).
- 9i
- I
- 7i
- 5i
Click here for the answer
Ans: (d) 5i
Explanation: √(-25) = √(25) × √(-1)
∴ the result is 5i [ i = √(-1) ]
Ques. Solve : i-37.
- 13/i
- 1/i
- 1
- 1/2i
Click here for the answer
Ans: (b) 1/i
Explanation: i-37 = 1/i37
⇒ 1/ (i4)9 . i
⇒ 1 / 1 × (i) [ i4 = 1]
∴ the result is 1/i
Ques: Add two complex numbers: z1 = 20+14i and z2 = 70+3i?
- 17+7i
- 17
- 90+17i
- 17 + 80i
Click here for the answer
Ans: (c) 90+17i
Explanation: Here a=20, b=14, c=70, d=3
⇒ z1+ z2 = (a+c)+(b+d)i
⇒ z1 + z2 = (20+70)+(14+3)i
∴ the result is z1 + z2 = 90+17i
Ques: Subtract two complex numbers: z1 = 10+19i and z2 = 8+i?
- 2+18i
- 17
- -1+7i
- 2+18i
Click here for the answer
Ans: (d) 2+18i
Explanation: Here a=10, b=19, c=8, d=1
⇒ z1 - z2 = (a − c) + (b − d) i.
⇒ z1 - z2 = (10- 8)+(19-1)i
∴ the result is z1 - z2 = 2+18i
Ques: Express the following complex expression in the form of a + bi: (3 – i) – (-3 + i6)
- 5 – 7i
- 6 – 7i
- 4 + 7i
- 3
Click here for the answer
Ans: (b) 6 – 7i
Explanation: (3 – i) – (-3 + i6) = 3 – i + 3 – i6
∴ the result is 6 – 7i
Ques: Find the conjugate of required complex numbers: √-7 + 4i2.
- √7i + 9
- 2
- √7i - 4
- √5i
Click here for the answer
Ans: (c)√7i - 4
Explanation: Simplify the expression -
∴ the result is √-7 + 4i2 = √7i - 4
Ques: Determine the value of √(-49).
- 9i
- I
- 7i
- 5i
Click here for the answer
Ans: (c) 7i
Explanation: √(-49) = √(49) × √(-1)
∴ the result is 7i [ i = √(-1) ]
Ques. Solve : i-36.
- 13/i
- 1/i
- 1
- 1/2i
Click here for the answer
Ans: (c) 1
Explanation: i-36 = 1/i36
⇒ 1/ (i4)9
⇒ 1 / 1 [ i4 = 1]
∴ the result is 1
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