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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
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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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