# NCERT solutions for Mathematics Exemplar Class 11 chapter 5 - Complex Numbers and Quadratic Equations [Latest edition]

## Chapter 5: Complex Numbers and Quadratic Equations

Solved ExamplesExercise
Solved Examples [Pages 78 - 90]

### NCERT solutions for Mathematics Exemplar Class 11 Chapter 5 Complex Numbers and Quadratic Equations Solved Examples [Pages 78 - 90]

Solved Examples | Q 1 | Page 78

Evaluate: (1 + i)6 + (1 – i)3

Solved Examples | Q 2 | Page 78

If (x + iy)^(1/3) = a + ib, where x, y, a, b ∈ R, show that x/a - y/b = – 2(a2 + b2)

Solved Examples | Q 3 | Page 78

Solve the equation z^2 = barz, where z = x + iy

Solved Examples | Q 4 | Page 79

If the imaginary part of (2z + 1)/(iz + 1) is – 2, then show that the locus of the point representing z in the argand plane is a straight line.

Solved Examples | Q 5 | Page 79

If |z2 – 1| = |z|2 + 1, then show that z lies on imaginary axis.

Solved Examples | Q 6 | Page 80

Let z1 and z2 be two complex numbers such that barz_1 + ibarz_2 = 0 and arg (z1 z2) = π. Then find arg (z1).

Solved Examples | Q 7 | Page 80

Let z1 and z2 be two complex numbers such that |z1 + z2| = |z1| + |z2|. Then show that arg (z1) – arg (z2) = 0.

Solved Examples | Q 8 | Page 80

If z1, z2, z3 are complex numbers such that |z_1| = |z_2| = |z_3| = |1/z_1 + 1/z_2 + 1/z_3| = 1, then find the value of |z1 + z2 + z3|.

Solved Examples | Q 9 | Page 81

If a complex number z lies in the interior or on the boundary of a circle of radius 3 units and centre (– 4, 0), find the greatest and least values of |z + 1|

Solved Examples | Q 10 | Page 81

Locate the points for which 3 < |z| < 4

Solved Examples | Q 11 | Page 81

Find the value of 2x4 + 5x3 + 7x2 – x + 41, when x = -2 - sqrt(3)"i".

Solved Examples | Q 12 | Page 82

Find the value of P such that the difference of the roots of the equation x2 – Px + 8 = 0 is 2.

Solved Examples | Q 13 | Page 82

Find the value of a such that the sum of the squares of the roots of the equation x2 – (a – 2)x – (a + 1) = 0 is least

Solved Examples | Q 14 | Page 82

Find the value of k if for the complex numbers z1 and z2, |1 - barz_1z_2|^2 - |z_1 - z_2|^2 = k(1 - |z_1|^2)(1 - |"z"_2|^2)

Solved Examples | Q 15 | Page 83

If z1 and z2 both satisfy "z" + bar"z" = 2|"z" - 1| arg ("z"_1 - "z"_2) = pi/4, then find "Im" ("z"_1 + "z"_2).

#### Fill in the blanks:

Solved Examples | Q 16.(i) | Page 83

The real value of ‘a’ for which 3i3 – 2ai2 + (1 – a)i + 5 is real is ______.

Solved Examples | Q 16.(ii) | Page 83

If |z| = 2 and arg (z) = pi/4, then z = ______.

Solved Examples | Q 16.(iii) | Page 83

The locus of z satisfying arg (z) = pi/3 is ______.

Solved Examples | Q 16.(iv) | Page 83

The value of (- sqrt(-1))^(4"n" - 3), where n ∈ N, is ______.

Solved Examples | Q 16.(v) | Page 84

The conjugate of the complex number (1 - i)/(1 + i) is ______.

Solved Examples | Q 16.(vi) | Page 84

If a complex number lies in the third quadrant, then its conjugate lies in the ______.

Solved Examples | Q 16.(vii) | Page 84

If (2 + i)(2 + 2i)(2 + 3i) ... (2 + ni) = x + iy, then 5.8.13 ... (4 + n2) = ______.

#### State true or false for the following:

Solved Examples | Q 17.(i) | Page 85

Multiplication of a non-zero complex number by i rotates it through a right angle in the anti-clockwise direction.

• True

• False

Solved Examples | Q 17.(ii) | Page 85

The complex number cosθ + i sinθ can be zero for some θ.

• True

• False

Solved Examples | Q 17.(iii) | Page 85

If a complex number coincides with its conjugate, then the number must lie on imaginary axis.

• True

• False

Solved Examples | Q 17.(iv) | Page 85

The argument of the complex number z = (1 + i sqrt(3))(1 + i)(cos theta + i sin theta) is (7pi)/12 + theta.

• True

• False

Solved Examples | Q 17.(v) | Page 85

The points representing the complex number z for which |z + 1| < |z − 1| lies in the interior of a circle.

• True

• False

Solved Examples | Q 17.(vi) | Page 85

If three complex numbers z1, z2 and z3 are in A.P., then they lie on a circle in the complex plane.

• True

• False

Solved Examples | Q 17.(vii) | Page 85

If n is a positive integer, then the value of in + (i)n+1 + (i)n+2 + (i)n+3 is 0.

• True

• False

#### Match the statements of column A and B.

Solved Examples | Q 18 | Page 86
 Column A Column B (a) The value of 1+ i2 + i4 + i6 + ... i20 is (i) purely imaginary complex number (b) The value of i^(-1097) is (ii) purely real complex number (c) Conjugate of 1 + i lies in (iii) second quadrant (d) (1 + 2i)/(1 - i) lies in (iv) Fourth quadrant (e) If a, b, c ∈ R and b2 – 4ac < 0, then the roots of the equation ax2 + bx + c = 0 are non real (complex) and (v) may not occur in conjugate pairs (f) If a, b, c ∈ R and b2 – 4ac > 0, and b2 – 4ac is a perfect square, then the roots of the equation ax2 + bx + c = 0 (vi) may occur in conjugate pairs
Solved Examples | Q 19 | Page 87

What is the value of (i^(4n + 1) -i^(4n - 1))/2?

Solved Examples | Q 20 | Page 87

What is the smallest positive integer n, for which (1 + i)2n = (1 – i)2n?

Solved Examples | Q 21 | Page 87

What is the reciprocal of 3 + sqrt(7)i

Solved Examples | Q 22 | Page 87

If z1 = sqrt(3) + i  sqrt(3) and z2 = sqrt(3) + i, then find the quadrant in which (z_1/z_2) lies.

Solved Examples | Q 23 | Page 88

What is the conjugate of (sqrt(5 + 12i) + sqrt(5 - 12i))/(sqrt(5 + 12i) - sqrt(5 - 12i))?

Solved Examples | Q 24 | Page 88

What is the principal value of amplitude of 1 – i?

Solved Examples | Q 25 | Page 88

What is the polar form of the complex number (i25)3?

Solved Examples | Q 26 | Page 88

What is the locus of z, if amplitude of z – 2 – 3i is pi/4?

Solved Examples | Q 27 | Page 89

If 1 – i, is a root of the equation x2 + ax + b = 0, where a, b ∈ R, then find the values of a and b.

#### Objective Type Questions from 28 to 33

Solved Examples | Q 28 | Page 89

1 + i2 + i4 + i6 + ... + i2n is ______.

• Positive

• Negative

• 0

• Can not be evaluated

Solved Examples | Q 29 | Page 89

If the complex number z = x + iy satisfies the condition |z + 1| = 1, then z lies on ______.

• X-axis

• Circle with centre (1, 0) and radius 1

• Circle with centre (–1, 0) and radius 1

• Y-axis

Solved Examples | Q 30 | Page 89

The area of the triangle on the complex plane formed by the complex numbers z, – iz and z + iz is ______.

• |z|2

• |barz|^2

• |z|^2/2

• None of these

Solved Examples | Q 31 | Page 90

The equation |z + 1 – i| = |z – 1 + i| represents a ______.

• Straight line

• Circle

• Parabola

• Hyperbola

Solved Examples | Q 32 | Page 90

Number of solutions of the equation z2 + |z|2 = 0 is ______.

• 1

• 2

• 3

• Infinitely many

Solved Examples | Q 33 | Page 90

The amplitude of sin  pi/5 + i(1 - cos  pi/5) is ______.

• (2pi)/5

• pi/5

• pi/15

• pi/10

Exercise [Pages 91 - 97]

### NCERT solutions for Mathematics Exemplar Class 11 Chapter 5 Complex Numbers and Quadratic Equations Exercise [Pages 91 - 97]

Exercise | Q 1 | Page 91

For a positive integer n, find the value of (1 - i)^n (1 - 1/i)^"n"

Exercise | Q 2 | Page 91

Evaluate sum_(n = 1)^13 (i^n + 1^(n + 1)), where n ∈ N.

Exercise | Q 3 | Page 91

If ((1 + i)/(1 - i))^3 - ((1 - i)/(1 + i))^3 = x + iy, then find (x, y).

Exercise | Q 4 | Page 91

If (1 + i)^2/(2 - i) = x + iy, then find the value of x + y.

Exercise | Q 5 | Page 91

If ((1 - i)/(1 + i))^100 = a + ib, then find (a, b).

Exercise | Q 6 | Page 91

If a = cos θ + i sin θ, find the value of (1 + "a")/(1 - "a").

Exercise | Q 7 | Page 91

If (1 + i)z = (1 - i)barz, then show that z = -ibarz.

Exercise | Q 8 | Page 91

If z = x + iy , then show that z  barz + 2(z + barz) + b = 0, where b ∈ R, represents a circle.

Exercise | Q 9 | Page 91

If the real part of (barz + 2)/(barz - 1) is 4, then show that the locus of the point representing z in the complex plane is a circle.

Exercise | Q 10 | Page 91

Show that the complex number z, satisfying the condition arg ((z - 1)/(z + 1)) = pi/4 lies on a circle.

Exercise | Q 11 | Page 91

Solve the equation |z| = z + 1 + 2i.

Exercise | Q 12 | Page 92

If |z +1| = z + 2(1 + i), then find z.

Exercise | Q 13 | Page 92

If arg (z – 1) = arg (z + 3i), then find x – 1: y. where z = x + iy.

Exercise | Q 14 | Page 92

Show that |(z - 2)/(z - 3)| = 2 represents a circle. Find its centre and radius.

Exercise | Q 15 | Page 92

If (z - 1)/(z + 1) is purely imaginary number (z ≠ – 1), then find the value of |z|.

Exercise | Q 16 | Page 92

z1 and z2 are two complex numbers such that |z1| = |z2| and arg (z1) + arg (z2) = π, then show that z1 = -barz_2.

Exercise | Q 17 | Page 92

If |z1| = 1(z1 ≠ –1) and z2 = (z_1 - 1)/(z_1 + 1), then show that the real part of z2 is zero.

Exercise | Q 18 | Page 92

If z1, z2 and z3, z4 are two pairs of conjugate complex numbers, then find arg (z_1/z_4) + arg (z_2/z_3).

Exercise | Q 19 | Page 92

If |z1| = |z2| = ... = |zn| = 1, then show that |z1 + z2 + z3 + ... + zn| = |1/z_1 + 1/z_2 + 1/z_3 + ... + 1/z_n|.

Exercise | Q 20 | Page 92

If for complex numbers z1 and z2, arg (z1) – arg (z2) = 0, then show that |z_1 - z_2| = |z_1| - |z_2|

Exercise | Q 21 | Page 92

Solve the system of equations Re(z2) = 0, z = 2.

Exercise | Q 22 | Page 92

Find the complex number satisfying the equation z + sqrt(2) |(z + 1)| + i = 0.

Exercise | Q 23 | Page 92

Write the complex number z = (1 - i)/(cos  pi/3 + i sin  pi/3) in polar form.

Exercise | Q 24 | Page 92

If z and w are two complex numbers such that |zw| =1 and arg (z) – arg (w) = pi/2, then show that barzw = – i.

#### Fill in the blanks of the following:

Exercise | Q 25.(i) | Page 93

For any two complex numbers z1, z2 and any real numbers a, b, |az1 – bz2|2 + |bz1 + az2|2 = ______.

Exercise | Q 25.(ii) | Page 93

The value of sqrt(-25) xx sqrt(-9) is ______.

Exercise | Q 25.(iii) | Page 93

The number (1 - i)^3/(1 - i^2) is equal to ______.

Exercise | Q 25.(iv) | Page 93

The sum of the series i + i2 + i3 + ... upto 1000 terms is ______.

Exercise | Q 25.(v) | Page 93

Multiplicative inverse of 1 + i is ______.

Exercise | Q 25.(vi) | Page 93

If z1 and z2 are complex numbers such that z1 + z2 is a real number, then z2 = ______.

Exercise | Q 25.(vii) | Page 93

arg (z) + arg barz  (barz ≠ 0) is ______.

Exercise | Q 25.(viii) | Page 93

If |z + 4| ≤ 3, then the greatest and least values of |z +1| are ______ and ______.

Exercise | Q 25.(ix) | Page 93

If |(z - 2)/(z + 2)| = pi/6, then the locus of z is ______.

Exercise | Q 25.(x) | Page 93

If |z| = 4 and arg (z) = (5pi)/6, then z = ______.

#### State whether the following is True or False:

Exercise | Q 26.(i) | Page 93

The order relation is defined on the set of complex numbers.

• True

• False

Exercise | Q 26.(ii) | Page 93

Multiplication of a non-zero complex number by – i rotates the point about origin through a right angle in the anti-clockwise direction.

• True

• False

Exercise | Q 26.(iii) | Page 93

For any complex number z the minimum value of |z + |z – 1| is 1.

• True

• False

Exercise | Q 26.(iv) | Page 93

The locus represented by |z – 1| = |z – i| is a line perpendicular to the join of (1, 0) and (0, 1).

• True

• False

Exercise | Q 26.(v) | Page 93

If z is a complex number such that z ≠ 0 and Re (z) = 0, then Im (z2) = 0.

• True

• False

Exercise | Q 26.(vi) | Page 93

The inequality  |z – 4| < |z – 2| represents the region given by x > 3.

• True

• False

Exercise | Q 26.(vii) | Page 93

Let z1 and z2 be two complex numbers such that |z1 + z2| = |z1| + |z2|, then arg (z1 – z2) = 0.

• True

• False

Exercise | Q 26.(viii) | Page 93

2 is not a complex number.

• True

• False

#### Match the statements of Column A and Column B.

Exercise | Q 27 | Page 94
 Column A Column B (a) The polar form of i + sqrt(3) is (i) Perpendicular bisector of segment joining (– 2, 0) and (2, 0) (b) The amplitude of -1 + sqrt(-3) is (ii) On or outside the circle having centre at (0, – 4) and radius 3. (c) If |z + 2| = |z − 2|, then locus of z is (iii) (2pi)/3 (d) If |z + 2i| = |z − 2i|, then locus of z is (iv) Perpendicular bisector of segment joining (0, – 2) and (0, 2). (e) Region represented by |z + 4i| ≥ 3 is (v) 2(cos  pi/6 + i sin  pi/6) (f) Region represented by |z + 4i| ≤ 3 is (vi) On or inside the circle having centre (– 4, 0) and radius 3 units. (g) Conjugate of (1 + 2i)/(1 - i) lies in (vii) First quadrant (h) Reciprocal of 1 – i lies in (viii) Third quadrant
Exercise | Q 28 | Page 94

What is the conjugate of (2 - i)/(1 - 2i)^2?

Exercise | Q 29 | Page 94

If |z1| = |z2| = , is it necessary that z1 = z2?

Exercise | Q 30 | Page 94

If (a^2 + 1)^2/(2a - i) = x + iy, what is the value of x2 + y2?

Exercise | Q 31 | Page 95

Find z if |z| = 4 and arg (z) = (5pi)/6.

Exercise | Q 32 | Page 95

Find |(1 + i) ((2 + i))/((3 + i))|.

Exercise | Q 33 | Page 95

Find principal argument of (1 + i sqrt(3))^2.

Exercise | Q 34 | Page 95

Where does z lie, if |(z - 5i)/(z + 5i)| = 1.

#### Objective Type Questions from 35 to 50

Exercise | Q 35 | Page 95

sinx + i cos 2x and cos x – i sin 2x are conjugate to each other for ______.

• x = nπ

• x = (n + 1/2) pi/2

• x = 0

• No value of x

Exercise | Q 36 | Page 95

The real value of α for which the expression (1 - i sin alpha)/(1 + 2i sin alpha) is purely real is ______.

• (n + 1)  pi/2

• (2n + 1)  pi/2

• None of these, where n ∈N

Exercise | Q 37 | Page 95

If z = x + iy lies in the third quadrant, then barz/z also lies in the third quadrant if ______.

• x > y > 0

• x < y < 0

• y < x < 0

• y > x > 0

Exercise | Q 38 | Page 95

The value of (z + 3)(barz + 3) is equivalent to ______.

• |z + 3|2

• |z – 3|

• z2 + 3

• None of these

Exercise | Q 39 | Page 95

If ((1 + i)/(1 - i))^x = 1, then ______.

• x = 2n + 1

• x = 4n

• x = 2n

• x = 4n + 1, where n ∈ N

Exercise | Q 40 | Page 96

A real value of x satisfies the equation ((3 - 4ix)/(3 + 4ix)) = α − iβ (α, β ∈ R) if α2 + β2 = ______.

• 1

• – 1

• 2

• – 2

Exercise | Q 41 | Page 96

Which of the following is correct for any two complex numbers z1 and z2

• |z1z2| = |z1|z2|

• arg (z1z2) = arg (z1) . arg (z2)

• |z1 + z2| = |z1| + |z2|

• |z1 + z2| ≥ |z1| – |z2|

Exercise | Q 42 | Page 96

The point represented by the complex number 2 – i is rotated about origin through an angle pi/2 in the clockwise direction, the new position of point is ______.

• 1 + 2i

• –1 – 2i

• 2 + i

• –1 + 2i

Exercise | Q 43 | Page 96

Let x, y ∈ R, then x + iy is a non-real complex number if ______.

• x = 0

• y = 0

• x ≠ 0

• y ≠ 0

Exercise | Q 44 | Page 96

If a + ib = c + id, then ______.

• a2 + c2 = 0

• b2 + c2 = 0

• b2 + d2 = 0

• a2 + b2 = c2 + d2

Exercise | Q 45 | Page 96

The complex number z which satisfies the condition |(i + z)/(i - z)| = 1 lies on ______.

• Circle x2 + y2 = 1

• The x-axis

• The y-axis

• The line x + y = 1.

Exercise | Q 46 | Page 96

If z is a complex number, then ______.

• |z2| > |z|

• |z2| = |z|2

• |z2| < |z|2

• |z2| ≥ |z|

Exercise | Q 47 | Page 96

|z1 + z2| = |z1| + |z2| is possible if ______.

• z_2 = barz_1

• z_2 = 1/z_1

• arg (z1) = arg (z2)

• |z1| = |z2

Exercise | Q 48 | Page 97

The real value of θ for which the expression (1 + i cos theta)/(1 - 2i cos theta) is a real number is ______.

• npi + pi/4

• npi + (-1)n  pi/4

• 2npi +-  pi/2

• None of these

Exercise | Q 49 | Page 97

The value of arg (x) when x < 0 is ______.

• 0

• pi/2

• π

• None of these

Exercise | Q 50 | Page 97

If f(z) = (7 - z)/(1 - z^2), where z = 1 + 2i, then |f(z)| is ______.

• |z|/2

• |z|

• 2|z|

• None of these

## Chapter 5: Complex Numbers and Quadratic Equations

Solved ExamplesExercise

## NCERT solutions for Mathematics Exemplar Class 11 chapter 5 - Complex Numbers and Quadratic Equations

NCERT solutions for Mathematics Exemplar Class 11 chapter 5 (Complex Numbers and Quadratic Equations) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics Exemplar Class 11 solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Mathematics Exemplar Class 11 chapter 5 Complex Numbers and Quadratic Equations are Argand Plane and Polar Representation, Quadratic Equations, Algebra of Complex Numbers - Equality, Algebraic Properties of Complex Numbers, Need for Complex Numbers, Square Root of a Complex Number, Algebra of Complex Numbers, The Modulus and the Conjugate of a Complex Number, Concept of Complex Numbers.

Using NCERT Class 11 solutions Complex Numbers and Quadratic Equations exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in NCERT Solutions are important questions that can be asked in the final exam. Maximum students of CBSE Class 11 prefer NCERT Textbook Solutions to score more in exam.

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