#### Online Mock Tests

#### Chapters

Chapter 2: Relations and Functions

Chapter 3: Trigonometric Functions

Chapter 4: Principle of Mathematical Induction

Chapter 5: Complex Numbers and Quadratic Equations

Chapter 6: Linear Inequalities

Chapter 7: Permutations and Combinations

Chapter 8: Binomial Theorem

Chapter 9: Sequences and Series

Chapter 10: Straight Lines

Chapter 11: Conic Sections

Chapter 12: Introduction to Three Dimensional Geometry

Chapter 13: Limits and Derivatives

Chapter 14: Mathematical Reasoning

Chapter 15: Statistics

Chapter 16: Probability

## Chapter 5: Complex Numbers and Quadratic Equations

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

Evaluate: (1 + i)^{6} + (1 – i)^{3}

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

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

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.

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

Let z_{1} and z_{2} be two complex numbers such that `barz_1 + ibarz_2` = 0 and arg (z_{1} z_{2}) = π. Then find arg (z_{1}).

Let z_{1} and z_{2} be two complex numbers such that |z_{1} + z_{2}| = |z_{1}| + |z_{2}|. Then show that arg (z_{1}) – arg (z_{2}) = 0.

If z_{1}, z_{2}, z_{3} 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 |z_{1} + z_{2} + z_{3}|.

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|

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

Find the value of 2x^{4} + 5x^{3} + 7x^{2} – x + 41, when x = `-2 - sqrt(3)"i"`.

Find the value of P such that the difference of the roots of the equation x^{2} – Px + 8 = 0 is 2.

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

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

If z_{1} and z_{2} 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:

The real value of ‘a’ for which 3i^{3} – 2ai^{2} + (1 – a)i + 5 is real is ______.

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

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

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

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

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

If (2 + i)(2 + 2i)(2 + 3i) ... (2 + ni) = x + iy, then 5.8.13 ... (4 + n^{2}) = ______.

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

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

True

False

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

True

False

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

True

False

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

True

False

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

True

False

If three complex numbers z_{1}, z_{2} and z_{3} are in A.P., then they lie on a circle in the complex plane.

True

False

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

True

False

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

Column A |
Column B |

(a) The value of 1+ i^{2} + i^{4} + i^{6} + ... i^{20} 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 b^{2} – 4ac < 0, then the roots of the equation ax ^{2} + bx + c = 0 are non real (complex) and |
(v) may not occur in conjugate pairs |

(f) If a, b, c ∈ R and b^{2} – 4ac > 0, and b ^{2} – 4ac is a perfect square, then the roots of the equation ax ^{2} + bx + c = 0 |
(vi) may occur in conjugate pairs |

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

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

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

If z_{1} = `sqrt(3) + i sqrt(3)` and z_{2} = `sqrt(3) + i`, then find the quadrant in which `(z_1/z_2)` lies.

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

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

What is the polar form of the complex number (i^{25})^{3}?

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

If 1 – i, is a root of the equation x^{2} + ax + b = 0, where a, b ∈ R, then find the values of a and b.

#### Objective Type Questions from 28 to 33

1 + i^{2} + i^{4} + i^{6} + ... + i^{2n} is ______.

Positive

Negative

0

Can not be evaluated

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

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

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

Straight line

Circle

Parabola

Hyperbola

Number of solutions of the equation z^{2} + |z|^{2} = 0 is ______.

1

2

3

Infinitely many

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

`(2pi)/5`

`pi/5`

`pi/15`

`pi/10`

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

#### Short Answer

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

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

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

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

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

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

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

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

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.

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

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

#### Long Answer

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

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

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

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

z_{1} and z_{2} are two complex numbers such that |z_{1}| = |z_{2}| and arg (z_{1}) + arg (z_{2}) = π, then show that z_{1} = `-barz_2`.

If |z_{1}| = 1(z_{1} ≠ –1) and z_{2} = `(z_1 - 1)/(z_1 + 1)`, then show that the real part of z_{2} is zero.

If z_{1}, z_{2} and z_{3}, z_{4} are two pairs of conjugate complex numbers, then find arg `(z_1/z_4)` + arg `(z_2/z_3)`.

If |z_{1}| = |z_{2}| = ... = |z_{n}| = 1, then show that |z_{1} + z_{2} + z_{3} + ... + z_{n}| = `|1/z_1 + 1/z_2 + 1/z_3 + ... + 1/z_n|`.

If for complex numbers z_{1} and z_{2}, arg (z_{1}) – arg (z_{2}) = 0, then show that `|z_1 - z_2| = |z_1| - |z_2|`

Solve the system of equations Re(z^{2}) = 0, z = 2.

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

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

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

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

For any two complex numbers z_{1}, z_{2} and any real numbers a, b, |az_{1} – bz_{2}|^{2} + |bz_{1} + az_{2}|^{2} = ______.

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

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

The sum of the series i + i^{2} + i^{3} + ... upto 1000 terms is ______.

Multiplicative inverse of 1 + i is ______.

If z_{1} and z_{2} are complex numbers such that z_{1} + z_{2} is a real number, then z_{2} = ______.

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

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

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

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

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

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

True

False

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

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

True

False

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

True

False

If z is a complex number such that z ≠ 0 and Re (z) = 0, then Im (z^{2}) = 0.

True

False

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

True

False

Let z_{1} and z_{2} be two complex numbers such that |z_{1} + z_{2}| = |z_{1}| + |z_{2}|, then arg (z_{1} – z_{2}) = 0.

True

False

2 is not a complex number.

True

False

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

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 |

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

If |z_{1}| = |z_{2}| = , is it necessary that z_{1 }= z_{2}?

If `(a^2 + 1)^2/(2a - i)` = x + iy, what is the value of x^{2} + y^{2}?

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

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

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

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

#### Objective Type Questions from 35 to 50

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

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`

nπ

None of these, where n ∈N

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

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

|z + 3|

^{2}|z – 3|

z

^{2}+ 3None of these

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

x = 2n + 1

x = 4n

x = 2n

x = 4n + 1, where n ∈ N

A real value of x satisfies the equation `((3 - 4ix)/(3 + 4ix))` = α − iβ (α, β ∈ R) if α^{2} + β^{2} = ______.

1

– 1

2

– 2

Which of the following is correct for any two complex numbers z^{1} and z^{2}?

|z

_{1}z_{2}| = |z_{1}|z_{2}|arg (z

_{1}z_{2}) = arg (z_{1}) . arg (z_{2})|z

_{1}+ z_{2}| = |z_{1}| + |z_{2}||z

_{1}+ z_{2}| ≥ |z_{1}| – |z_{2}|

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

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

x = 0

y = 0

x ≠ 0

y ≠ 0

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

a

^{2}+ c^{2}= 0b

^{2}+ c^{2}= 0b

^{2}+ d^{2}= 0a

^{2}+ b^{2}= c^{2}+ d^{2}

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

Circle x

^{2}+ y^{2}= 1The x-axis

The y-axis

The line x + y = 1.

If z is a complex number, then ______.

|z

^{2}| > |z||z

^{2}| = |z|^{2}|z

^{2}| < |z|^{2}|z

^{2}| ≥ |z|^{2 }

|z_{1} + z_{2}| = |z_{1}| + |z_{2}| is possible if ______.

`z_2 = barz_1`

`z_2 = 1/z_1`

arg (z

_{1}) = arg (z_{2})|z

_{1}| = |z_{2}|

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

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

0

`pi/2`

π

None of these

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

## 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.

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