Advertisements
Online Mock Tests
Chapters
2: Relations and Functions
3: Trigonometric Functions
4: Principle of Mathematical Induction
▶ 5: Complex Numbers and Quadratic Equations
6: Linear Inequalities
7: Permutations and Combinations
8: Binomial Theorem
9: Sequences and Series
10: Straight Lines
11: Conic Sections
12: Introduction to Three Dimensional Geometry
13: Limits and Derivatives
14: Mathematical Reasoning
15: Statistics
16: Probability
![NCERT Exemplar solutions for Mathematics [English] Class 11 chapter 5 - Complex Numbers and Quadratic Equations - Shaalaa.com](/images/mathematics-english-class-11_6:5f2b1b2038084cf381bfa42c826a928c.jpg "NCERT Exemplar solutions for Mathematics [English] Class 11 chapter 5 - Complex Numbers and Quadratic Equations")
Advertisements
Solutions for Chapter 5: Complex Numbers and Quadratic Equations
Below listed, you can find solutions for Chapter 5 of CBSE NCERT Exemplar for Mathematics [English] Class 11.
NCERT Exemplar solutions for Mathematics [English] Class 11 5 Complex Numbers and Quadratic Equations Solved Examples [Pages 78 - 90]
Short Answer Type
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(a2 + b2)
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 |z2 – 1| = |z|2 + 1, then show that z lies on imaginary axis.
Let z1 and z2 be two complex numbers such that `barz_1 + ibarz_2` = 0 and arg(z1 z2) = π. Then find arg (z1).
Let z1 and z2 be two complex numbers such that |z1 + z2| = |z1| + |z2|. Then show that arg(z1) – arg(z2) = 0.
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|.
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 2x4 + 5x3 + 7x2 – x + 41, when x = `-2 - sqrt(3)"i"`.
Find the value of P such that the difference of the roots of the equation x2 – Px + 8 = 0 is 2.
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.
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)`
If z1 and z2 both satisfy `z + barz = 2|z - 1|` arg`(z_1 - z_2) = pi/4`, then find `"Im" (z_1 + z_2)`.
Objective Type Questions Fill in the blanks:
The real value of ‘a’ for which 3i3 – 2ai2 + (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 + n2) = ______.
State true or false for the following:
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
State true or false for the following:
The complex number cosθ + isinθ can be zero for some θ.
True
False
State true or false for the following:
If a complex number coincides with its conjugate, then the number must lie on imaginary axis.
True
False
State true or false for the following:
The argument of the complex number z = `(1 + i sqrt(3))(1 + i)(cos theta + i sin theta)` is `(7pi)/12 + theta`.
True
False
State true or false for the following:
The points representing the complex number z for which |z + 1| < |z − 1| lies in the interior of a circle.
True
False
State true or false for the following:
If three complex numbers z1, z2 and z3 are in A.P., then they lie on a circle in the complex plane.
True
False
State true or false for the following:
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.
Match the statements of column A and B.
| 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 |
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 z1 = `sqrt(3) + i sqrt(3)` and z2 = `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 (i25)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 x2 + ax + b = 0, where a, b ∈ R, then find the values of a and b.
Choose the correct options out of given four options in each of the Examples from 28 to 33 (M.C.Q.).
1 + i2 + i4 + i6 + ... + i2n 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 z2 + |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 Exemplar solutions for Mathematics [English] Class 11 5 Complex Numbers and Quadratic Equations Exercise [Pages 91 - 97]
Short Answer Type
For a positive integer n, find the value of `(1 - i)^n (1 - 1/i)^"n"`
Evaluate `sum_(n = 1)^13 (i^n + i^(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 = cosθ + isinθ, 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 Type
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|.
z1 and z2 are two complex numbers such that |z1| = |z2| and arg(z1) + arg(z2) = π, then show that z1 = `-barz_2`.
If |z1| = 1(z1 ≠ –1) and z2 = `(z_1 - 1)/(z_1 + 1)`, then show that the real part of z2 is zero.
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)`.
If |z1| = |z2| = ... = |zn| = 1, then show that |z1 + z2 + z3 + ... + zn| = `|1/z_1 + 1/z_2 + 1/z_3 + ... + 1/z_n|`.
If for complex numbers z1 and z2, arg (z1) – arg (z2) = 0, then show that `|z_1 - z_2| = |z_1| - |z_2|`.
Solve the system of equations Re(z2) = 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.
Objective Type Questions Fill in the blanks of the following:
For any two complex numbers z1, z2 and any real numbers a, b, |az1 – bz2|2 + |bz1 + az2|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 + i2 + i3 + ... upto 1000 terms is ______.
Multiplicative inverse of 1 + i is ______.
If z1 and z2 are complex numbers such that z1 + z2 is a real number, then z2 = ______.
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 True or False for the following:
The order relation is defined on the set of complex numbers.
True
False
State True or False for the following:
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
State True or False for the following:
For any complex number z the minimum value of |z| + |z – 1| is 1.
True
False
State True or False for the following:
The locus represented by |z – 1| = |z – i| is a line perpendicular to the join of (1, 0) and (0, 1).
True
False
State True or False for the following:
If z is a complex number such that z ≠ 0 and Re(z) = 0, then Im(z2) = 0.
True
False
State True or False for the following:
The inequality |z – 4| < |z – 2| represents the region given by x > 3.
True
False
State True or False for the following:
Let z1 and z2 be two complex numbers such that |z1 + z2| = |z1| + |z2|, then arg(z1 – z2) = 0.
True
False
State True or False for the following:
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 + 4| ≤ 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 |z1| = |z2|, is it necessary that z1 = z2?
If `(a^2 + 1)^2/(2a - i)` = x + iy, what is the value of x2 + y2?
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.
Choose the correct answer from the given four options indicated against each of the Exercises from 35 to 50 (M.C.Q)
sinx + icos2x and cosx – isin2x 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|
z2 + 3
None 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 α2 + β2 = ______.
1
–1
2
–2
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|
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 ______.
a2 + c2 = 0
b2 + c2 = 0
b2 + d2 = 0
a2 + b2 = c2 + d2
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.
If z is a complex number, then ______.
|z2| > |z|2
|z2| = |z|2
|z2| < |z|2
|z2| ≥ |z|2
|z1 + z2| = |z1| + |z2| is possible if ______.
`z_2 = barz_1`
`z_2 = 1/z_1`
arg(z1) = arg(z2)
|z1| = |z2|
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
Solutions for 5: Complex Numbers and Quadratic Equations
NCERT Exemplar solutions for Mathematics [English] Class 11 chapter 5 - Complex Numbers and Quadratic Equations
Shaalaa.com has the CBSE Mathematics Mathematics [English] Class 11 CBSE solutions in a manner that help students grasp basic concepts better and faster. The detailed, step-by-step solutions will help you understand the concepts better and clarify any confusion. NCERT Exemplar solutions for Mathematics Mathematics [English] Class 11 CBSE 5 (Complex Numbers and Quadratic Equations) include all questions with answers and detailed explanations. This will clear students' doubts about questions and improve their application skills while preparing for board exams.
Further, we at Shaalaa.com provide such solutions so students can prepare for written exams. NCERT Exemplar textbook solutions can be a core help for self-study and provide excellent self-help guidance for students.
Concepts covered in Mathematics [English] 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, Algebraic Operations of Complex Numbers, The Modulus and the Conjugate of a Complex Number, Concept of Complex Numbers.
Using NCERT Exemplar Mathematics [English] Class 11 solutions Complex Numbers and Quadratic Equations exercise by students is an easy way to prepare for the exams, as they involve solutions arranged chapter-wise and also page-wise. The questions involved in NCERT Exemplar Solutions are essential questions that can be asked in the final exam. Maximum CBSE Mathematics [English] Class 11 students prefer NCERT Exemplar Textbook Solutions to score more in exams.
Get the free view of Chapter 5, Complex Numbers and Quadratic Equations Mathematics [English] Class 11 additional questions for Mathematics Mathematics [English] Class 11 CBSE, and you can use Shaalaa.com to keep it handy for your exam preparation.
