Table Of ContentSuper Course in Mathematics
aLGEBRa ii
for IIT-JEE
Volume 2
Trishna Knowledge Systems
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Triumphant Institute of Management Education Pvt. Ltd
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Contents
Preface iv
Chapter 1 Complex Numbers 1.1—1.88
sTUDY MATERIAL
• Complex Numbers—Introduction • Algebra of Complex Numbers • Modulus
Amplitude form Representation of a Complex Number (or Polar form Representation
of a Complex Number) • Geometrical Representation of Complex Numbers-
Argand Diagram • Geometrical Representations of the Sum and Difference of Two
Complex Numbers • De Moivre’s Theorem • Euler’s Formula • nth Roots of
Unity • Logarithm of a Complex Number
Chapter 2 Matrices and Determinants 2.1—2.124
sTUDY MATERIAL
• Introduction • Definition of a Matrix • Algebra of Matrices • Transpose of a
Matrix • Conjugation Operation • Concept of a Determinant • Properties of
Determinants • Solution of Linear System of Equations • Homogeneous Linear
System of Equations • Product of Two Determinants • Solution of a Non-Homogenous
Linear System of Equations Using Matrices • Derivative of a Determinant
Chapter 3 Permutations, Combinations and Binomial Theorem 3.1—3.83
STUDY MATERIAL
• Introduction • Formula for np • Permutations—When Some of the Objects (or Things)
r
are not Distinct (i.e., Some Objects are Alike) • Circular Permutations • Formula for
n
nCr or (r) • Method of Induction • Binomial Theorem • Expansion of (1 + x)n Where,
n is a Positive Integer • Binomial Series
Chapter 4 Theory of Probability 4.1—4.93
STUDY MATERIAL
• Introduction • Events or Outcomes • Definition of Probability • Probability–
Axiomatic Approach • Conditional Probability • Independent Events • Bayes’
Theorem or Bayes’ Formula • Binomial Distribution [or Binomial Model] • Geometric
Probability or Probability in Continuum
Preface
The IIT-JEE, the most challenging amongst national level engineering entrance examinations, remains on the top of the
priority list of several lakhs of students every year. The brand value of the IITs attracts more and more students every year,
but the challenge posed by the IIT-JEE ensures that only the best of the aspirants get into the IITs. Students require thorough
understanding of the fundamental concepts, reasoning skills, ability to comprehend the presented situation and exceptional
problem-solving skills to come on top in this highly demanding entrance examination.
The pattern of the IIT-JEE has been changing over the years. Hence an aspiring student requires a step-by-step study
plan to master the fundamentals and to get adequate practice in the various types of questions that have appeared in the
IIT-JEE over the last several years. Irrespective of the branch of engineering study the student chooses later, it is important
to have a sound conceptual grounding in Mathematics, Physics and Chemistry. A lack of proper understanding of these
subjects limits the capacity of students to solve complex problems thereby lessening his/her chances of making it to the top-
notch institutes which provide quality training.
This series of books serves as a source of learning that goes beyond the school curriculum of Class XI and Class XII
and is intended to form the backbone of the preparation of an aspiring student. These books have been designed with the
objective of guiding an aspirant to his/her goal in a clearly defined step-by-step approach.
• Master the Concepts and Concept Strands!
This series covers all the concepts in the latest IIT-JEE syllabus by segregating them into appropriate units. The theories
are explained in detail and are illustrated using solved examples detailing the different applications of the concepts.
• Let us First Solve the Examples—Concept Connectors!
At the end of the theory content in each unit, a good number of “Solved Examples” are provided and they are designed
to give the aspirant a comprehensive exposure to the application of the concepts at the problem-solving level.
• Do Your Exercise—Daily!
Over 200 unsolved problems are presented for practice at the end of every chapter. Hints and solutions for the same are
also provided. These problems are designed to sharpen the aspirant’s problem-solving skills in a step-by-step manner.
• Remember, Practice Makes You Perfect!
We recommend you work out ALL the problems on your own – both solved and unsolved – to enhance the effective-
ness of your preparation.
A distinct feature of this series is that unlike most other reference books in the market, this is not authored by an in-
dividual. It is put together by a team of highly qualified faculty members that includes IITians, PhDs etc from some of the
best institutes in India and abroad. This team of academic experts has vast experience in teaching the fundamentals and
their application and in developing high quality study material for IIT-JEE at T.I.M.E. (Triumphant Institute of Manage-
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experienced team is what is presented in this self-preparatory series. While the contents of these books have been organized
keeping in mind the specific requirements of IIT-JEE, we are sure that you will find these useful in your preparation for
various other engineering entrance exams also.
We wish you the very best!
c h a p t e r COMPLEX
NUMBERS
1
nnn Chapter Outline
Preview
sTUDY MATERIAL ToPIC GRIP
• subjective Questions (10)
Complex Numbers—Introduction
• straight objective Type Questions (5)
• Concept strand (1)
• Assertion–Reason Type Questions (5)
Algebra of Complex Numbers
• Linked Comprehension Type Questions (6)
• Concept strands (2-11)
• Multiple Correct objective Type Questions (3)
Modulus Amplitude form Representation of a • Matrix-Match Type Question (1)
Complex Number (or Polar form Representation
IIT AssIGNMENT ExERCIsE
of a Complex Number)
• Concept strands (12-13) • straight objective Type Questions (80)
• Assertion–Reason Type Questions (3)
Geometrical Representation of Complex Numbers-
• Linked Comprehension Type Questions (3)
Argand Diagram
• Multiple Correct objective Type Questions (3)
Geometrical Representations of the sum and
• Matrix-Match Type Question (1)
Difference of Two Complex Numbers
ADDITIoNAL PRACTICE ExERCIsE
De Moivre’s Theorem
• subjective Questions (10)
Euler’s Formula
• straight objective Type Questions (40)
• Concept strands (14-15)
• Assertion–Reason Type Questions (10)
nth Roots of Unity
• Linked Comprehension Type Questions (9)
Logarithm of a Complex Number • Multiple Correct objective Type Questions (8)
• Matrix-Match Type Questions (3)
CoNCEPT CoNNECToRs
• 25 Connectors
1.2 Complex Numbers
Set of complex numbers was introduced in the unit ‘Quadrat- explain the polar form (or modulus amplitude form)
ic Equations and Expressions’ as an extension of the set of of representation of a complex number. Geometrical
real numbers for accommodating the case where the discrim- representation, Argand diagram are dealt with a number
inant of the quadratic equation is a negative number. The de- of illustrative examples. The linkage between complex
velopment of the topic ‘Complex numbers’, has led to its use numbers, circular functions and coordinate geometry are
in many other branches of mathematics. Also, it has found highlighted. De Moivre’s theorem and its applications,
applications in various problems in science and engineering. nth roots of a number, particularly the nth roots of
In the sequel, after defining a complex number, unity, and their geometrical representation are then
the algebra of complex numbers is discussed. We then discussed.
COmplex numbers—intrOduCtiOn
A number of the form x + iy where x and y are real numbers algebra of complex numbers
(i.e., x, y ∈ R, the set of real numbers) and i stands for −1
(or i is such that i2 = – 1) is called a complex number. Let z = x + iy and z = x + iy represent two complex
1 1 1 2 2 2
If z denotes this complex number z = x + iy, x is called numbers.
the real part of z denoted by Re(z). y is called the imaginary
part of z denoted by Im (z) (i) Equality
Consider the following examples:
z = z if and only if x = x and y = y
(i) z = 2 + 3i 1 2 1 2 1 2
Real part = 2
(ii) Addition
Imaginary part = 3
(ii) z = −1−3 2i z + z = (x + x) + i(y + y)
1 2 1 2 1 2
Real part = – 1
(iii) Multiplication by a real number
Imaginary part = −3 2
If k is a real number, kz = kx + iky
(iii) z = 4 1 1 1
Re(z) = 4
(iv) Subtraction
Im(z) = 0
(iv) z = – 5i z – z = z + (–1)z = (x – x) + i(y – y).
1 2 1 2 1 2 1 2
Re(z) = 0
Im(z) = – 5
(v) Multiplication of two complex numbers
(v) z = 3 - 7i
Re(z) = 3 zz = (xx – yy) + i(xy + xy)
1 2 1 2 1 2 1 2 2 1
Im(z) = - 7 The multiplication rule is such that we may treat zz as the
1 2
(vi) z = 0 product of the two factors (x + iy) and (x + iy). We use
1 1 2 2
Real part = 0 = Imaginary part. the ordinary rule for multiplication of two algebraic expres-
sions and replace i2 by (–1).
The set of complex numbers is denoted by C.
If y = 0, z is real.
(vi) Complex Conjugate
If x = 0, z is said to be purely imaginary.
If z = x + iy, the complex conjugate of z, denoted by zis
This clearly shows that the set of real numbers R is a
subset of the set of complex numbers, or R ⊂ C. defined as z = x – iy.
Complex Numbers 1.3
For example, if z = (5 + 7i), z = (5−7i) Graphical Illustration
if z = – 4i, z = 4i Case (i): D > 0
if z = 7, z = 7 = z
We observe that zz = (x +iy)(x −iy) = (x2 + y2), a > 0 a < 0
y y
which is real and positive. Also, conjugate of z is z. We say
that z and z constitute a conjugate pair.
We will be having a detailed study of complex numbers
O
and its applications in another module. α β x O α β x
Referring to example (iv) under “Nature of roots of
a quadratic equation”, we note that the two roots of the
equation x2 + 2x + 2 = 0 are complex numbers. Roots are
−2+ −4 −2− −4 −2+2i
given by x = and , i.e., and Fig. 1.1
2 2 2
−2−2i
or (–1 + i) and (–1 – i). Also note that the two
2 Case (ii): D = 0
roots form a conjugate pair.
In a quadratic equation ax2+ bx + c = 0 with real coef- a > 0 a < 0
ficients, if b2 - 4ac < 0, the roots are complex in nature and y y
they occur in conjugate pairs.
To sum up the above observations, if D denotes the dis-
criminant (b2 – 4ac) of the quadratic equation ax2 + bx + c
= 0, [where a, b, c are rational], then the nature of the roots α
of the quadratic equation will be as given in the table below:
α O x O x
Table 1.1
Nature of D Nature of the roots
D positive real and distinct (different) Fig. 1.2
D positive and is a real, distinct and rational
perfect square
Case (iii): D < 0
D positive but is not a real, distinct and irrational
perfect square or, the roots are of the form There are no real roots for ax2 + bx + c = 0, i.e., there are no
p± q , where p and q are x-intercepts for the curve y = ax2 + bx + c.
rational
D zero real and equal a > 0 a < 0
y y
D negative complex or the roots are of the
form p±iq
If we consider the graph of y = ax2 + bx + c, the real
roots of the quadratic equation ax2 + bx + c = 0 are the O x O x
x-coordinates of the points of intersection of the graph
with the x-axis (y = 0).
We can therefore have a graphical illustration of the
results given in the above table. The curve y = f(x) = ax2 + Fig. 1.3
bx + c is shown in the following graphs.
1.4 Complex Numbers
We hasten to add that if the coefficients of a quadratic Again, if the coefficients of a quadratic equation are
equation are not rational, and if D is greater than zero but not real, then its roots will not be of the form p±iq, where
not a perfect square, the roots of the equation will not be of p, q are real and i stands for −1.
the form p± q, where p and q are rational. Consider the equation x2 – 4x + (1 + 4i) = 0
It can be verified that i and (4 – i) satisfy the above
( )
Consider the equation x2 −5x + 3+ 3 = 0. equation and therefore, the roots are i and (4 – i). However,
these do not form a conjugate pair.
( ) A number of the form x + iy, where x and y are real
5± 25− 12+ 4 3
Its roots are given by x = numbers (i.e., x, y ∈ R, the set of real numbers) and i stands
2 for −1 (or i is such that i2 = –1) is called a complex num-
ber.
( )
5± 2 3 −1
If z denotes this complex number, z = x + iy. x is called
=
2 the real part of z denoted by Re(z). y is called the imaginary
part of z denoted by Im(z).
= 2+ 3or3− 3
ConCept Strand
Concept strand 1 Solution
Write the real part and imaginary parts of the following (i) z = 4 + 9i ⇒ Re(z) = 4, Im(z) = 9
numbers: (ii) z = 7 3 − 5i ⇒ Re(z) =7 3 , Im(z) = − 5
(iii) z = - 8 ⇒ Re(z) = -8, Im(z) = 0
(i) z = 4 + 9i
(iv) z = 10 i ⇒ Re(z) = 0, Im(z) = 10
(ii) z = 7 3 − 5i (v) z = 0 ⇒ Re(z) = 0, Im(z) = 0
(iii) z = -8
The set of complex numbers is denoted by C. If y = 0,
(iv) z = 10 i
z is real which means that the set of real numbers is a sub-
(v) z = 0
set of the set of complex numbers. In other words, R ⊂ C.
If x = 0, z is pure imaginary.
algebra Of COmplex numbers
Let z = x + iy and z = x + iy represent two complex (iii) Multiplication by a real number
1 1 1 2 2 2
numbers.
If k is a real number, kz = kx + iky
1 1 1
(i) Equality
(iv) Subtraction
z = z , if and only if z = x + iy = 0 implies
1 2
x = x, y = y; x = 0, y = 0. z – z = z + (–1)z = (x – x) + i(y – y).
1 2 1 2 1 2 1 2 1 2 1 2
(ii) Addition (v) Multiplication of two complex numbers
z + z = (x + x) + i(y + y) zz = (xx – yy) + i(xy + xy)
1 2 1 2 1 2 1 2 1 2 1 2 1 2 2 1
Complex Numbers 1.5
The multiplication rule is such that we may treat zz as the (vii) Division of two complex numbers
1 2
product of two factors (x + iy) and (x + iy). We use the
1 1 2 2 z
ordinary rule for multiplication of two algebraic expres- 1 where z ≠ 0
sions and replace i2 by (–1). z 2
2
z z z (x +iy )(x −iy )
1 = 1 2 = 1 1 2 2
(vi) Conjugate of a complex number z z z (x2 + y2)
2 2 2 2 2
x x + y y x y − x y
Complex Conjugate of z = x + iy, denoted by z is defined = 1 2 1 2 +i 2 1 1 2
(x2 + y2) (x2 + y2)
as z = x – iy. 2 2 2 2
Note that zz = (x + iy) (x – iy) = x2 + y2 = real and We can easily see that the set of complex numbers is
positive. closed under addition and multiplication. The addition and
The conjugate of z is z. multiplication operations satisfy associative property.
For example, if z = 4 – 3i, z= 4 + 3i; If z = 7i, z= -7i. Also, the distributive property, z (z +z ) = z z +
If z is real, i.e., z = x (x real), then z= x. ⇒ If z is real, 1 2 3 1 2
its conjugate is itself. z1z3where z1, z2, z3 are complex numbers is satisfied.
ConCept StrandS
Concept strand 2 (iii) 4z2 + 2z3 +z z + zz
1 2 1 3 2 3
= 4 (3 + 7i)2 + 2 (-2i)3 + (3 + 7i) (4 + 6i) +
If 2x + 3iy = 2 + 9i, find x and y.
(-2i) (4 + 6i)
= 4 (9 + 49i2 + 42i) - 16i3 + (12 + 18i + 28i + 42i2)
Solution - (8i + 12i2)
= 4 (9 - 49 + 42i) + 16i + (12 + 46i - 42) -
Equating real and imaginary parts, 2x = 2, 3y = 9 giving
(8i - 12)
x = 1, y = 3.
= -178 + 222i
Concept strand 3
Concept strand 5
Express (2+ 3i)2 in the form x + iy.
Find the value of 1 + i2 + i4 + i6 + i8 + i10.
Solution Solution
Since the above expression is a geometric series with first
(2 + 3i)2 = 4 + 9i2 + 12i
term 1, common ratio i2 and number of terms 6, sum of the
= 4 - 9 + 12i = -5 + 12i
1−(i2)6 1−(−1)6
series = = = 0
1−i2 (1+1)
Concept strand 4
Concept strand 6
If z = 3 + 7i, z = -2i, z = 4 + 6i, find
1 2 3
2+3i
(i) z - 3z + z, Express in the form x + iy.
(ii) z1z2, 2 3 −5− 4i
1 2
(iii) 4z2 + 2z3 +z z + zz Solution
1 2 1 3 2 3
2+3i (2+3i)(−5+ 4i)
Solution = ,
−5− 4i (−5)2 + 42
(i) z - 3z + z = 3 + 7i - 3(-2i) + 4 + 6i = 7 + 19i. (on multiplying both numerator and denominator by the
1 2 3
(ii) zz2 = (3 + 7i) (-2i)2 = (3 + 7i) × (-4) = -12 -28i conjugate of -5 - 4i)
1 2
