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Trang 2Arihant Prakashan (Series), Meerut
JEE Main & Advanced
Trang 3All Rights Reserved
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Trang 41 Complex Numbers 1-28
Trang 6UNIT I Sets, Relations and Functions
Sets and their representation, Union, intersection and
complement of sets and their algebraic properties, Power
set, Relation, Types of relations, equivalence relations,
functions, one-one, into and onto functions, composition
of functions.
UNIT II Complex Numbers and
Quadratic Equations
Complex numbers as ordered pairs of reals,
Representation of complex numbers in the form a+ib and
their representation in a plane, Argand diagram, algebra
of complex numbers, modulus and argument (or
amplitude) of a complex number, square root of a
complex number, triangle inequality, Quadratic
equations in real and complex number system and their
solutions Relation between roots and co-efficients,
nature of roots, formation of quadratic equations with
given roots.
UNIT III Matrices and Determinants
Matrices, algebra of matrices, types of matrices,
determinants and matrices of order two and three
Properties of determinants, evaluation of deter-minants,
area of triangles using determinants Adjoint and
evaluation of inverse of a square matrix using
determinants and elementary transformations, Test of
consistency and solution of simultaneous linear
equations in two or three variables using determinants
and matrices.
UNIT IV Permutations and Combinations
Fundamental principle of counting, permutation as an
arrangement and combination as selection, Meaning of
P(n,r) and C (n,r), simple applications.
UNIT V Mathematical Induction
Principle of Mathematical Induction and its simple applications.
UNIT VI Binomial Theorem and its Simple
Applications Binomial theorem for a positive integral index, general term and middle term, properties of Binomial coefficients and simple applications.
UNIT VII Sequences and Series
Arithmetic and Geometric progressions, insertion of arithmetic, geometric means between two given numbers Relation between AM and GM Sum upto n
terms of special series: ∑ n, ∑ n , ∑n Arithmetico -
Geometric progression.
UNIT VIII Limit, Continuity and Differentiability
Real valued functions, algebra of functions, polynomials, rational, trigonometric, logarithmic and exponential functions, inverse functions Graphs of simple functions Limits, continuity and differenti-ability Differentiation of the sum, difference, product and quotient of two functions Differentiation of trigonometric, inverse trigonometric, logarithmic, exponential, composite and implicit functions; derivatives of order upto two Rolle's and Lagrange's Mean Value Theorems Applications of derivatives: Rate of change of quantities, monotonic - increasing and decreasing functions, Maxima and minima
of functions of one variable, tangents and normals.
UNIT IX Integral Calculus
Integral as an anti - derivative Fundamental integrals involving algebraic, trigonometric, exponential and logarithmic functions Integration by substitution, by
JEE MAIN
SYLLABUS
Trang 7the type
Integral as limit of a sum Fundamental Theorem of
Calculus Properties of definite integrals Evaluation of
definite integrals, determining areas of the regions
bounded by simple curves in standard form.
UNIT X Differential Equations
Ordinary differential equations, their order and degree
Formation of differential equations Solution of
differential equations by the method of separation of
variables, solution of homogeneous and linear differential
equations
of the type
UNIT XI Coordinate Geometry
Cartesian system of rectangular coordinates in a plane,
distance formula, section formula, locus and its equation,
translation of axes, slope of a line, parallel and
perpendicular lines, intercepts of a line on the coordinate
axes.
Straight lines
Various forms of equations of a line, intersection of lines,
angles between two lines, conditions for concurrence of
three lines, distance of a point from a line, equations of
internal and external bisectors of angles between two
lines, coordinates of centroid, orthocentre and
circumcentre of a triangle, equation of family of lines
passing through the point of intersection of two lines.
Circles, Conic sections
Standard form of equation of a circle, general form of the
equation of a circle, its radius and centre, equation of a
origin and condition for a line to be tangent to a circle, equation of the tangent Sections of cones, equations of conic sections (parabola, ellipse and hyperbola) in standard forms, condition for y=mx + c to be a tangent and point (s) of tangency.
UNIT XII Three Dimensional Geometry
Coordinates of a point in space, distance between two points, section formula, direction ratios and direction cosines, angle between two intersecting lines Skew lines, the shortest distance between them and its equation Equations of a line and a plane in different forms, intersection of a line and a plane, coplanar lines.
UNIT XIII Vector Algebra
Vectors and scalars, addition of vectors, components of a vector in two dimensions and three dimensional space, scalar and vector products, scalar and vector triple product.
UNIT XIV Statistics and Probability
Measures of Dispersion: Calculation of mean, median, mode of grouped and ungrouped data Calculation of standard deviation, variance and mean deviation for grouped and ungrouped data.
Probability: Probability of an event, addition and multiplication theorems of probability, Baye's theorem, probability distribution of a random variate, Bernoulli trials and Binomial distribution.
UNIT XV Trigonometry
Trigonometrical identities and equations
Trigonometrical functions Inverse trigonometrical functions and their properties Heights and Distances.
UNIT XVI Mathematical Reasoning
Statements, logical operations And, or, implies, implied
by, if and only if Understanding of tautology, contradiction, converse and contra positive.
Trang 8Arithmetic, geometric and harmonic progressions, arithmetic, geometric and harmonic means, sums of finite
arithmetic and geometric progressions, infinite geometric series, sums of squares and cubes of the first n natural numbers
Logarithms and their Properties, Permutations and combinations, Binomial theorem for a positive integral index, properties of binomial coefficients
Matrices
Matrices as a rectangular array of real numbers, equality of matrices, addition, multiplication by a scalar and product
of matrices, transpose of a matrix, determinant of a square matrix of order up to three, inverse of a square matrix of order up to three, properties of these matrix operations, diagonal, symmetric and skew-symmetric matrices and their properties, solutions of simultaneous linear equations in two or three variables
Probability
Addition and multiplication rules of probability, conditional probability, independence of events, computation of probability of events using permutations and combinations
Trigonometry
Trigonometric functions, their periodicity and graphs, addition and subtraction formulae, formulae involving
multiple and sub-multiple angles, general solution of trigonometric equations
Relations between sides and angles of a triangle, sine rule, cosine rule, half-angle formula and the area of a triangle, inverse trigonometric functions (principal value only)
Analytical Geometry
Two Dimensions Cartesian oordinates, distance between two points, section formulae, shift of origin
Equation of a straight line in various forms, angle between two lines, distance of a point from a line Lines through the point of intersection of two given lines, equation of the bisector of the angle between two lines, concurrency of lines, centroid, orthocentre, incentre and circumcentre of a triangle
Equation of a circle in various forms, equations of tangent, normal and chord
Parametric equations of a circle, intersection of a circle with a straight line or a circle, equation of a circle through the points of intersection of two circles and those of a circle and a straight line
Equations of a parabola, ellipse and hyperbola in standard form, their foci, directrices and eccentricity, parametric equations, equations of tangent and normal
JEE ADVANCED
Trang 9Differential Calculus
Real valued functions of a real variable, into, onto and one-to-one functions, sum, difference, product and
quotient of two functions, composite functions, absolute value, polynomial, rational, trigonometric, exponential and logarithmic functions
Limit and continuity of a function, limit and continuity of the sum, difference, product and quotient of two functions, l'Hospital rule of evaluation of limits of functions
Even and odd functions, inverse of a function, continuity of composite functions, intermediate value property of continuous functions
Derivative of a function, derivative of the sum, difference, product and quotient of two functions, chain rule, derivatives of polynomial, rational, trigonometric, inverse trigonometric, exponential and logarithmic functions Derivatives of implicit functions, derivatives up to order two, geometrical interpretation of the derivative, tangents and normals, increasing and decreasing functions, maximum and minimum values of a function, applications of Rolle's Theorem and Lagrange's Mean Value Theorem
Trang 10Topic 1 Complex Number in Iota Form
Objective Questions I (Only one correct option)
1. Let z∈Cwith Im ( )z =10 and it satisfies2
z n
z n i
−+ = −
for some natural number n, then (2019 Main, 12 April II)
(a) n=20 and Re( )z = −10 (b) n=40 and Re( )z =10
(c) n=40 and Re( )z = −10 (d) n=20 and Re( )z =10
2. All the points in the set S i
(a) circle whose radius is 2
(b) straight line whose slope is−1
(c) circle whose radius is 1
(d) straight line whose slope is 1
3. Let z∈Cbe such that| |z<1 Ifω = +
−
5 3
5 1
z z
i x iy(i= −1 , where x and y are real)
numbers, then y−xequals (2019 Main, 11 Jan I)
2
3 2
1 2, : sin
sin is purely imaginary
sinsin
(One or more than one correct option)
10. Let a b x , , and y be real numbers such that a− =b 1 and
y≠0 If the complex number z= +x iy satisfies
Im az b
++
1 , then which of the following is(are)
possible value(s) of x? (2017 Adv.)
(a) 1− 1+ y2 (b)− −1 1− y2
(c) 1+ 1+ y2 (d)− +1 1−y2
Topic 2 Conjugate and Modulus of a Complex Number
Objective Questions I (Only one correct option)
1. The equation|z− =i| |z−1|,i= −1, represents
(a) a circle of radius1
5, then z isequal to (2019 Main, 10 April I)
(a) 15
35
− i (b) − −1
5
3
5i(c) − +1
Trang 113. Let z1and z2be two complex numbers satisfying|z1|=9
and |z2− −3 4i|=4 Then, the minimum value of
|z1−z2|is (2019 Main, 12 Jan II)
α(α ∈R is a purely imaginary number and)
| |z =2, then a value ofαis (2019 Main, 12 Jan I)
6. A complex number z is said to be unimodular, if z ≠1
If z1 and z2are complex numbers such that z z
z z
1 2
22
–– is
unimodular and z2is not unimodular
Then, the point z1lies on a (2015 Main)
(a) straight line parallel to X-axis
(b) straight line parallel to Y -axis
(b) lies in the interval (1, 2)
(c) is strictly greater than 5/2
(d) is strictly greater than 3/2 but less than 5/2
8. Let complex numbers α and 1 /α lies on circles
9. Let z be a complex number such that the imaginary part
of z is non-zero and a=z2+ +z 1 is real Then, a cannot
10. Let z= +x iy be a complex number where, x and y are
integers Then, the area of the rectangle whose vertices
are the root of the equation zz3+zz3 =350, is (2009)
14. For all complex numbers z z1, 2 satisfying | |z1 =12and
|z2− −3 4i|=5, the minimum value of|z1−z2|is
(a) equal to 1 (b) less than 1 (2000, 2M)
(c) greater than 3 (d) equal to 3
16. For positive integers n n1, 2 the value of expression(1+ i)n1 + +(1 i3)n1 +(1+ i5)n2 +(1+i7)n2, here
i= −1 is a real number, if and only if (1996, 2M)
22. The complex numbers z= +x iy which satisfy theequation z i
z i
−+
Trang 12Objective Questions II
(One or more than one correct option)
23. Let S be the set of all complex numbers z satisfying
|z2+ +z 1|=1 Then which of the following statements
(d) The set S has exactly four elements
24. Let s, t, r be non-zero complex numbers and L be the set of
solutions z= +x iy ( ,x y R i∈ , = −1) of the equation
sz+tz+ =r 0, where z= −x iy Then, which of the
following statement(s) is (are) TRUE? (2018 Adv.)
(a) If L has exactly one element, then| | | | s≠ t
(b) If| | | |s =t , then L has infinitely many elements
(c) The number of elements in L∩{ :|z z− + =1 i| 5 is at most 2}
(d) If L has more than one element, then L has infinitely many
elements
25. Let z1 and z2be complex numbers such that z1≠z2and
| | | |z1 = z2 If z1has positive real part and z2has negative
imaginary part, thenz z
(a) zero (b) real and positive
(c) real and negative (d) purely imaginary
26. If z1= +a ib and z2= +c id are complex numbers such
that | | | |z1 = z2 =1 and Re (z z1 2)=0, then the pair of
complex numbers w1= +a ic and w2= +b idsatisfies
(a)| |w1 =1 (b)| |w2 =1 (1985, 2M)
(c) Re (w w1 2)=0 (d) None of these
Passage Based Problems
Read the following passages and answer the questions
29. Let z be any point in A∩B∩C and let w be any point
satisfying |w− − <2 i| 3 Then, | | | |z − w +3 liesbetween
(a)−6 and 3 (b)−3 and 6(c)−6 and 6 (d)−3 and 9
Match the Columns
32. Match the statements of Column I with those ofColumn II
Here, z takes values in the complex plane and Im ( ) z
and Re ( )z denote respectively, the imaginary part and the real part of z (2010)
A. The set of points z satisfying
|z−i z| || | = z+i z| || is contained in or equal to
p an ellipse with eccentricity 4/5
B. The set of points z satisfying
|z+ 4 | + |z− = 4 | 0 is contained in or equal to
q. the set of points z
= + 1is contained in or equal to
Fill in the Blanks
33. Ifα β γ, , are the cube roots of p p, <0, then for any x y,
34. For any two complex numbers z z1, 2 and any real
numbers a and b,| az1−bz2| |2+ bz1+ az2|2=K
(1988, 2M)
Trang 1336. If three complex numbers are in AP Then, they lie on a
circle in the complex plane (1985 M)
37. If the complex numbers, z z1, 2 and z3 represent the
vertices of an equilateral triangle such that
|z1| |= z2| |= z3|,then z1+z2+z3 =0 (1984, 1M)
38. For complex numbers z1=x1+ iy1 and z2=x2+iy2, we
write z1∩z2, if x1≤x2and y1≤y2 Then, for all complex
numbers z with 1∩z, we have1
−+ ∩
z
z . (1981, 2M)
Analytical & Descriptive Questions
39. Find the centre and radius of the circle formed by all the
points represented by z= +x iysatisfying the relation
β ( 1 , where) α and β are the constant
complex numbers given byα α= 1+iα β β2, = 1+ iβ2
41. If z1 and z2 are two complex numbers such that
| |z1 < <1 | |z2, then prove that 1 1 2 1
Show that R is an equivalence relation. (1982, 2M)
46. Find the real values of x and y for which the following
+ −+ +
− +
− =
i x i i
Integer & Numerical Answer Type Question
49. If z is any complex number satisfying | z− −3 2i|≤2,then the maximum value of|2z− +6 5i|is …… (2011)
Topic 3 Argument of a Complex Number
Objective Questions I (Only one correct option)
1. If z z1, 2are complex numbers such that Re( ) |z1 = z1−1|,
Re( ) |z2 = z2−1|and arg(z1 z2)
6
− = π, then Im(z1+z2) isequal to (2020 Main, 3 Sep II)
=3 +
2
23
1
2 2
5. Let z and w be two complex numbers such that| | z ≤1,
| |w ≤1 and|z+i w|=|z−iw|=2 , then z equals
(1995, 2M)
(a) 1 or i (b) i or−i
(c) 1 or−1 (d) i or−1
6. Let z and w be two non-zero complex numbers
such that | | | |z = w and arg ( )z +arg ( )w = π, then z
(1987, 2M)
(a)− π (b)− π
2(c) 0 (d) π
2
Trang 148. If a b c, , and u v w, , are the complex numbers
representing the vertices of two triangles such that
c=(1−r a) +rb and w=(1−r u) + rv , where r is a
complex number, then the two triangles (1985, 2M)
(a) have the same area (b) are similar
(c) are congruent (d) None of these
Objective Questions II
(One or more than one correct option)
9. For a non-zero complex number z, let arg( ) z denote the
principal argument with− <π arg( )z ≤π Then, which of
the following statement(s) is (are) FALSE ? (2018 Adv.)
(a) arg (− −1 )= ,
4
i π where i= −1
(b) The function f R: → − π, π]( , defined by
f t( )=arg (− +1 it) for all t∈R, is continuous at all
points of R, where i= −1
(c) For any two non-zero complex numbers z1 and z2,
arg z arg ( ) arg ( )
(d) For any three given distinct complex numbers z1,z2and
z3, the locus of the point z satisfying the condition
π, lies on a straight line
10. Let z1and z2be two distinct complex numbers and let
z= −(1 t z) 1+tz2for some real number t with 0< <t 1 If
arg (w) denotes the principal argument of a non-zero complex number w, then (2010)
(a) |z−z1| |+ z−z2| |= z1−z2|(b) arg (z−z1)=arg (z−z2)(c) z z z z
Match the Columns
11. Match the conditions/expressions in Column I with
statement in Column II (z≠0 is a complex number)
Analytical & Descriptive Questions
12. | |z ≤1,| |w ≤1 then show that,
|z−w|2≤(| | | |)z − w 2+ ( z− w)2
13. Let z1=10+6i and z2= +4 6i If z is any complex
number such that the argument of (z−z1) / (z−z2)is
π/4, then prove that|z− −7 9i|=3 2 (1991, 4M)
Y
X O
Topic 4 Rotation of a Complex Number
Objective Questions I (Only one correct option)
1. Let S be the set of all complex numbers z satisfying
|z− +2 i|≥ 5 If the complex number z0is such that
respectively denote the real and imaginary parts of z,
(a) R z( )>0 and I z( )>0 (b) I z( )=0
(c) R z( )<0 and I z( )>0 (d) R z( )= −3
3. A particle P starts from the point z0 = +1 2i, where
i= −1 It moves first horizontally away from origin by
5 units and then vertically away from origin by 3 units
to reach a point z1 From z1the particle moves 2 units
in the direction of the vector$i+ $jand then it moves
he walks a distance of 4 units towards the North-West
(N 45° W) direction to reach a point P Then, the position of P in the Argand plane is (2007, 3M)
(a) 3e iπ/4+4i
(b) (3−4i e) iπ/4(c) (4+3i e)iπ/4(d) (3+4i e) iπ/4
5. The shaded region, where P= −( 1 0 Q, ), = − +( 1 2, 2)
< <α π is a fixed angle If P=(cos ,sin )θ θ and
Q={cos(α θ− ),sin(α θ− )}, then Q is obtained from P by
(2002, 2M)
(a) clockwise rotation around origin through an angleα
(b) anti-clockwise rotation around origin through an angleα
(c) reflection in the line through origin with slope tanα
(d) reflection in the line through origin with slope tanα
2
Trang 157. The complex numbers z z1, 2 and z3 satisfying
− = − are the vertices of a triangle which is(2001, 1M)
(a) of area zero (b) right angled isosceles
(c) equilateral (d) obtuse angled isosceles
(c) the X-axis for a≠0,b=0
(d) the Y-axis for a=0,b≠0
π (d) 5
6
π
Fill in the Blanks
10. Suppose z z z1, 2, 3 are the vertices of an equilateral
triangle inscribed in the circle | |z =2 If z 1= +1 i 3,
then z2=K, z3 =… (1994, 2M)
11. ABCD is a rhombus Its diagonals AC and BD intersect
at the point M and satisfy BD=2AC If the points D and
M represent the complex numbers 1+ i and 2−i
respectively, then A represents the complex number
12. If a and b are real numbers between 0 and 1 such that
the points z1= +a i z, 2= +1 bi and z3 =0 form an
equilateral triangle, then a=Kand b=K (1990, 2M)
Analytical & Descriptive Questions
13. If one of the vertices of the square circumscribing thecircle |z− =1| 2 is 2+ 3i Find the other vertices of
14. Let bz+ bz=c , b≠0, be a line in the complex plane,
where b is the complex conjugate of b If a point z1is the
reflection of the point z2through the line, then show
that c=z b1 +z b2 (1997C, 5M)
15. Let z1and z2be the roots of the equation z2+ pz+ =q 0,
where the coefficients p and q may be complex numbers Let A and B represent z1and z2in the complex plane If
∠AOB= ≠α 0 and OA=OB , where O is the origin prove that p2 4q 2
17. Show that the area of the triangle on the argand
diagram formed by the complex number z iz, and z+iz
is12
2
| |z
(1986, 2 1
2 M)
18. Prove that the complex numbers z z1, 2and the origin
form an equilateral triangle only if z12+z22−z z1 2=0
(1983, 2M)
19. Let the complex numbers z z1, 2and z3be the vertices of
an equilateral triangle Let z0be the circumcentre of the
triangle Then, prove that z1+ z2+z3 =3z0 (1981, 4M)
Integer & Numerical Answer Type Questions
20. For a complex number z, let Re( ) z denote the real part of
z Let S be the set of all complex numbers z satisfying
z4 −| |z4=4i z2, where i= −1 Then the minimumpossible value of|z1−z2|, where z z1, 2∈Swith Re( )z1 >0and Re( )z2 <0is …… (2020 Adv.)
21. For any integer k, letαk π
4 1 1
3
4 2
is
(2016 Adv.)
Trang 16Topic 5 De-Moivre’s Theorem, Cube Roots and nth Roots of Unity
Objective Questions I (Only one correct option)
1. The value of
9
29
9
29
3
+ ++ −
π π
π π
i i
2( 3−i) (d)1
2(1−i 3)
2. If z and w are two complex numbers such that | zw|=1
and arg( )z −arg( )w = π
2, then (2019 Main, 10 April II)
4. Let z0be a root of the quadratic equation, x2+ + =x 1 0, If
z= +3 6iz081−3iz093, then arg z is equal to
(2019 Main, 9 Jan II)
6. The minimum value of |a+bω+ cω2|,where a, b and c
are all not equal integers andω(≠1 is a cube root of)
unity, is
(2005, 1M)
(a) 3 (b)1
2 (c) 1 (d) 0
7. Ifω(≠1 be a cube root of unity and () 1+ω2)n=(1+ω4)n,
then the least positive value of n is (2004, 1M)
(a) 2 (b) 3 (c) 5 (d) 6
8. Let ω = − +1
2
32
i , then value of the determinant
32
12. Ifω(≠1 is a cube root of unity and () 1+ω)7=A+B ,ω
then A and B are respectively
Match the Columns
14. Let z k=cos2k +isin k ;k= , ,…
Q. There exists a k∈ { , , 1 2 … , } 9 such that
z1⋅ =z z k has no solution z in the set of
(a) (i) (ii) (iv) (iii)(b) (ii) (i) (iii) (iv)(c) (i) (ii) (iii) (iv)(d) (ii) (i) (iv) (iii)
Fill in the Blanks
15. Letωbe the complex number cos2 sin
3
23
+++
=
1
11
Trang 1716. The value of the expression
17. The cube roots of unity when represented on Argand
diagram form the vertices of an equilateral triangle
(1988, 1M)
Analytical & Descriptive Questions
18. Let a complex numberα α, ≠1, be a root of the equation
z p+q−z p−z q+ =1 0
where, p and q are distinct primes Show that either
1+ +α α2+ +αp−1=0
or 1+ +α α2+ +αq−1=0
but not both together (2002, 5M)
19. If 1, a a1, 2, ,a n−1are the n roots of unity, then show
that (1−a1) (1−a2) (1−a3)K(1−a n−1)=n (1984, 2M)
20. It is given that n is an odd integer greater than 3, but n
is not a multiple of 3 Prove that x3 +x2+xis a factor of(x+1)n−x n−1 (1980, 3M)
21. If x= +a b , y=aα+ bβ, z=aβ+bα, where α β, arecomplex cube roots of unity, then show that
Integer & Numerical Answer Type Questions
22. Letω ≠1 be a cube root of unity Then the minimum ofthe set {|a+bω+ cω2 2| : a b c, , distinct non-zerointegers} equals (2019 Adv.)
23. Let ω=e i /3π and a b c x y z, , , , , be non-zero complex
numbers such that a+ + =b c x a, + bω+cω2=y,
a+bω2+ cω=z.Then, the value of| | | | | |
Trang 18Topic 1 Complex Number in Iota Form
⇒x+iy= +i i
+ =
− ++
(α ) ( ) ( )
α
α αα
αα
2
11
2
αα
Now, x2+y2= α
α
αα
2
2 2
2
2
11
21
−+
(( )
α ;α R lies on a circle with radius 1.
3. Given complex number
On equating real and imaginary part, we get
sinsin
θθ
θθ
(rationalising the denominator)
θ θθ
i
[Qa2−b2=(a+ b a)( −b)andi2= −1]
= −+
sinsin
θθ
sinsin
θθ
= + +
− +
( sin ( sin )( sin ) ( sin
– 3/2 √
√ 3/2
Trang 19θ θθ
= −+ + +
sinsin
θθ
θθ
333
11
i i
11
11
n n
i i n
21
2 21
i a i a
ai a
2
++ =
a a
1
25
[Qif z= +x iy , then z= −x iy]
3. Clearly| |z1 =9, represents a circle having centre C1( , )0 0
and radius r1=9.and |z2− −3 4i|=4 represents a circle having centre
C2( , ) and radius r3 4 2=4.The minimum value of |z1−z2|is equals to minimumdistance between circles| |z1 =9and|z2− −3 4i|=4
Q C C1 2= (3 0− )2+ −(4 0)2= 9+16= 25=5and |r1− = − = ⇒r2| |9 4| 5 C C1 2= −|r1 r2|
∴Circles touches each other internally
Hence, |z1−z2|min=0
4. Since, the complex number z
z
−+
α
α (α ∈R is purely)
imaginary number, therefore
z z
z z
−+ + −+ =
αα
Trang 20∴ Point z1lies on a circle of radius 2.
7. | |z ≥2 is the region on or outside circle whose centre is
2
= − +
21 2 + =0
32
2
Geometrically Min z+ 1 =AD
2
Hence, minimum value of z+1
2 lies in the interval(1, 2)
8. PLAN Intersection of circles, the basic concept is to solve the
equations simultaneously and using properties of modulus of complex numbers.
For roots to be real b2− 4ac≥ 0
Description of Situation As imaginary part of
Trang 211− 2=1 2 2
+
− +
cos sin(cos sin )
θ θ
θ θ θ
i i
2 2 2
= +
− +
cos sinsin (cos sin )
11
[n C n C i n C i K]
+2 2 + 2 + 2 +
2 4 4
Trang 2217. Since, (sinx+ icos2x)=cosx−isin2x
⇒ sinx i− cos2x=cosx i− sin2x
⇒ sinx=cosxand cos2x=sin2x
⇒ tan x=1 and tan 2x=1
⇒ x= π /4 and x= π /8 which is not possible at same
time
Hence, no solution exists
18. Since, z z z1, 2, 3,z4are the vertices of parallelogram
∴It is a perpendicular bisector of ( , )0 1 and ( ,0 −1)
i.e X-axis Thus, z lies on the real axis.
20. Given,|z−4| |< z−2|
Since, |z−z1| |> z−z2|represents the region on right
side of perpendicular bisector of z1and z2
bisector of z1and z2]
∴Perpendicular bisector of (0, 5) and (0, – 5) is X-axis.
23. It is given that the complex number Z, satisfying
12
34
74
2
72
Q |z1+z2| || | | ||≥ z1 − z2
∴ z+ 1 ≥ z −
2
12
| |
⇒ | |z −1 ≤ +z ≤
2
12
72
⇒ − 7≤ − ≤
2
12
72
| |z
⇒ 1 7
2
7 12
Trang 23Q z+ 1 + ≤ z+ +
2
34
12
34
⇒ 1 1
2
34
12
72
| |s2−| |t2≠0 ⇒ | | | |s ≠ t
It is true
(b) If| | | |,s= t then rt−rsmay or may not be zero
So, z may have no solutions.
∴L may be an empty set.
It is false
(c) If elements of set L represents line, then this line
and given circle intersect at maximum two point
Hence, it is true
(d) In this case locus of z is a line, so L has infinite
elements Hence, it is true
⇒ a
b
d c
−+ = −
Set B consists of points lying on the circle, centre at
(2, 1) and radius 3
i.e x2+ y2−4x−2y=4 …(ii) Set C consists of points lying on the x+ =y 2 …(iii)
Clearly, there is only one point of intersection of the line
Y
Y′
X
X′
Trang 242
52cosθ + isinθ
//
D Let w=cosθ+isinθ
+ + =
+ +
( ) ( ) ( )( )
= + ++ +
sinxsinx+cosxcosx+ sinxcosx=
⇒sinx sinx cosx cos x sin x cosx
Thus, f t( ) changes sign from negative to positive in theinterval (1, 2)
∴Let t=kbe the root for which
36. Since, z z z1, 2, 3 are in AP
z
− −+ + ∩
x iy x iy
Trang 252 2
α2 2β 2 2
∴ Centre for Eq (i)= −
k k
1
⇒ | |z >1,
3 which contradicts …(ii)
∴There exists no complex number z such that
z z
1 2 1 2
If zw=1, then zw=1 and
LHS=zz w−ww z= ⋅ − ⋅z 1 w 1
= −z w= −z w= =0 RHS
Hence proved.
Trang 26= ++
= ⇒ zw=zw …(i)Again, | |z w2 −| |w z2 = z −w
NOTE It is a compound equation, therefore we can generate
from it more than one primary equations.
On equating real and imaginary parts, we get
Here, let z1=x1+iy z1, 2=x2+iy2and z3 =x3 +iy3
1 1 2 2
= (i)Similarly, z Rz2 3
y
x y
2
2 3
1 1 3 3
= ⇒ z Rz1 3Thus, z Rz1 2and z Rz2 3⇒z Rz1 3 [transitive]
Hence, R is an equivalence relation.
46. (1 ) 2 ( )
3
2 33
+ −+ +
− +
− =
i x i i
2 2 2
2 2sin sin cos
sin cossin c
θ θ θ
θ θθ
Trang 27θ θ
θ 2 θ θ
222
θ θ
θ2 θ
222
θ
θθ
1 2 1 2
Topic 3 Argument of a Complex Number
1. Let the complex numbers
2. (*) Given, 3| |z1 =4| |z2 ⇒| |
| |
z z
1
2
43
1 2 1 2
= θandz
z
z
z e i
2 1 2 1
=3 +
2
23
1 2 2 1
[Q e±iθ=(cosθ±isin ]θ)
=5 +
2
32cosθ isinθ
⇒ | |z = 52 + = =
32
344
172
z z=arg 1 arg ( )
1 1
++
O –θ
π − θ
r
( )z Y
Y′
X
X′
Trang 285. Given, |z+iw| |= z−iw|=2
⇒ |z− −( iw)| |= z−(iw)|=2
⇒ |z− −( iw)| |= z− −( iw)|
∴ z lies on the perpendicular bisector of the line joining
−iw and−iw Since,−iw is the mirror image of−iw in
the X-axis, the locus of z is the X-axis.
∴ z=1 or −1 is the correct option
6. Since,| | | |z = w and arg ( )z = −π arg ( )w
Let w=re iθ, then w=re–iθ
∴ z=re i( π θ − )=re iπ⋅e−iθ= −re−iθ= −w
7. Given,|z1+ z2| | | | |= z1 + z2
On squaring both sides, we get
| |z1 +| |z2 +2| || | cos (z1 z2 argz1−argz2)
=| |z1 +| |z2 +2| || |z1 z2
⇒ 2| || |cos (z1 z2 argz1−argz2)=2| || |z1 z2
⇒ cos (argz1−argz2)=1
c (1 r a) rb
u v
w−(1−r u) −rv −( −r)−r
11
u v
0 0
110
0 [from Eq (i)]
9. (a) Let z= − −1 iand arg(z)= θ
This function is discontinuous at t=0
(c) We have,
arg z
1 2
=arg( )z1 −arg( )z2 +2nπ−arg( )z1 +arg( )z2 =2nπ
So, given expression is multiple of 2π.(d) We have, arg ( ) ( )
Trang 29Clearly, z divides z1 and z2 in the ratio of t: (1−t),
0< <t 1
⇒ AP+BP=AB i.e |z−z1| |+ −z z2| |= z1−z2|
⇒ Option (a) is true
and arg (z−z1)=arg (z2−z)=arg (z2−z1)
⇒ Option (b) is false and option (d) is true
Also, arg (z−z1)=arg (z2−z1)
⇒ a= = −b 2 3 [Qa b, ←( , )]0 1
12. Let z=r1(cosθ1+ isinθ1)and w=r2(cosθ2+ isinθ2)
We have,| |z =r1,| |w =r2,arg ( )z = θ1and arg ( )w = θ2
=r1 cos2θ1+ r2cos2θ2−2r r1 2cosθ1cosθ2
+ r1sin2θ1 +r2sin2θ2−2r r1 2sinθ1sinθ2
=r1(cos2θ1 +sin2θ1)+ r2(cos2θ2+sin2θ2)
−2r r (cos1 2 θ1cosθ2+sinθ1sinθ2)
π represents locus of z is a circle
shown as from the figure whose centre is (7, y) and
∠AOB=90 , clearly OC° =9 ⇒ OD= + =6 3 9
∴ Centre=( , )7 9 and radius= 6 =
2 3 2
⇒ Equation of circle is|z−(7+9i)|=3 2
Topic 4 Rotation of a Complex Number
1. The complex number z satisfying| z− + ≥2 i| 5, whichrepresents the region outside the circle (including thecircumference) having centre ( ,2 −1)and radius 5 units
Now, for z0∈S 1
1
0
|z − |is maximum.
When |z0−1|is minimum And for this it is required
that z0∈S , such that z0 is collinear with the points( ,2 −1) and ( , )1 0 and lies on the circumference of thecircle|z− + =2 i| 5
t: (1 - )t
z C
Trang 30So let z0= +x iy, and from the figure 0< <x 1and y>0.
−+
( )( )
x
x y
=cos5π+ sin π +cos π − sin π
6
56
56
56
[Q e iθ=cosθ+isin ]θ
=2 5
6cos π
4. Let OA=3, so that the complex number associated with
A is 3 e iπ/4 If z is the complex number associated with P,
then
z e
i i
43
As we know equation of circle
having centre z0and radius r,
|arg z( +1)|≤π/4
6. In the Argand plane, P is represented by e i0 and Q is represented by e i(α θ− )
Now, rotation about a line with angleα is given by
eθ→e(α θ− ) Therefore, Q is obtained from P by reflection
in the line making an angleα/2
= −+
1 3
2 1 3
2
i i
=+
2 2 2 and y bt
a b t
= −+
2 2 2
⇒ y
x
bt a
5 (6,2)
z0 (1 )
Y P
Trang 31On putting x a
a b t
=+
2 2 2
∴Option (a) is correct
For a≠0 and b=0,
x iy a
9. PLAN It is the simple representation of points on Argand plane and
to find the angle between the points.
76
=cosπ+ sin , cosπ π+ sin π,cos π +i sin7
Now, the triangle z z1, 2and z3being an equilateral and
the sides z z1 2and z z1 3make an angle 2π/ at the centre.3Therefore, ∠POz2= + =
3
23
π π π
and ∠POz3 = + + =
3
23
23
53
32
Whenever vertices of an equilateral triangle having
centroid is given its vertices are of the form z z, ω ω,z 2
∴If one of the vertex is z1= +1 i 3, then other twovertices are (z1ω), (z1ω2)
A D(1+ )i
C B
M
(2 i− )
Trang 32Now, let coordinate of A be ( x+iy).
But in a rhombus AD= AB, therefore we have
Since, M is the centre of rhombus.
∴ By rotating D about M through an angle of± π/2 , we
get possible position of A.
14. Let Q be z2and its reflection be the point P z( )1 in the
given line If O z( ) be any point on the given line then by
definition OR is right bisector of QP.
∴ OP=OQ or |z−z1| |= z−z2|
⇒ |z−z1| =|z−z|
2
2 2
⇒ (z−z1) (z−z1)=(z−z2) (z−z2)
⇒ z z( 1−z2)+z z( 1−z2)=z z1 1−z z2 2Comparing with given line zb+zb=c
z z b
z z b
z z z z c
(1, 0)
Y
Trang 331 2
11
+
− =
+ ++ −
cos sincos sin
⇒ p2cosec ( / )2α2 =4qcot ( / )2α2 ⇒ p2=4qcos2α/2
16. Since, triangle is a right angled isosceles triangle
∴ Rotating z2 about z3 in anti-clockwise direction
through an angle ofπ/ ,2 we get
17. We have, iz=ze iπ /2 This implies that iz is the vector
obtained by rotating vector z in anti-clockwise direction
through 90° Therefore, OA⊥AB So,
Area of∆OAB=1 ×
2OA OB=1 =
2
12
2
| || |z iz | |z
18. Since, z z1, 2and origin form an equilateral triangle
Q if z z z z
⇒ z=0, which is not possible according to givenconditions
Case-II, if (z−z) (z+z)=4 and z i = +x iy
So, (2iy) (2x)=4i
⇒xy=1 is an equation of rectangular hyperbola and forminimum value of |z1−z2|2, the z1 and z2 must bevertices of the rectangular hyperbola
k
π π
∴ αkare vertices of regular polygon having 14 sides
Let the side length of regular polygon be a.
∴ αk+ 1−αk =length of a side of the regular polygon
=a …(i)and α4k− 1−α4k− 2 = length of a side of the regularpolygon
4 1 4 2 1
3
12
( )( )
z + iz Y
Y′
Trang 34Topic 5 De-Moivre’s Theorem, Cube Roots
and nth Roots of Unity
1. Given expression
9
29
9
29
π π
π π
i i
29
291
π i π π i π
29
29
29
29
29
3 2
2. It is given that, there are two complex numbers z and w,
such that|z w|=1 and arg( )z −arg( )w = π/2
∴ | || |z w =1 [ |Q z z1 2| |= z1||z2|]
and arg( )z =π +arg(w)
2Let| |z =r, then| |w
[Q if z= +x iy is a complex number, then it can be
written as z=re iθwhere, r=| |andz θ =arg ( )z ]
r e i
23
56
56cos π isin π cos π isin π
+ +
cos11 sin6
116
32
32
12
=cos3π+isin3π [Q for any natural number ‘n’
(cosθ+isin )θn=cos(nθ)+ isin(nθ)]
1
15 1
Trang 3515 2222
Because difference of integers=integer
⇒(b−c)2≥1 {as minimum difference of two consecutive
integers is (±1 also ()} c−a)2≥1
and we have taken a=b ⇒(a−b)2=0
From Eq (i), z2≥1 + +
2(0 1 1) ⇒ z2≥1Hence, minimum value of| |z is 1
121
111
2
2
2
ωω
ωω
2 1
32
2
11
2
2
11
k
=
2 10
π
( )
z kis 10th root of unity
⇒ z kwill also be 10th root of unity
Taking, z as z , we have z ⋅z =1 (True)
Trang 36(i) 1 − cos 2 θ = 2 sin2θ
(ii) sin 2 θ = 2 sin cos θ θ and
i k = 2
10 sin πk
Now, required product is
810
91010
9sin π ⋅sin π⋅sin π Ksin π⋅sin π
310
410
51010
2
10 110
2510
5 14
A zl
+++
= + =
1
11
12
32
simultaneously as p and q are distinct primes, so neither p divides q nor q divides p, which is the
Trang 3720. Since, n is not a multiple of 3, but odd integers and
Qa b, and c are distinct non-zero integers.
For minimum value a=1, b=2 and c=3
π
Then,| | | | | | |( ) ( ) | |
π
, then only integer solution exists.
Trang 38Topic 1 Quadratic Equations
Objective Questions I (Only one correct option)
1. Suppose a b, denote the distinct real roots of the
quadratic polynomial x2+20x−2020and suppose c d,
denote the distinct complex roots of the quadratic
polynomial x2−20x+2020 Then the value of
ac a( − +c) ad a( −d)+ bc b( − +c) bd b( −d)is (2020 Adv.)
(a) 0 (b) 8000
(c) 8080 (d) 16000
2. Letλ ≠0 be in R Ifαandβare the roots of the equation,
x2− +x 2λ=0andαandγare the roots of the equation,
3x2−10x+27λ =0, thenβγ
λ is equal to(2020 Main, 4 Sep II)
(a) 36 (b) 9
(c) 27 (d) 18
3. The set of all real values ofλ for which the quadratic
equations, (λ2+1)x2−4λx+ =2 0always have exactly
one root in the interval (0, 1) is (2020 Main, 3 Sep II)
(a) (0, 2) (b) (− −3, 1)
(c) ( , ]2 4 (d) ( , ]1 3
4. If α and β are the roots of the quadratic equation,
x2+ xsinθ−2sinθ=0,θ∈0,π2, then
6 12
(sinθ + )(c) 2
12 6
(sinθ − )
5. Let p, q∈R If2− 3is a root of the quadratic equation,
x2+ px+ =q 0, then (2019 Main, 9 April I)
(a) q2−4p−16=0 (b) p2−4q−12=0
(c) p2−4q+12=0 (d) q2+ 4p+14=0
6. If m is chosen in the quadratic equation
(m2+1)x2−3x+(m2+1)2=0 such that the sum of its
roots is greatest, then the absolute difference of the
cubes of its roots is (2019 Main, 9 April II)
(a) 10 5 (b) 8 5 (c) 8 3 (d) 4 3
7. Ifαandβare the roots of the equation x2−2x+ =2 0,
then the least value of n for which α
(a) 3 (b) infinitely many(c) 1 (d) 2
9. The number of integral values of m for which the
quadratic expression, (1+2m x) 2−2 1( +3m)
x+4 1( +m ), x∈R, is always positive, is
(2019 Main, 12 Jan II)
(a) 6 (b) 8 (c) 7 (d) 3
10. Ifλbe the ratio of the roots of the quadratic equation in
x, 3 m x2 2+m m( −4)x+ =2 0, then the least value of m for
whichλ
λ+ 1=1, is (2019 Main, 12 Jan I)
(a)− +2 2 (b) 4−2 3(c) 4−3 2 (d) 2− 3
11. If one real root of the quadratic equation
81x2+kx+256=0is cube of the other root, then a value
of k is (2019 Main, 11 Jan I)
(a) 100 (b) 144 (c) −81 (d) −300
12. If 5 5 5, r, r are the lengths of the sides of a triangle, then2
r cannot be equal to (2019 Main, 10 Jan I)
13. The value ofλsuch that sum of the squares of the roots
of the quadratic equation, x2+(3−λ)x+ =2 λ has theleast value is (2019 Main, 10 Jan II)
(a)4
9 (b) 1 (c)
15
8 (d) 2
14. The number of all possible positive integral values ofα
for which the roots of the quadratic equation,
6x2−11x+ =α 0are rational numbers is
(2019 Main, 9 Jan II)
(a) 5 (b) 2 (c) 4 (d) 3
Theory of Equations
2
Trang 3915. Letαandβbe two roots of the equation x2+2x+ =2 0,
thenα + β15 15is equal to (2019 Main, 9 Jan I)
(a) 256 (b) 512
(c) −256 (d) −512
16. Let S={x∈R:x≥0 and 2| x− +3| x( x−6)+ =6 0
(a) is an empty set
(b) contains exactly one element
(c) contains exactly two elements
(d) contains exactly four elements
17. If α β, ∈C are the distinct roots of the equation
x2− + =x 1 0, thenα101+β107is equal to (2018 Main)
6 12 Supposeα1andβ1are the roots of the
equation x2−2xsecθ+ =1 0, andα2andβ2are the roots
of the equation x2+2xtanθ− =1 0 If α1>β1 and
α2>β2, thenα1+β2equals (2016 Adv.)
(a) 2(secθ−tan )θ (b) 2secθ
(c)−2tanθ (d) 0
21 Letα andβbe the roots of equation x2−6x− =2 0 If
a n=αn−βn , for n≥1, then the value of a a
a
9
22
− is
(2015 Main)
(a) 6 (b) – 6 (c) 3 (d) – 3
22. In the quadratic equation p x( )=0 with real coefficients
has purely imaginary roots Then, the equation
p p x[ ( )]=0 has (2014 Adv.)
(a) only purely imaginary roots
(b) all real roots
(c) two real and two purely imaginary roots
(d) neither real nor purely imaginary roots
23. Letα andβ be the roots of equation px2+ qx+ =r 0,
p≠0 If p q, and r are in AP and 1 1 4
2 139
24. Letαandβbe the roots of x2−6x− =2 0, withα β> If
a n=αn−βn for n≥1 , then the value ofa a
a
9
22
− is
(a) 1 (b) 2 (c) 3 (d) 4 (2011)
25. Let p and q be real numbers such that p≠0, p3 ≠qand
p3 ≠ −q If α and β are non-zero complex numberssatisfyingα β+ = −p andα3+β3 =q, then a quadratic
26. Letα,βbe the roots of the equation x2−px+ =r 0and
27. If a b c, , are the sides of a triangle ABC such that
x2−2(a+ +b c x) +3λ (ab+bc+ca)=0 has real roots,
3
53,
28. If one root is square of the other root of the equation
x2+ px+ =q 0, then the relation between p and q is
31. For the equation 3x2+ px+ =3 0,p>0, if one of the root
is square of the other, then p is equal to (2000, 1M)
34. The equation x x x
3 4
5 4
Trang 4035. Ifαandβare the roots of x2+ px+ =q 0andα β4, 4are
the roots of x2−rx+ =s 0, then the equation
x2−4qx+2q2− =r 0has always (1989, 2M)
(a) two real roots (b) two positive roots
(c) two negative roots
(d) one positive and one negative root
(a) no root (b) one root
(c) two equal roots (d) infinitely many roots
37. For real x, the function( )( )
(a) positive (b) negative
(c) real (d) None of these
40. Let a>0,b>0and c>0 Then, both the roots of the
equation ax2+ bx+ =c 0 (1979, 1M)
(a) are real and negative (b) have negative real parts
(c) have positive real parts (d) None of the above
Assertion and Reason
For the following question, choose the correct answer
from the codes (a), (b), (c) and (d) defined as follows :
(a) Statement I is true, Statement II is also true;
Statement II is the correct explanation of Statement I
(b) Statement I is true, Statement II is also true;
Statement II is not the correct explanation of
Statement I
(c) Statement I is true; Statement II is false
(d) Statement I is false; Statement II is true
41. Let a b c p q, , , , be the real numbers Supposeα β, are the
roots of the equation x2+2px+ =q 0
Fill in the Blanks
42. The sum of all the real roots of the equation
|x−2|2+|x−2|− =2 0is…… (1997, 2M)
43. If the products of the roots of the equation
x2−3kx+2e2logk− =1 0is 7, then the roots are real for
44. If 2+i 3is a root of the equation x2+ px+ =q 0, where
p and q are real, then ( , ) p q =(…,…) (1982, 2M)
45. The coefficient of x99in the polynomial(x−1)(x−2) (x−100 is ) (1982, 2M)
Analytical & Descriptive Questions
48. If x2−10ax−11b=0have roots c and d x2−10cx−11d=0
have roots a and b, then find a+ + +b c d (2006, 6M)
49. If α β, are the roots of ax2+ bx+ =c 0, (a≠0) and
α δ β δ+ , + are the roots of Ax2+Bx+C=0, (A≠0) forsome constantδ, then prove that
b ac a
A
2
2 2
2
− = −
(2000, 4M)
50. Let f x( )=Ax2+Bx+C where, A B C, , are real
numbers prove that if f x ( ) is an integer whenever x is
an integer, then the numbers 2 A A, + B and C are all
integers Conversely, prove that if the numbers
2 A A, +B and C are all integers, then f x( ) is an integer
56. If one root of the quadratic equation ax2+bx+ =c 0 is
equal to the nth power of the other, then show that
57. Ifαandβare the roots of x2+ px+ =q 0andγ δ, are the
roots of x2+rx+ =s 0, then evaluate (α γ β γ− ) ( − )(α δ− )(β δ− )in terms of p q r , , and s. (1979, 2M)
58. Solve 2logx a+logax a+3logb a=0,
where a>0,b=a x2 (1978 , 3M )
59. If α and β are the roots of the equation
x2+ px+ =1 0; ,γ δare the roots of x2+qx+ =1 0, then
q2−p2=(α γ β γ α δ β δ− )( − )( + )( + ) (1978, 2M)