1. Complex Number 2
2. Theory of Equation (Quadratic Equation) 3
3. Sequence & Progression(AP, GP, HP, AGP, Spl. Series) 4
4. Permutation & Combination 5
5. Determinant 6
6. Matrices 7
7. Logarithm and their properties 9
8. Probability 9
9. Function 11
10 Inverse Trigonometric Functions 14
11. Limit and Continuity & Differentiability of Function 16
12. Differentiation & L' hospital Rule 18
13. Application of Derivative (AOD)14. Integration (efinite 20
& Indefinite) 22
15. Area under curve (AUC) 24
16. Differential Equation 25
17. Straight Lines & Pair of Straight Lines 26
18. Circle 28
19. Conic Section (Parabola 30, Ellipse 32, Hyperbola 33) 30
20. Binomial Theorem and Logarithmic Series 35
, (vi) There exists a one-one correspondence be
1.COMPLEX NUMBERS (b)
complex numbers.
Trignometric / Polar Representation :
1. DEFINITION : Complex numbers are definited as expressions of the form a + ib where a, b ∈ R &
z = r (cos θ + i sin θ) where | z | = r ; ar
i = −1 . It is denoted by z i.e. z = a + ib. ‘a’ is called as real part of z (Re z) and ‘b’ is called as
Note: cos θ + i sin θ is also written as C
imaginary part of z (Im z). www.MathsBySuhag.com , www.TekoClasses.com
EVERY COMPLEX NUMBER CAN BE REGARDED AS eix + e −ix eix − e
Also cos x = & sin x =
2 2
Purely real Purely imaginary Imaginary (c) Exponential Representation :
if b = 0 if a = 0 if b ≠ 0 z = reiθ ; | z | = r ; arg z = θ ; z = re
Note : 6. IMPORTANT PROPERTIES OF CON
(a) The set R of real numbers is a proper subset of the Complex Numbers. Hence the Complete Number If z , z1 , z2 ∈ C then ;
system is N ⊂ W ⊂ I ⊂ Q ⊂ R ⊂ C.
(b) Zero is both purely real as well as purely imaginary but not imaginary. (a) z + z = 2 Re (z) ; z − z = 2 i Im (z)
(c) i = −1 is called the imaginary unit. Also i² = − l ; i3 = −i ; i4 = 1 etc.
(d) z1 − z 2 = z1 − z 2 ; z1 z 2 = z1 . z 2
a b = a b only if atleast one of either a or b is non-negative.www. Maths By Suhag .com
2. CONJUGATE COMPLEX :
If z = a + ib then its conjugate complex is obtained by changing the sign of its imaginary part & (b) | z | ≥ 0 ; | z | ≥ Re (z) ; | z | ≥ Im (z) ;
is denoted by z . i.e. z = a − ib.
z1 |z
Note that : www.MathsBySuhag.com , www.TekoClasses.com | z1 z2 | = | z1 | . | z2 | ; =
(i) z + z = 2 Re(z) (ii) z − z = 2i Im(z) (iii) z z = a² + b² which is real z2 |z
(iv) If z lies in the 1st quadrant then z lies in the 4th quadrant and − z lies in the 2nd quadrant.
3. ALGEBRAIC OPERATIONS : | z1 + z2 |2 + | z1 – z2 |2 = 2 [| z1 |2 + | z 2 |2 ]
The algebraic operations on complex numbers are similiar to those on real numbers treating i as a
polynomial. Inequalities in complex numbers are not defined. There is no validity if we say that complex z1− z2 ≤ z1 + z2 ≤ z1 + z
number is positive or negative. (c) (i) amp (z1 . z2) = amp z1 + amp z2 +
e.g. z > 0, 4 + 2i < 2 + 4 i are meaningless . z
However in real numbers if a2 + b2 = 0 then a = 0 = b but in complex numbers, (ii) amp 1 = amp z1 − amp z2 + 2 k
z2
z12 + z22 = 0 does not imply z1 = z2 = 0.www.MathsBySuhag.com , www.TekoClasses.com (iii) amp(zn) = n amp(z) + 2kπ .
4. EQUALITY IN COMPLEX NUMBER : where proper value of k must be c
Two complex numbers z1 = a1 + ib1 & z2 = a2 + ib2 are equal if and only if their real & imaginary (7) VECTORIAL REPRESENTATION OF
parts coincide. Every complex number can be considered
5. REPRESENTATION OF A COMPLEX NUMBER IN VARIOUS FORMS: →
(a) Cartesian Form (Geometric Representation) : represents the complex number z then, O
Every complex number z = x + i y can be represented by a point on the → →
cartesian plane known as complex plane (Argand diagram) by the ordered NOTE :(i) If OP = z = r ei θ then OQ = z1 = r
, = 0. This is also the condition for three co
− 1+ i 3 − 1− i 3
9. CUBE ROOT OF UNITY :(i)The cube roots of unity are 1 , , . (G) Complex equation of a straight line
2 2
(ii) If w is one of the imaginary cube roots of unity then 1 + w + w² = 0. In general z (z1 − z 2 ) − z (z1 − z 2 ) + (z1z 2 − z1z 2 ) = 0, w
1 + wr + w2r = 0 ; where r ∈ I but is not the multiple of 3. α z + α z + r = 0 where r
(iii) In polar form the cube roots of unity are : (H) The equation of circle having centre z
2π 2π 4π 4π z z − z0 z − z 0 z + z 0 z0 − ρ² = 0 which is of
cos 0 + i sin 0 ; cos + i sin , cos + i sin
3 3 3 3
(iv) The three cube roots of unity when plotted on the argand plane constitute the verties of an equilateral αα −r . Circle will be real if α α −
triangle.www.MathsBySuhag.com , www.TekoClasses.com (I) The equation of the circle described on th
(v) The following factorisation should be remembered :
(a, b, c ∈ R & ω is the cube root of unity) z − z2 π
(i) arg = ± or (z − z1) ( z −
a3 − b3 = (a − b) (a − ωb) (a − ω²b) ; x2 + x + 1 = (x − ω) (x − ω2) ; z − z1 2
a3 + b3 = (a + b) (a + ωb) (a + ω2b) ; (J) Condition for four given points z1 , z2 , z3
a3 + b3 + c3 − 3abc = (a + b + c) (a + ωb + ω²c) (a + ω²b + ωc) z 3 − z1 z 4 − z 2
10. nth ROOTS OF UNITY :www.MathsBySuhag.com , www.TekoClasses.com . is real. Hence the equatio
z 3 − z 2 z 4 − z1
If 1 , α1 , α2 , α3 ..... αn − 1 are the n , nth root of unity then :
(i) They are in G.P. with common ratio ei(2π/n) & (z − z 2 ) (z3 − z1 ) (z −
⇒
(ii) 1p + α 1p + α 2p + .... +α pn − 1
= 0 if p is not an integral multiple of n
taken as
(z − z1 ) (z3 − z 2 ) is real (z −
= n if p is an integral multiple of n 13.(a) Reflection points for a straight line :
(iii) (1 − α1) (1 − α2) ...... (1 − αn − 1) = n & given straight line if the given line is the right bisec
(1 + α1) (1 + α2) ....... (1 + αn − 1) = 0 if n is even and 1 if n is odd. complex numbers z1 & z2 will be the refle
(iv) 1 . α1 . α2 . α3 ......... αn − 1 = 1 or −1 according as n is odd or even. ; α z + α z + r = 0 , where r is real and α
11. THE SUM OF THE FOLLOWING SERIES SHOULD BE REMEMBERED : 1 2
(b) Inverse points w.r.t. a circle :www.Mat
sin (nθ 2 ) n +1 Two points P & Q are said to be inverse w
cos θ + cos 2 θ + cos 3 θ + ..... + cos n θ = cos θ.
sin (θ 2 )
(i)
2 (i) the point O, P, Q are collinear and on
Note that the two points z1 & z2 will be t
sin (nθ 2 ) n + 1 zz + αz+αz + r =0 if and only if z1 z 2 + αz1
sin θ + sin 2 θ + sin 3 θ + ..... + sin n θ = sin θ.
sin (θ 2 )
(ii)
2 14. PTOLEMY’S THEOREM :www.Math
Note : If θ = (2π/n) then the sum of the above series vanishes. It states that the product of the length
12. STRAIGHT LINES & CIRCLES IN TERMS OF COMPLEX NUMBERS : circle is equal to the sum of the lengt
nz + mz 2 i.e. z1 − z3 z2 − z4 = z1 − z2 z
(A) If z1 & z2 are two complex numbers then the complex number z = 1 divides the joins of z1
m+n 15. LOGARITHM OF A COMPLEX QU
& z2 in the ratio m : n. 1
Note:(i) If a , b , c are three real numbers such that az1 + bz2 + cz3 = 0 ; where a + b + c = 0 (i) Loge (α + i β) = Loge (α² + β²) + i 2nπ
2
and a,b,c are not all simultaneously zero, then the complex numbers z1 , z2 & z3 are collinear.
, root must be the conjugate of it i.e. β = p − q & vice versa. a h
4. A quadratic equation whose roots are α & β is (x − α)(x − β) = 0 i.e. abc + 2 fgh − af2 − bg2 − ch2 = 0 OR h b
x2 − (α + β) x + α β = 0 i.e. x2 − (sum of roots) x + product of roots = 0. g
5.Remember that a quadratic equation cannot have three different roots & if it has, it becomes an identity. 11. THEORY OF EQUATIONS : If α1
6. Consider the quadratic expression , y = ax² + bx + c , a ≠ 0 & a , b , c ∈ R then
f(x) = a0xn + a1xn-1 + a2xn-2 + .... + an-1x + an
(i) The graph between x , y is always a parabola . If a > 0 then the shape of the
parabola is concave upwards & if a < 0 then the shape of the parabola is concave downwards.
a2 a
(ii) ∀ x ∈ R , y > 0 only if a > 0 & b² − 4ac < 0 (figure 3) . ∑ α1 α2 = + , ∑ α1 α2 α3 = − 3 , .....,
a0 a0
(iii) ∀ x ∈ R , y < 0 only if a < 0 & b² − 4ac < 0 (figure 6) .
Note : (i) If α is a root of the equation f(x)
Carefully go through the 6 different shapes of the parabola given below. (x − α) is a factor of f(x) and conv
Fig. 1 Fig. 2
(ii) Every equation of nth degree (n ≥ 1
y y y
it is an identity.
a>0 (iii) If the coefficients of the equation f(
a>0 a>0 root. i.e. imaginary roots occur in
(iv) If the coefficients in the equation ar
a root where α, β ∈ Q & β is not
(v) If there be any two real number
x1 O x2 x O x O x signs, then f(x) = 0 must have atleast one real
(vi)Every eqtion f(x) = 0 of degree odd has atle
Roots are real & Roots are Roots are complex 12. LOCATION OF ROOTS :
Let f (x) = ax2 + bx + c, where a > 0 & a
Fig. 4 Fig. 5 (i) Conditions for both the roots of
y b2 − 4ac ≥ 0; f (d) > 0 & (− b/2a)
y y
(ii) Conditions for both roots of f (x)
the number ‘d’ lies between the roo
O x O x (iii) Conditions for exactly one root
a<0 − 4ac > 0 & f (d) . f (e) < 0.
x2 a<0 (iv) Conditions that both
x1
are (p < q). b2 − 4ac ≥ 0; f (p) > 0
a<0
O x 13. LOGARITHMIC INEQUALITIES
(i) For a > 1 the inequality 0 < x < y
Roots are real & Roots are Roots are complex (ii) For 0 < a < 1 the inequality 0 < x
7. SOLUTION OF QUADRATIC INEQUALITIES: (iii) If a > 1 then loga x < p ⇒
ax2 + bx + c > 0 (a ≠ 0). (iv) If a > 1 then logax > p ⇒
(i) If D > 0, then the equation ax2 + bx + c = 0 has two different roots x1 < x2. (v) If 0 < a < 1 then loga x < p ⇒
Then a > 0 ⇒ x ∈ (−∞, x1) ∪ (x2, ∞) (vi) If 0 < a < 1 then logax > p ⇒
a < 0 ⇒ x ∈ (x1, x2)www.MathsBySuhag.com , www.TekoClasses.com www.MathsBySuhag.com , www.TekoClasses.co
2. Theory of Equation (Quadratic Equation) 3
3. Sequence & Progression(AP, GP, HP, AGP, Spl. Series) 4
4. Permutation & Combination 5
5. Determinant 6
6. Matrices 7
7. Logarithm and their properties 9
8. Probability 9
9. Function 11
10 Inverse Trigonometric Functions 14
11. Limit and Continuity & Differentiability of Function 16
12. Differentiation & L' hospital Rule 18
13. Application of Derivative (AOD)14. Integration (efinite 20
& Indefinite) 22
15. Area under curve (AUC) 24
16. Differential Equation 25
17. Straight Lines & Pair of Straight Lines 26
18. Circle 28
19. Conic Section (Parabola 30, Ellipse 32, Hyperbola 33) 30
20. Binomial Theorem and Logarithmic Series 35
, (vi) There exists a one-one correspondence be
1.COMPLEX NUMBERS (b)
complex numbers.
Trignometric / Polar Representation :
1. DEFINITION : Complex numbers are definited as expressions of the form a + ib where a, b ∈ R &
z = r (cos θ + i sin θ) where | z | = r ; ar
i = −1 . It is denoted by z i.e. z = a + ib. ‘a’ is called as real part of z (Re z) and ‘b’ is called as
Note: cos θ + i sin θ is also written as C
imaginary part of z (Im z). www.MathsBySuhag.com , www.TekoClasses.com
EVERY COMPLEX NUMBER CAN BE REGARDED AS eix + e −ix eix − e
Also cos x = & sin x =
2 2
Purely real Purely imaginary Imaginary (c) Exponential Representation :
if b = 0 if a = 0 if b ≠ 0 z = reiθ ; | z | = r ; arg z = θ ; z = re
Note : 6. IMPORTANT PROPERTIES OF CON
(a) The set R of real numbers is a proper subset of the Complex Numbers. Hence the Complete Number If z , z1 , z2 ∈ C then ;
system is N ⊂ W ⊂ I ⊂ Q ⊂ R ⊂ C.
(b) Zero is both purely real as well as purely imaginary but not imaginary. (a) z + z = 2 Re (z) ; z − z = 2 i Im (z)
(c) i = −1 is called the imaginary unit. Also i² = − l ; i3 = −i ; i4 = 1 etc.
(d) z1 − z 2 = z1 − z 2 ; z1 z 2 = z1 . z 2
a b = a b only if atleast one of either a or b is non-negative.www. Maths By Suhag .com
2. CONJUGATE COMPLEX :
If z = a + ib then its conjugate complex is obtained by changing the sign of its imaginary part & (b) | z | ≥ 0 ; | z | ≥ Re (z) ; | z | ≥ Im (z) ;
is denoted by z . i.e. z = a − ib.
z1 |z
Note that : www.MathsBySuhag.com , www.TekoClasses.com | z1 z2 | = | z1 | . | z2 | ; =
(i) z + z = 2 Re(z) (ii) z − z = 2i Im(z) (iii) z z = a² + b² which is real z2 |z
(iv) If z lies in the 1st quadrant then z lies in the 4th quadrant and − z lies in the 2nd quadrant.
3. ALGEBRAIC OPERATIONS : | z1 + z2 |2 + | z1 – z2 |2 = 2 [| z1 |2 + | z 2 |2 ]
The algebraic operations on complex numbers are similiar to those on real numbers treating i as a
polynomial. Inequalities in complex numbers are not defined. There is no validity if we say that complex z1− z2 ≤ z1 + z2 ≤ z1 + z
number is positive or negative. (c) (i) amp (z1 . z2) = amp z1 + amp z2 +
e.g. z > 0, 4 + 2i < 2 + 4 i are meaningless . z
However in real numbers if a2 + b2 = 0 then a = 0 = b but in complex numbers, (ii) amp 1 = amp z1 − amp z2 + 2 k
z2
z12 + z22 = 0 does not imply z1 = z2 = 0.www.MathsBySuhag.com , www.TekoClasses.com (iii) amp(zn) = n amp(z) + 2kπ .
4. EQUALITY IN COMPLEX NUMBER : where proper value of k must be c
Two complex numbers z1 = a1 + ib1 & z2 = a2 + ib2 are equal if and only if their real & imaginary (7) VECTORIAL REPRESENTATION OF
parts coincide. Every complex number can be considered
5. REPRESENTATION OF A COMPLEX NUMBER IN VARIOUS FORMS: →
(a) Cartesian Form (Geometric Representation) : represents the complex number z then, O
Every complex number z = x + i y can be represented by a point on the → →
cartesian plane known as complex plane (Argand diagram) by the ordered NOTE :(i) If OP = z = r ei θ then OQ = z1 = r
, = 0. This is also the condition for three co
− 1+ i 3 − 1− i 3
9. CUBE ROOT OF UNITY :(i)The cube roots of unity are 1 , , . (G) Complex equation of a straight line
2 2
(ii) If w is one of the imaginary cube roots of unity then 1 + w + w² = 0. In general z (z1 − z 2 ) − z (z1 − z 2 ) + (z1z 2 − z1z 2 ) = 0, w
1 + wr + w2r = 0 ; where r ∈ I but is not the multiple of 3. α z + α z + r = 0 where r
(iii) In polar form the cube roots of unity are : (H) The equation of circle having centre z
2π 2π 4π 4π z z − z0 z − z 0 z + z 0 z0 − ρ² = 0 which is of
cos 0 + i sin 0 ; cos + i sin , cos + i sin
3 3 3 3
(iv) The three cube roots of unity when plotted on the argand plane constitute the verties of an equilateral αα −r . Circle will be real if α α −
triangle.www.MathsBySuhag.com , www.TekoClasses.com (I) The equation of the circle described on th
(v) The following factorisation should be remembered :
(a, b, c ∈ R & ω is the cube root of unity) z − z2 π
(i) arg = ± or (z − z1) ( z −
a3 − b3 = (a − b) (a − ωb) (a − ω²b) ; x2 + x + 1 = (x − ω) (x − ω2) ; z − z1 2
a3 + b3 = (a + b) (a + ωb) (a + ω2b) ; (J) Condition for four given points z1 , z2 , z3
a3 + b3 + c3 − 3abc = (a + b + c) (a + ωb + ω²c) (a + ω²b + ωc) z 3 − z1 z 4 − z 2
10. nth ROOTS OF UNITY :www.MathsBySuhag.com , www.TekoClasses.com . is real. Hence the equatio
z 3 − z 2 z 4 − z1
If 1 , α1 , α2 , α3 ..... αn − 1 are the n , nth root of unity then :
(i) They are in G.P. with common ratio ei(2π/n) & (z − z 2 ) (z3 − z1 ) (z −
⇒
(ii) 1p + α 1p + α 2p + .... +α pn − 1
= 0 if p is not an integral multiple of n
taken as
(z − z1 ) (z3 − z 2 ) is real (z −
= n if p is an integral multiple of n 13.(a) Reflection points for a straight line :
(iii) (1 − α1) (1 − α2) ...... (1 − αn − 1) = n & given straight line if the given line is the right bisec
(1 + α1) (1 + α2) ....... (1 + αn − 1) = 0 if n is even and 1 if n is odd. complex numbers z1 & z2 will be the refle
(iv) 1 . α1 . α2 . α3 ......... αn − 1 = 1 or −1 according as n is odd or even. ; α z + α z + r = 0 , where r is real and α
11. THE SUM OF THE FOLLOWING SERIES SHOULD BE REMEMBERED : 1 2
(b) Inverse points w.r.t. a circle :www.Mat
sin (nθ 2 ) n +1 Two points P & Q are said to be inverse w
cos θ + cos 2 θ + cos 3 θ + ..... + cos n θ = cos θ.
sin (θ 2 )
(i)
2 (i) the point O, P, Q are collinear and on
Note that the two points z1 & z2 will be t
sin (nθ 2 ) n + 1 zz + αz+αz + r =0 if and only if z1 z 2 + αz1
sin θ + sin 2 θ + sin 3 θ + ..... + sin n θ = sin θ.
sin (θ 2 )
(ii)
2 14. PTOLEMY’S THEOREM :www.Math
Note : If θ = (2π/n) then the sum of the above series vanishes. It states that the product of the length
12. STRAIGHT LINES & CIRCLES IN TERMS OF COMPLEX NUMBERS : circle is equal to the sum of the lengt
nz + mz 2 i.e. z1 − z3 z2 − z4 = z1 − z2 z
(A) If z1 & z2 are two complex numbers then the complex number z = 1 divides the joins of z1
m+n 15. LOGARITHM OF A COMPLEX QU
& z2 in the ratio m : n. 1
Note:(i) If a , b , c are three real numbers such that az1 + bz2 + cz3 = 0 ; where a + b + c = 0 (i) Loge (α + i β) = Loge (α² + β²) + i 2nπ
2
and a,b,c are not all simultaneously zero, then the complex numbers z1 , z2 & z3 are collinear.
, root must be the conjugate of it i.e. β = p − q & vice versa. a h
4. A quadratic equation whose roots are α & β is (x − α)(x − β) = 0 i.e. abc + 2 fgh − af2 − bg2 − ch2 = 0 OR h b
x2 − (α + β) x + α β = 0 i.e. x2 − (sum of roots) x + product of roots = 0. g
5.Remember that a quadratic equation cannot have three different roots & if it has, it becomes an identity. 11. THEORY OF EQUATIONS : If α1
6. Consider the quadratic expression , y = ax² + bx + c , a ≠ 0 & a , b , c ∈ R then
f(x) = a0xn + a1xn-1 + a2xn-2 + .... + an-1x + an
(i) The graph between x , y is always a parabola . If a > 0 then the shape of the
parabola is concave upwards & if a < 0 then the shape of the parabola is concave downwards.
a2 a
(ii) ∀ x ∈ R , y > 0 only if a > 0 & b² − 4ac < 0 (figure 3) . ∑ α1 α2 = + , ∑ α1 α2 α3 = − 3 , .....,
a0 a0
(iii) ∀ x ∈ R , y < 0 only if a < 0 & b² − 4ac < 0 (figure 6) .
Note : (i) If α is a root of the equation f(x)
Carefully go through the 6 different shapes of the parabola given below. (x − α) is a factor of f(x) and conv
Fig. 1 Fig. 2
(ii) Every equation of nth degree (n ≥ 1
y y y
it is an identity.
a>0 (iii) If the coefficients of the equation f(
a>0 a>0 root. i.e. imaginary roots occur in
(iv) If the coefficients in the equation ar
a root where α, β ∈ Q & β is not
(v) If there be any two real number
x1 O x2 x O x O x signs, then f(x) = 0 must have atleast one real
(vi)Every eqtion f(x) = 0 of degree odd has atle
Roots are real & Roots are Roots are complex 12. LOCATION OF ROOTS :
Let f (x) = ax2 + bx + c, where a > 0 & a
Fig. 4 Fig. 5 (i) Conditions for both the roots of
y b2 − 4ac ≥ 0; f (d) > 0 & (− b/2a)
y y
(ii) Conditions for both roots of f (x)
the number ‘d’ lies between the roo
O x O x (iii) Conditions for exactly one root
a<0 − 4ac > 0 & f (d) . f (e) < 0.
x2 a<0 (iv) Conditions that both
x1
are (p < q). b2 − 4ac ≥ 0; f (p) > 0
a<0
O x 13. LOGARITHMIC INEQUALITIES
(i) For a > 1 the inequality 0 < x < y
Roots are real & Roots are Roots are complex (ii) For 0 < a < 1 the inequality 0 < x
7. SOLUTION OF QUADRATIC INEQUALITIES: (iii) If a > 1 then loga x < p ⇒
ax2 + bx + c > 0 (a ≠ 0). (iv) If a > 1 then logax > p ⇒
(i) If D > 0, then the equation ax2 + bx + c = 0 has two different roots x1 < x2. (v) If 0 < a < 1 then loga x < p ⇒
Then a > 0 ⇒ x ∈ (−∞, x1) ∪ (x2, ∞) (vi) If 0 < a < 1 then logax > p ⇒
a < 0 ⇒ x ∈ (x1, x2)www.MathsBySuhag.com , www.TekoClasses.com www.MathsBySuhag.com , www.TekoClasses.co
