"JEE Math Short Notes" are concise summaries or key points covering important mathematical concepts relevant to the Joint Entrance Examination (JEE) in India. These notes typically include fundamental formulas, theorems, definitions, and problem-solving strategies required for success in the math s...
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
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