TALHA SIDDIQUI’s ACADEMY OF COMMERCE & SCIENCES 021-35248855
Sir. Talha Siddiqui 0345-3093759
r
XI – MATHEMATICS FORMULAE De Morgan’s Laws:(a) (AUB)/ = A/ ∩ B/ (b) (A ∩ B)/ = A/ U B/ Note:- A/ = U – A, B/ = U – B. System of Complex Number:(a , b) = a + ιb For (a , b) , (c , d) CC We define (1)
Equality:(a , b) = (c , d) if and only if a = c , b = d
(2)
Addition:(a , b) + (c , d) = (a + c , b + d) CC
(3)
Multiplication:(a , b) . (c , d) = (ac + bd , ad + bc) CC The additive inverse of (a , b) CC is (−a , −b) The Multiplicative inverse of (a , b) CC is b a , 2 2 2 a b a b2 The Complex Conjugate of z = a + ιb is z = a – ιb
Let z = a + ιb then z a 2 b 2
Quadratic Formula: b b2 4ac 2a The Cube Roots of Unity:The cube roots of unity are:1, ω, ω2 where, 1 3 1 3 and 2 are the complex cube roots of unity. 2 2 Note: 1 + ω + ω2 = 0 ω3 = 1 Nature of Roots :For the nature of roots, we have the following formula. Discriminate: D = b2 – 4ac. Note: If D = 0, the roots are real equal If D > 0, the roots are real and unequal If D < 0, the roots are complex and unequal If D is a perfect square, the roots are rational and unequal; otherwise they are irrational. Sum of Roots b Coefficient of x a Coefficient of x2 Product of Roots c Cons tan t a Coefficient of x2 To form a quadratic equation whose roots are given, we have the following formula: x2 – (sum of roots)x + product of roots = 0 x
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TALHA SIDDIQUI’s ACADEMY OF COMMERCE & SCIENCES 021-35248855
Sir. Talha Siddiqui 0345-3093759
Determinant For 2 × 2 order matrix a a a a If A 11 12 , then A 11 12 a 21 a 22 a 21 a 22 For 3 × 3 order matrix a11 a11 a12 a13 If A a 21 a 22 a 23 , then A a 21 a 31 a 32 a 33 a 31 The Adjoint Matrix a11 a12 Let A a 21 a 22 a 31 a 32 A11 A12 Adj A A 21 A 22 A31 A32 The Inverse Matrix Adj A A 1 . A
Note:-
a11a 22 a12a 21 a12 a 22 a 32
a13
a a 23 = a11 22 a 32 a 33
a 23 a a a a a12 21 23 a13 21 22 a 33 a 31 a 33 a 31 a 32
a13 a 23 then, a 33 A13 A 23 A33
If A 0, then A1 does not exist.
Arithmetic Progression(A.P.):The standard form of an A.P is: a , a + d , a + 2d , a + 3d , . . . , a + (n – 1)d General Term of an A.P:Tn a n 1d where, a = 1st term, d = common difference, n = no. of terms Sum of Arithmetic Series:n Sn 2a n 1d 2 where, a = 1st term, d = common difference, n = no. of terms Arithmetic Mean:ab A.M. 2 Note:- To find more than one A.M.’s we need to find the Common difference ‘d’ which is given by the following formula: ba d n 1 where, a = 1st term, b = last term, n = no. of A.M’s
Geometric Progression(G.P.):The standard form of an G.P is: a , ar , ar2 , ar3 , . . . , arn – 1
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TALHA SIDDIQUI’s ACADEMY OF COMMERCE & SCIENCES 021-35248855
Sir. Talha Siddiqui 0345-3093759
General Term of G.P:-
Tn ar n 1 where, a = 1st term,
r = common ratio,
n = no. of terms
Sum of Geometric Series:a 1 rn Sn , when r < 1 1 r a rn 1 Sn , when r < 1 r 1 where, a = 1st term, r = common ratio, n = no. of terms Sum of Infinite Geometric Series:a S 1 r Geometric Mean G ab Note:- To find more than one G.M.’s we need to find the Common ratio ‘r’ which is given by the following formula:
1
b n 1 r a where, a = 1st term,
b = last term,
n = no. of G.M’s
Harmonic Progression(H.P.):The standard form of an G.P is: 1 1 1 1 , , ,......, a a d a 2d a (n 1)d Harmonic Mean 2ab a, H, b H ab where, a = 1st term, b = last term Permutation n! n Pr ( n r )! Combination n! n Cr (n r )! r! Sum of First n Natural Numbers n (n 1) n 2 Sum of Squares of First n Natural Numbers n (n 1)(2n 1) n2 6 Sum of Cubes of First n Natural Numbers
n (n 1) n3 2
2
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TALHA SIDDIQUI’s ACADEMY OF COMMERCE & SCIENCES 021-35248855
Sir. Talha Siddiqui 0345-3093759
Binomial Formula
a bn a n na n 1b n(n 1) a n 2b2 n(n 1)(n 2) a n 3b3 ...... n(n 1)...(n r 1) a n r br ...... bn 1 x n
2! 3! n (n 1) 2 n (n 1)...(n r 1) r 1 nx x ...... x ...... 2! r!
Relation Between Arc-Length, Radius And General Angle s = rθ Where, s = arc length, r = radius, θ = central angle(in radians) Signs of the Trigonometric Functions in the Four Quadrants II sin, cosec +ve all other –ve tan, cot +ve all other –ve III
I All +ve
cos, sec +ve all other −ve IV
Trigonometric Identities sin2θ + cos2θ = 1 sin2θ = 1 − cos2θ cos2θ = 1 – sin2θ 1 + tan2θ = sec2θ 1 + cot2θ = cosec2θ The Distance Formula AB
x 2 x1 2 y 2 y1 2
Deductions From The Fundanmental Law cos(−β) = cosβ cos sin 2 sin(−β) = −sinβ tan(−β) = −tanβ cot(−β) = −cotβ sin cos 2 tan cot 2 cot tan 2 Sum And Difference Formula cos(α − β) = cosα cosβ + sinα sinβ cos(α + β) = cosα cosβ − sinα sinβ sin(α + β) = sinα sinβ + cosα cosβ sin(α − β) = sinα sinβ − cosα cosβ tan tan tan 1 tan tan
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r!
TALHA SIDDIQUI’s ACADEMY OF COMMERCE & SCIENCES 021-35248855
Sir. Talha Siddiqui 0345-3093759
tan tan 1 tan tan cot cot 1 cot cot cot cot cot 1 cot cot cot cot cot 1 cot cot cot tan
Double Angle Identity
cos2θ = cos2θ – sin2θ = 2cos2θ – 1 = 1 – 2sin2θ =
sin2θ = 2sinθcosθ =
tan 2
2 tan
2 tan
1 tan 2 1 tan 2
1 tan 2
1 tan 2 Half Angle Identity 1 cos sin 2 2 1 cos 2 2 sin 1 cos tan 2 1 cos sin Sum And Product Formulae For Trigonometric Functions 1 sin cos sin sin 2 1 cos sin sin sin 2 1 sin sin cos cos 2 1 cos cos cos cos 2 uv uv sin u sin v 2 sin cos 2 2 uv uv sin u sin v 2 cos sin 2 2 uv uv cos u cos v 2 cos cos 2 2 uv uv sin u cosv 2 sin sin 2 2 Periodic Function:A function y = f(x) is called a periodic function of periodic p if f(x + p) = f(x). The Law of Sine a b c sin sin sin The Law of Cosine a2 = b2 + c2 − 2bc cosα b2 = a2 + c2 − 2ac cosβ c2 = a2 + b2 − 2ab cosγ
cos
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TALHA SIDDIQUI’s ACADEMY OF COMMERCE & SCIENCES 021-35248855
Sir. Talha Siddiqui 0345-3093759
The Law of Tangents tan 2 ab a b tan 2 tan 2 bc bc tan 2 tan 2 ca ca tan 2 Half Angle Formulae in terms of Lengths of Sides (s b)(s c) sin 2 bc
sin
2
(s a )(s c) ac
sin
2
(s a )(s b) ab
cos
2
s(s a ) bc
cos
s(s b) 2 ac
cos
2
tan
(s b)(s c) 2 s(s a )
tan
(s a )(s c) 2 s(s b)
tan
(s a )(s b) 2 s(s c)
cot
s(s a ) 2 (s b)(s c)
cot
s(s b) 2 (s a )(s c)
cot
s(s c) 2 (s a )(s b)
s(s c) ab
Area of Triangle Case 1:- When measures of two sides and the measure of the included angle are given: 1 ab sin 2 1 ac sin 2 1 bc sin 2 -6-
TALHA SIDDIQUI’s ACADEMY OF COMMERCE & SCIENCES 021-35248855
Sir. Talha Siddiqui 0345-3093759
Case 2:- When measures of two angles and the measure of one side are known: 1 sin sin a2 2 sin 1 2 sin sin b 2 sin 1 sin sin c2 2 sin Case 3:- When measures of three sides are known: abc where, s ss a s bs c 2 Case 4:- When measures of base and the measure of height is known 1 base height 2
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