Mathematical Formula Samacheer X

MATHEMATICS – FORMULAE

NUMBER WORK

General form of A.P:  a, a+d, a+2d, …    

                            

t_n = a+ [n-1]d

         

s_n =n/2 [2a + [n-1]d]    or n/2 [a+l]

No. of terms in AP is n= (l-a /d )+ 1

 

General form of G.P: a, ar, ar², … 

   

t_n = ar^n-1         

       

s_n = a(rⁿ -1) / r-1 if r>1

s_n = a(1-rⁿ ) / r-1 if r<1

s_n = an if r =1

s_α = a / 1-r  if r<1

εn = n(n+1) / 2

εn² = n(n+1) (2n +1) / 2

εn³ = n² (n+1) ² / 4

 MENSURATION

CYLINDER:

Volume = πr² h cu.units

Curved surface area = 2πrh sq.units

Total surface area == 2πr[r+h] sq.units

Hollow Cylinder:

Volume = πh [R+r] [R –r] cu.units

Curved surface area = 2πh [R+r] sq.units

Total surface area =  2π [R+r] [R-r+h] sq.units

l² = h² +r²

Right circular cone:

Volume of the cone = 1/3r²h cu.units

Curved surface area = πrl sq.units

Total surface area = πr (l+r) sq.units

Radius of the cone r = Ө / 360* l

Volume of Frustum of a cone = 1/3 πh [ R²+Rr +r²] cu.units

Sphere:                                                                       HOLLOW SPHERE

Volume of the sphere = 4/3 πr³ cu.units                     4/3 π (R³ – r³)

Surface of the sphere = 4πr² sq.units

Hemisphere:                                                               HOLLOW H.SPHERE

Volume = 2/3 πr³ cu.units                                           2/3 π (R³ – r³)

Curved surface area = 2πr² sq.units                            2π (R² – r²)

Total surface area = 3 πr² sq.units                              2π (R² + r²) +π (R² – r²)

           

SETLANGUAGE

AU(B C) = (AUB) ∩    (AUC)

A (BUC) = (A∩   B) U (A∩   C)

n (AUBUC) = n(A) + n(B) + n(C) – n(A∩ B) –n(B∩ C) – n(C∩ A) + n(A∩ B ∩C)

De Morgan’s Laws for set difference

A\ (BUC) = (A/B) ∩ (A/C)

A\ (B∩C) = (A/B) U (A/C)

Complementation: (AUB)¹ = A ∩ B              (A∩B) = A U B

Symmetric differences A∆B = (A\ B) U (B \ A)

ALGEBRA

If [x – a] divides p(x) the remainder id p(a)

If [x – a] is a factor of  p(x) then  p(a) = 0

p (x) * q (x ) = GCD * LCM

Sum of the roots = -b/a ; Product of the roots = c / a

If Δ = b² -4ac < 0 the roots are REAL

  

   Δ = 0 the roots are REAL and EQUAL

   Δ  > 0 not a perfect square i.e., the roots are REAL, UNEQUAL and IRRATIONAL

   Δ  > 0 a perfect square i.e., the roots are REAL, UNEQUAL and RATIONAL

MATRICES

Þ    Matrix is a rectangular array of rows and columns

Þ    All scalar matrix is also called Diagonal matrix

Þ    All Diagonal matrix need not be scalar

Þ    Unit matrix is also scalar matrix

Þ    2 matrix can be multiplied by using row – by- column mode

Þ    The multiplicative  identity is unit matrix

CO-ORDINATE GEOMETRY

Distance formula d =  √ (x_{2 –}x₁)² + (y_{2 –}y₁)²

Mid – point = [x₂ + x₁ / 2  , y₂ +y₁ / 2]

Centroid = [x₂ + x₁ + x₃ / 3 , y₂ +y₁+y₃ /3]

Area of the Triangle = ½ [ x₁ (y₂ – y₃) + x₂ (y₃ – y₁) +x₃ (y₁ – y₂) ]

If the 3 points are collinear (x ₁y₂ +x₂ y₃ + x₃ y₂) – ( x₂ y₃ + x₃ y₁ + x₁ y₂ )  = 0

Slope of a Straight line: m = tan Ө

Slope of the line joining points (x , y ) = y – y₁ / x – x₁

Given the slope m and y intercept, the eq. is y = mx + c

Given the slope m and (x , y ) intercept, the eq. is y – y₁ = m (x – x₁ )

Eq. of the line joining 2 points is y – y₁ / y₂ – y₁  = x – x₁ / x₂ – x₁

If x – intercept and y – intercept are given then line eq. is x /a + y / b = 1 

Two lines are parallel if m₁ = m₂

Two lines are perpendicular if m_1* m₂ = -1

Any line parallel to X –axis  y = k

Any line parallel to Y –axis  x = c

TRIGNOMETRY

Sin ² Ө  + Cos ² Ө = 1

1 + tan² Ө  = Sec² Ө     

              

1 + cot² Ө  = Cosec² Ө           

STATISTICS

Arithmetic mean = x = εx /n and εfd /f

Arithmetic mean =  A + εfd / f * C [c- class intervals]

Standard Deviation s = √εd² / n  d = x- x [x is the mean]

                                      √εd² /n – (εd /n)²  [d = x –A, A is assumed mean]

                                      √εd² /n – (εd /n)²  * C  [d = x –A/ C]                     

 Variance = s²

The co-eff. Of variation = s /x * 100

If CV is more the consistence of given data is less

If CV is less the consistence of given data is more

s Remains unchanged when each value is added / subtracted by the same quantity.

PROBABILITY

P(A) = n(A) / n(S) [ n(S) > n(A)]

For any event A, P(A) ≥ 0

For any sure event P(S) = P(A) + P(A’) =1

If two events are mutually EXCLUSIVE events of A,B  P(AUB) = P(A) + P(B)

If two events are mutually NON- EXCLUSIVE events of A,B

    

                                                   P (AUB) = P (A) + P (B) – P (A ∩ B)