ONE-SCHOOL.NET
Add Maths Formulae List :Form 4 01 Function Absolute Value Function
Inverse Function
f ( x ), if f ( x ) ≥ 0 f ( x)
If
y = f ( x ) , then f −1 ( y ) = x
− f ( x), if f ( x ) < 0
02 Quadratic Equation General Form
Quadratic Formula
ax 2 + bx + c = 0
−b ± b 2 − 4ac x= 2a
where a, b, and c are constants and a ≠ 0. *Note that the highest power of an unknown of a quadratic equation is 2.
Nature of Roots
Forming Quadratic Equation From its Roots: If α and β are the roots of a quadratic equation
α +β =−
b a
αβ =
c a
b 2 − 4ac b 2 − 4ac b 2 − 4ac b 2 − 4ac
The Quadratic Equation
x 2 − (α + β ) x + αβ = 0 or x 2 − ( SoR ) x + ( PoR ) = 0 SoR = Sum of Roots PoR = Product of Roots
http://www.one-school.net/notes.html
1
>0 =0 <0 ≥0
⇔ two real and distinct roots ⇔ two real and equal roots ⇔ no real roots ⇔ the roots are real
ONE-SCHOOL.NET Indices and Logarithm Fundamental if Indices
Laws of Indices
Zero Index,
a0 = 1
a m × a n = a m+n
Negative Index,
a −1 =
1 a
a m ÷ a n = a m−n ( a m ) n = a m× n
a b ( ) −1 = b a Fractional Index
1 an
= a
m an
= a
( ab) n = a n b n
n
n
a n an ( ) = n b b
m
Fundamental of Logarithm
Law of Logarithm
log a mn = log a m + log a n
log a y = x ⇔ a x = y log a a = 1
log a
log a 1 = 0
m = log a m − log a n n
log a mn = n log a m Changing the Base
http://www.one-school.net/notes.html
2
log a b =
log c b log c a
log a b =
1 log a b
ONE-SCHOOL.NET Coordinate Geometry Distance and Gradient
Distance Between Point A and C =
(x1 − x2 )2 + (x1 − x2 )2 Gradient of line AC, m = Or Gradient of a line = −
y2 − y1 x2 − x1
y − int ercept x − int ercept
When 2 lines are perpendicular to each other,
When 2 lines are parallel,
m1 = m2 .
m1 × m2 = −1 m1 = gradient of line 1 m2 = gradient of line 2
Midpoint
A point dividing a segment of a line
⎛ x1 + x2 y1 + y2 ⎞ , ⎟ 2 ⎠ ⎝ 2
A point dividing a segment of a line
Midpoint, M = ⎜
http://www.one-school.net/notes.html
⎛ nx + mx2 ny1 + my2 ⎞ P =⎜ 1 , ⎟ m+n ⎠ ⎝ m+n
3
ONE-SCHOOL.NET Area of triangle:
Area of Triangle, A =
1 ( x1 y2 + x2 y3 + x3 y1 ) − ( x2 y1 + x3 y2 + x1 y3 ) 2
Equation of Straight Line Gradient (m) and 1 point (x1, y1) 2 points, (x1, y1) and (x2, y2) given given y − y1 y2 − y1 y − y1 = m( x − x1 ) = x − x1 x2 − x1
x-intercept and y-intercept given
x y + =1 a b a = x-intercept b = y-intercept
Equation of Locus The equation of the locus of a moving point P ( x, y ) which is always at a constant distance (r) from a fixed point ( x1 , y1 ) is 2
2
( x − x1 ) + ( y − y1 ) = r
2
The equation of the locus of a moving point P ( x, y ) which is always at a constant distance from two fixed points ( x1 , y1 ) and
( x2 , y 2 ) with a ratio m : n is
http://www.one-school.net/notes.html
( x − x1 ) 2 + ( y − y1 ) 2 m 2 = ( x − x2 ) + ( y − y 2 ) 2 n 2
4
The equation of the locus of a moving point P ( x, y ) which is always equidistant from two fixed points A and B is the perpendicular bisector of the straight line AB.
ONE-SCHOOL.NET Statistic Measure of Central Tendency Ungrouped Data Mean
x=
Σx N
x = mean Σx = sum of x x = value of the data
Median
Without Class Interval Σ fx x= Σf
Grouped Data With Class Interval Σ fx x= Σf
x = mean Σx = sum of x f = frequency x = value of the data
Median, m = Tn +1
x = mean Σx = sum of x f = frequency x = value of the data
Median, m = Tn +1
2
⎛ 1N −F⎞ ⎟⎟ C m = L + ⎜⎜ 2 ⎝ fm ⎠
2
m = median L = Lower boundary of median class N = Number of data F = Total frequency before median class fm = Total frequency in median class c = Class size = (Upper boundary – lower boundary)
Measure of Dispersion Ungrouped Data
x2 ∑ σ = 2
variance
N
−x
2
fx 2 ∑ σ = ∑f 2
σ = variance Standard Deviation
σ=
σ=
Σ(x − x ) N
Without Class Interval
−x
Grouped Data With Class Interval
2
σ = variance
fx 2 ∑ σ = ∑f 2
−x
2
σ = variance Σ f (x − x) σ= Σf
2
σ=
Σx 2 − x2 N
Σ fx 2 − x2 Σf
The variance is a measure of the mean for the square of the deviations from the mean. The standard deviation refers to the square root for the variance.
http://www.one-school.net/notes.html
5
2
ONE-SCHOOL.NET Effects of data changes on Measures of Central Tendency and Measures of dispersion
Measures of Mean, median, mode Central Tendency Range , First Quartile, Third Quartile, Interquartile Range Measures of Standard Deviation dispersion Variance
Data are changed uniformly with +k −k ×k ÷k +k −k ×k ÷k
No changes
×k
÷k
No changes No changes
×k × k2
÷k ÷ k2
Circular Measure Terminology
Convert degree to radian:
xo = ( x ×
π 180
Convert radian to degree:
x radians = ( x ×
)radians
180
π
) degrees
Length and Area
r = radius A = area s = arc length θ = angle l = length of chord
Arc Length:
s = rθ
Length of chord:
l = 2r sin
θ 2
http://www.one-school.net/notes.html
Area of Sector:
A=
1 2 rθ 2
6
Area of Triangle:
A=
1 2 r sin θ 2
Area of Segment:
A=
1 2 r (θ − sin θ ) 2
ONE-SCHOOL.NET Differentiation Differentiation of a Function III
Gradient of a tangent of a line (curve or straight)
y = ax n dy = anx n−1 dx
dy δy = lim ( ) dx δ x →0 δ x
Example y = 2 x3 Differentiation of Algebraic Function Differentiation of a Constant
y=a dy =0 dx
dy = 2(3) x 2 = 6 x 2 dx
a is a constant Differentiation of a Fractional Function 1 xn Rewrite y=
Example y=2 dy =0 dx
y = x−n dy −n = − nx − n−1 = n+1 dx x
Differentiation of a Function I
y=x
Example 1 y= x y = x −1 −1 dy = −1x −2 = 2 dx x
n
dy = nx n−1 dx Example y = x3
dy = 3x 2 dx
Law of Differentiation Sum and Difference Rule
Differentiation of a Function II
y =u±v u and v are functions in x dy du dv = ± dx dx dx
y = ax dy = ax1−1 = ax 0 = a dx
Example y = 2 x3 + 5 x 2
Example y = 3x dy =3 dx
http://www.one-school.net/notes.html
dy = 2(3) x 2 + 5(2) x = 6 x 2 + 10 x dx
7
ONE-SCHOOL.NET Chain Rule
y = un
Quotient Rule
u and v are functions in x
y=
dy dy du = × dx du dx
dy = dx
Example y = (2 x 2 + 3)5
= 5(2 x 2 + 3) 4 × 4 x = 20 x(2 x 2 + 3) 4
Or differentiate directly y = (2 x 2 + 3)5
=
dy = 5(2 x 2 + 3)4 × 4 x = 20 x(2 x 2 + 3) 4 dx
v
du dv −u dx dx v2
4 x2 + 2 x − 2 x2 2 x2 + 2 x = (2 x + 1) 2 (2 x + 1) 2
Or differentiate directly x2 y= 2x +1 dy (2 x + 1)(2 x) − x 2 (2) = dx (2 x + 1) 2
Product Rule
y = uv u and v are functions in x dy du dv = v +u dx dx dx
=
Example y = (2 x + 3)(3 x 3 − 2 x 2 − x) v = 3x3 − 2 x 2 − x
du dv =2 = 9 x2 − 4x − 1 dx dx dy du dv =v +u dx dx dx 3 2 =(3 x − 2 x − x)(2) + (2 x + 3)(9 x 2 − 4 x − 1)
Or differentiate directly y = (2 x + 3)(3x3 − 2 x 2 − x)
dy = (3x3 − 2 x 2 − x)(2) + (2 x + 3)(9 x 2 − 4 x − 1) dx
http://www.one-school.net/notes.html
u and v are functions in x
Example x2 y= 2x +1 u = x2 v = 2x +1 du dv = 2x =2 dx dx du dv −u v dy = dx 2 dx dx v dy (2 x + 1)(2 x) − x 2 (2) = (2 x + 1) 2 dx
du u = 2 x + 3, therefore = 4x dx dy y = u 5 , therefore = 5u 4 du dy dy du = × dx du dx = 5u 4 × 4 x 2
u = 2x + 3
u v
8
4 x2 + 2 x − 2 x2 2 x2 + 2 x = (2 x + 1) 2 (2 x + 1) 2
ONE-SCHOOL.NET Gradients of tangents, Equation of tangent and Normal
If A(x1, y1) is a point on a line y = f(x), the gradient of the line (for a straight line) or the gradient of the dy tangent of the line (for a curve) is the value of dx when x = x1. Gradient of tangent at A(x1, y1):
m=
dy dx
Gradient of normal at A(x1, y1): m=−
1 dy dx
Maximum and Minimum Point
At maximum point (local maximum),
dy =0 dx
At maximum point (local maximum),
d2y <0 dx 2
http://www.one-school.net/notes.html
d2y >0 dx 2
9
dy =0 dx
ONE-SCHOOL.NET Related Rates of Change
Small Changes and Approximation Small Change:
dA dA dr = × dt dr dt
δ y dy dy ≈ ⇒ δ y ≈ ×δ x dx δ x dx Approximation: ynew = yoriginal + δ y
= yoriginal +
dy ×δ x dx
δ x = small change in x δ y = small change in y
Solution of Triangle
Sine Rule:
Area of triangle:
Cosine Rule:
a b c = = sin A sin B sin C
a2 = b2 + c2 – 2bc cosA b2 = a2 + c2 – 2ac cosB c2 = a2 + b2 – 2ab cosC
Case 1: When a < b sin A CB is too short to reach the side opposite to C.
Outcome: No solution
http://www.one-school.net/notes.html
A=
1 a b sin C 2
Case 2: When a = b sin A CB just touch the side opposite to C
Outcome: 1 solution
10
ONE-SCHOOL.NET Case 3: When a > b sin A but a < b. CB cuts the side opposite to C at 2 points
Case 4: When a > b sin A and a > b. CB cuts the side opposite to C at 1 points
Outcome: 2 solution
Outcome: 1 solution
Index Number Price Index
Composite index
P I = 1 × 100 P0
I=
I = Price index
I = Composite Index W = Weightage I = Price index
P1 = Price at the base time P2 = Price at a Specific Time
http://www.one-school.net/notes.html
Σ Wi I i Σ Wi
11