SPM-Add-Maths-Formula-List-Form4.pdf

ONE-SCHOOL.NET

Add Maths Formulae List: Form 4 (Update 18/9/08) 01 Functions Absolute Value Function

Inverse Function If

f ( x ), if f ( x ) ≥ 0 f ( x)

y = f ( x ) , then f −1 ( y ) = x

Remember: Object = the value of x Image = the value of y or f(x) f(x) map onto itself means f(x) = x

− f ( x), if f ( x ) < 0

02 Quadratic Equations 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.

When the equation can not be factorized. 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 − ( SoR ) x + ( PoR ) = 0 SoR = Sum of Roots PoR = Product of Roots 2

http://www.one-school.net/notes.html

1

>0 =0 <0 ≥0

⇔ two real and different roots ⇔ two real and equal roots ⇔ no real roots ⇔ the roots are real

ONE-SCHOOL.NET 03 Quadratic Functions Completing the square:

General Form

f ( x) = ax 2 + bx + c

f ( x) = a ( x + p)2 + q

where a, b, and c are constants and a ≠ 0.

(i) (ii) (iii) (iv)

*Note that the highest power of an unknown of a quadratic function is 2.

the value of x, x = − p min./max. value = q min./max. point = (− p, q) equation of axis of symmetry, x = − p

Alternative method: a > 0 ⇒ minimum ⇒ ∪ (smiling face)

f ( x) = ax 2 + bx + c

a < 0 ⇒ maximum ⇒ ∩ (sad face)

Quadratic Inequalities a > 0 and f ( x) > 0

the value of x, x = −

(ii)

min./max. value = f (−

(iii)

equation of axis of symmetry, x = −

b

x < a or x > b

b ) 2a b 2a

Nature of Roots a > 0 and f ( x) < 0

⇔ intersects two different points at x-axis 2 b − 4ac = 0 ⇔ touch one point at x-axis b 2 − 4ac < 0 ⇔ does not meet x-axis b 2 − 4ac > 0

a

b 2a

(i)

a

b a< x
04 Simultaneous Equations To find the intersection point ⇒ solves simultaneous equation. Remember: substitute linear equation into non- linear equation.


2

ONE-SCHOOL.NET 05 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 y = x ⇔ a x = y

log a mn = log a m + log a n

log a a = 1

log a

log a a x = x

log a mn = n log a m

log a 1 = 0


m = log a m − log a n n

Changing the Base

3

log a b =

log c b log c a

log a b =

1 logb a

ONE-SCHOOL.NET 06 Coordinate Geometry Distance and Gradient

Distance Between Point A and C =

(x1 − x2 )2 + (x1 − x2 )2 Gradient of line AC, m =

y2 − y1 x2 − x1

Or ⎛ y − int ercept ⎞ Gradient of a line, m = − ⎜ ⎟ ⎝ x − int ercept ⎠

Parallel Lines

Perpendicular Lines

When 2 lines are parallel,

When 2 lines are perpendicular to each other,

m1 = m2 .

m1 × m2 = −1

m1 = gradient of line 1 m2 = gradient of line 2

Midpoint

A point dividing a segment of a line

A point dividing a segment of a line

⎛ x1 + x2 y1 + y2 ⎞ , ⎟ 2 ⎠ ⎝ 2

Midpoint, M = ⎜


⎛ nx + mx2 ny1 + my2 ⎞ P =⎜ 1 , ⎟ m+n ⎠ ⎝ m+n

4

ONE-SCHOOL.NET Area of triangle:

Area of Triangle

=

1 2

A=

1 ( x1 y2 + x2 y3 + x3 y1 ) − ( x2 y1 + x3 y2 + x1 y3 ) 2

Form of Equation of Straight Line General form ax + by + c = 0

Gradient form

Intercept form

y = mx + c

x y + =1 a b

m = gradient c = y-intercept

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

a = x-intercept b = y-intercept

m=−

b a

x-intercept and y-intercept given x y + =1 a b

Equation of perpendicular bisector ⇒ gets midpoint and gradient of perpendicular line. Information in a rhombus: A

D

B

C


(i) (ii)

same length ⇒ AB = BC = CD = AD parallel lines ⇒ mAB = mCD or mAD = mBC

(iii)

diagonals (perpendicular) ⇒ mAC × mBD = −1

(iv)

share same midpoint ⇒ midpoint AC = midpoint BD any point ⇒ solve the simultaneous equations

(v)

5

ONE-SCHOOL.NET Remember:

y-intercept ⇒ x = 0 cut y-axis ⇒ x = 0 x-intercept ⇒ y = 0 cut x-axis ⇒ y = 0 **point lies on the line ⇒ satisfy the equation ⇒ substitute the value of x and of y of the point into the equation.

Equation of Locus ( use the formula of distance) The equation of the locus of a moving point P ( x, y ) which is always at a constant distance (r) from a fixed point A ( x1 , y1 ) is

The equation of the locus of a moving point P ( x, y ) which is always at a constant distance from two fixed points A ( x1 , y1 ) and B ( x2 , y 2 ) with a ratio m : n is

PA = r

PA m = PB n

( x − x1 ) 2 + ( y − y1 ) 2 = r 2

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. PA = PB ( x − x1 ) + ( y − y1 ) 2 = ( x − x2 ) 2 + ( y − y2 ) 2 2

( x − x1 ) 2 + ( y − y1 ) 2 m 2 = ( x − x2 ) + ( y − y 2 ) 2 n 2

More Formulae and Equation List: SPM Form 4 Physics - Formulae List SPM Form 5 Physics - Formulae List SPM Form 4 Chemistry - List of Chemical Reactions SPM Form 5 Chemistry - List of Chemical Reactions

All at One-School.net


6

ONE-SCHOOL.NET 07 Statistics Measure of Central Tendency Ungrouped Data Mean x=

Without Class Interval

Σx N

x=

x = mean Σx = sum of x x = value of the data N = total number of the data

Median

Σ fx Σf

2

2

=

2

TN + TN 2

When N is an odd number.

TN + T N

+1

m=

2

When N is an even number.

2

2

Σ fx Σf

x = mean f = frequency x = class mark

m = TN +1

When N is an odd number.

m=

x=

x = mean Σx = sum of x f = frequency x = value of the data

m = TN +1

Grouped Data With Class Interval

+1

2

When N is an even number.

(lower limit+upper limit) 2

⎛ 1N −F⎞ ⎟⎟ C m = L + ⎜⎜ 2 ⎝ fm ⎠ 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 = Size class = (Upper boundary – lower boundary)

Measure of Dispersion Ungrouped Data

variance

x2 ∑ σ = 2

N

−x

2

σ = variance Standard Deviation

Σ(x − x ) σ= N

σ=

2

Σx 2 − x2 N


Grouped Data Without Class Interval With Class Interval

fx 2 ∑ σ = ∑f 2

−x

σ = variance σ= σ=

Σ(x − x )

2

N Σx 2 − x2 N

7

2

fx 2 ∑ σ = ∑f 2

−x

σ = variance Σ f (x − x) σ= Σf

σ=

2

Σ fx 2 − x2 Σf

2

ONE-SCHOOL.NET 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. Effects of data changes on Measures of Central Tendency and Measures of dispersion Data are changed uniformly with +k −k ×k ÷k +k −k ÷k ×k

Measures of Mean, median, mode Central Tendency Range , Interquartile Range Measures of Standard Deviation dispersion Variance

×k ×k × k2

No changes No changes No changes

08 Circular Measures Terminology

Convert degree to radian: Convert radian to degree:

×

xo = ( x ×

180

π

radians

π

)radians 180 180 ) degrees x radians = ( x ×

D

π

degrees

×

π 180D

Remember:

180D = π rad ??? 360 = 2π rad D


O ???

0.7 rad

8

1.2 rad

÷k ÷k ÷ k2

ONE-SCHOOL.NET 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

Area of Sector:

θ

A=

2

Area of Triangle:

1 2 rθ 2

A=

1 2 r sin θ 2

Area of Segment:

A=

09 Differentiation Differentiation of a Function I

Gradient of a tangent of a line (curve or straight)

y = xn dy = nx n−1 dx

dy δy = lim ( ) dx δ x →0 δ x

Example y = x3 Differentiation of Algebraic Function Differentiation of a Constant

y=a dy =0 dx

dy = 3x 2 dx

a is a constant Differentiation of a Function II y = ax dy = ax1−1 = ax 0 = a dx

Example y=2 dy =0 dx


Example y = 3x dy =3 dx

9

1 2 r (θ − sin θ ) 2

ONE-SCHOOL.NET Chain Rule

Differentiation of a Function III

y = ax n

y = un

dy = anx n−1 dx

dy dy du = × dx du dx

Example y = 2 x3

Example y = (2 x 2 + 3)5

dy = 2(3) x 2 = 6 x 2 dx

u = 2 x 2 + 3, y = u5 ,

Differentiation of a Fractional Function

u and v are functions in x

therefore

therefore

du = 4x dx

dy = 5u 4 du

dy dy du = × dx du dx = 5u 4 × 4 x

1 xn Rewrite y=

= 5(2 x 2 + 3) 4 × 4 x = 20 x(2 x 2 + 3) 4

y = x−n

Or differentiate directly y = (ax + b) n

dy −n = − nx − n−1 = n+1 dx x

dy = n.a.(ax + b) n −1 dx

Example 1 y= x y = x −1 dy −1 = −1x −2 = 2 dx x

y = (2 x 2 + 3)5 dy = 5(2 x 2 + 3) 4 × 4 x = 20 x(2 x 2 + 3) 4 dx

Law of Differentiation Sum and Difference Rule

y =u±v u and v are functions in x dy du dv = ± dx dx dx Example y = 2 x3 + 5 x 2

dy = 2(3) x 2 + 5(2) x = 6 x 2 + 10 x dx


10

ONE-SCHOOL.NET Product Rule

Quotient Rule

y = uv u and v are functions in x dy du dv = v +u dx dx dx

y=

u v

dy = dx

Example y = (2 x + 3)(3 x 3 − 2 x 2 − x)

u and v are functions in x v

du dv −u dx dx v2

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) = dx (2 x + 1) 2

u = 2x + 3 v = 3x3 − 2 x 2 − x du dv =2 = 9 x2 − 4 x − 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

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 4 x2 + 2 x − 2 x2 2 x2 + 2 x = = (2 x + 1) 2 (2 x + 1) 2


11

ONE-SCHOOL.NET Gradients of tangents, Equation of tangent and Normal Gradient of tangent at A(x1, y1):

dy = gradient of tangent dx Equation of tangent: y − y1 = m( x − x1 ) Gradient of normal at A(x1, y1): mnormal = −

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.

1 − dy

1 mtangent

= gradient of normal

dx Equation of normal : y − y1 = m( x − x1 )

Maximum and Minimum Point

Turning point ⇒ At maximum point, dy =0 dx

2

d y <0 dx 2


dy =0 dx

At minimum point , dy =0 dx

12

d2y >0 dx 2

ONE-SCHOOL.NET Rates of Change

Chain rule

Small Changes and Approximation Small Change:

dA dA dr = × dt dr dt

dx =5 dt Decreases/leaks/reduces ⇒ NEGATIVES values!!!

If x changes at the rate of 5 cms -1 ⇒

δ y dy dy ≈ ⇒ δ y ≈ ×δ x δ x dx dx Approximation: ynew = yoriginal + δ y

= yoriginal +

dy ×δ x dx

δ x = small changes in x δ y = small changes in y If x becomes smaller ⇒ δ x = NEGATIVE


13

ONE-SCHOOL.NET 10 Solution of Triangle

Sine Rule:

Cosine Rule:

Use, when given 2 sides and 1 non included angle 2 angles and 1 side

a A b

B

a

a2 = b2 + c2 – 2bc cosA b2 = a2 + c2 – 2ac cosB c2 = a2 + b2 – 2ab cosC

a b c = = sin A sin B sin C

a

Area of triangle:

cos A =

C b

b2 + c2 − a 2 2bc

A=

Use, when given 2 sides and 1 included angle 3 sides

A 180 – (A+B)

a

a


b

14

C is the included angle of sides a and b. c

A b

1 a b sin C 2

ONE-SCHOOL.NET If ∠C, the length AC and length AB remain unchanged, the point B can also be at point B′ where ∠ABC = acute and ∠A B′ C = obtuse. If ∠ABC = θ, thus ∠AB′C = 180 – θ .

Case of AMBIGUITY A 180 - θ

Remember : sinθ = sin (180° – θ)

θ

C B B′ Case 1: When a < b sin A CB is too short to reach the side opposite to C.

Case 2: When a = b sin A CB just touch the side opposite to C

Outcome: No solution Case 3: When a > b sin A but a < b. CB cuts the side opposite to C at 2 points

Outcome: 1 solution 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

Useful information: c

b a

θ

In a right angled triangle, you may use the following to solve the problems. (i) Phythagoras Theorem: c = a 2 + b2 (ii)

Trigonometry ratio: sin θ = bc , cos θ = ac , tan θ =

(iii) Area = ½ (base)(height)


15

b a

ONE-SCHOOL.NET 11 Index Number Price Index

Composite index

I =

P1 × 100 P0

I=

I = Price index / Index number

I = Composite Index W = Weightage I = Price index

P0 = Price at the base time P1 = Price at a specific time

I A, B × I B ,C = I A,C ×100


Σ Wi I i Σ Wi

16

SPM-Add-Maths-Formula-List-Form4.pdf

Recommend Documents