A-PDF Watermark DEMO: Purchase from www.A-PDF.com to remove the watermark 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 Quadratic Formula General Form 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<b 04 Simultaneous Equations To find the intersection point ⇒ solves simultaneous equation. Remember: substitute linear equation into non- linear equation. http://www.one-school.net/notes.html 2 ONE-SCHOOL.NET 05 Indices and Logarithm Laws of Indices Fundamental if 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 http://www.one-school.net/notes.html 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 = ⎜ http://www.one-school.net/notes.html ⎛ 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 http://www.one-school.net/notes.html same length ⇒ AB = BC = CD = AD parallel lines ⇒ mAB = mCD or mAD = mBC diagonals (perpendicular) ⇒ mAC × mBD = −1 share same midpoint ⇒ midpoint AC = midpoint BD any point ⇒ solve the simultaneous equations (i) (ii) (iii) (iv) (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 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: At http://www.one-school.net/notes.html 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. 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 − x2 N 2 http://www.one-school.net/notes.html Grouped Data Without Class Interval With Class Interval fx 2 ∑ σ = ∑f 2 −x σ = variance σ= σ= Σ(x − x ) fx 2 ∑ σ = ∑f 2 Σx 2 − x2 N −x σ = variance 2 N 7 2 σ= σ= Σ f (x − x) 2 Σf 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 x radians = ( x × ) degrees D π degrees × π 180D Remember: 180D = π rad ??? 360 = 2π rad D http://www.one-school.net/notes.html 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 http://www.one-school.net/notes.html Example y = 3x dy =3 dx 9 1 2 r (θ − sin θ ) 2 ONE-SCHOOL.NET Differentiation of a Function III Chain Rule 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 −1 dy = −1x −2 = 2 x dx 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 http://www.one-school.net/notes.html 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) = (2 x + 1) 2 dx u = 2x + 3 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 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 http://www.one-school.net/notes.html 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 http://www.one-school.net/notes.html 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 http://www.one-school.net/notes.html 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 http://www.one-school.net/notes.html 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 + b 2 (ii) Trigonometry ratio: sin θ = bc , cos θ = ac , tan θ = (iii) Area = ½ (base)(height) http://www.one-school.net/notes.html 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 http://www.one-school.net/notes.html Σ Wi I i Σ Wi 16