NOTES AND FORMULAE SPM MATHEMATICS Cone

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1 FORM 3 NOTES. SOLID GEOMETRY (a) Area ad perimeter Triagle NOTES AND FORMULAE SPM MATHEMATICS Coe V = 3 r h A = base height = bh Trapezium A = (sum of two parallel sides) height = (a + b) h Circle Area = r Circumferece = r Sphere V = 3 r 3 Pyramid V = 3 base area Prism height V = Area of cross sectio legth Sector Area of sector = r Legth of arc = r 3 3. CIRCLE THEOREM Agle at the cetre = agle at the circumferece = y Cylider Curve surface area = rh Agles i the same segmet are equal = y Sphere Solid ad Volume Cube: Curve surface area = r V = = 3 Cuboid: V = l b h = lbh Cylider V = r h Agle i a semicircle ACB = 9 o Sum of opposite agles of a cyclic quadrilateral = 8 o a + b = 8 o The eterior agle of a cyclic quadrilateral is equal to the iterior opposite agle. b = a Agle betwee a taget ad a radius = 9 o OPQ = 9 o

2 The agle betwee a taget ad a chord is equal to the agle i the alterate segmet. = y If PT ad PS are tagets to a circle, PT = PS TPO = SPO TOP = SOP 3. POLYGON (a) The sum of the iterior agles of a sided polygo = ( ) 8 o Sum of eterior agles of a polygo = 3 o Each eterior agle of a regular sided polygo = 3 (d) Regular petago + 3 = 3 ( + 3) = = ( y)( + y) = + y y y = y. LAW OF INDICES (a) m = m + (d) - = (e) (f) m = m ( m ) = m m ( ) m (e) (f) Each eterior agle = 7 o Each iterior agle = 8 o Regular heago Each eterior agle = o Each iterior agle = o Regular octago Each eterior agle = o Each iterior agle = 3 o (g) = 7. ALGEBRAIC FRACTION Epress k as a fractio i its simplest k k form. k 3 k ( k) k k k = 3k k k ( k ) k k k k 3k 8. LINEAR EQUATION Give that of. (3 + ) = (3 + ) =, calculate the value (3 + ) = ( ). FACTORISATION (a) y + z = (y + z) y = ( y)( + y) y + z + ay + az = (y + z) + a (y + z) = (y + z)( + a) (d) = ( + 3)( + ). EXPANSION OF ALGERBRAIC EXPRESSIONS (a) 3 + = + = 3 = = 9. SIMULTANEOUS LINEAR EQUATIONS (a) Substitutio Method: y = () + y = () Substitute () ito () + = 7 = = 3 Substitute = 3 ito (), y = = Elimiatio Method: Solve: 3 + y = () y = () () + (), =, = 3 Substitute ito () 9 + y = y = 9 =

3 y =. ALGEBRAIC FORMULAE Give that k (m + ) = 3m, epress m i terms of k. k (m + ) = 3m k m = 3m k = 3m + m = m k m =. LINEAR INEQUALITIES. Solve the liear iequality 3 >. 3 > 3 > + 3 > >. List all iteger values of which satisfy the liear iequality + < + < Subtract, + < < =,, 3. Solve the simultaeous liear iequalities p 3 p ad p + p p 3 p p p 3 3p 3 p p + p, p + p p p p The solutio is p.. STATISTICS sum of data Mea = umber of data Mea = sum of(frequecy data), whe the data sum of frequecy has frequecy. Mode is the data with the highest frequecy Media is the middle data which is arraged i ascedig/descedig order.. 3, 3,,, 8 Mea = Mode = 3 Media =.,,, 8, 9,, there is o middle umber, the media is the mea of the two middle umbers. 8 Media = = 7. A pictograph uses symbols to represet a set of data. Each symbol is used to represet certai frequecy of the data. Jauary February March Represets books 3. A bar chart uses horizotal or vertical bars to represet a set of data. The legth or the height of each bar represets the frequecy of each data.. A pie chart uses the sectors of a circle to represet the frequecy/quatitiy of data. A pie chart showig the favourite driks of a group of studets. FORM FOUR NOTES. SIGNIFICANT FIGURES AND STANDARD FORM Sigificat Figures. Zero i betwee umbers are sigificat. Eample: 3 ( sigificat figures). Zero betwee whole umbers are ot sigificat figures. Eample: (3 sigificat figures) 3. Zero i frot of decimal umbers are ot sigificat. Eample:.3 ( 3 sigificat figures). Zero behid decimal umbers are sigificat. Eample:. ( sigificat figures) Stadard Form Stadard form are umbers writte i the form A, where A < ad are itegers. Eample: 3 = 3.. =. -. QUADRATIC EXPRESSION AND QUADRATIC EQUATIONS. Solve quadratic equatios by factorizatio. Eample: Solve k 8 k 3 k 8 = k k k 8 = (k + )(k ) = k =,. Solve qudratic equatio by formula: Eample: Solve 3 = = b b ac = (3)( ) a = 8 =.,.8 3. SET (a) Symbol - itersectio - uio - subset - uiversal set - empty set - is a member of 3

(A) umber of elemet i set A. A Complemet of set A. Ve Diagram Type III Premise : If A, the B Premise : Not B is true. Coclusio: Not A is true.

+ + = 9 (A B) = Gradiet Form: y = m + c m = gradiet c = y-itercept. MATHEMATICAL REASONING (a) Statemet A mathematical setece which is either true or false but ot both.

Three types of argumet: Type I Premise : All A are B Premise : C is A Coclusio: C is B Type II Premise : If A, the B Premise : A is true Coclusio: B is true.

4 (A) umber of elemet i set A. A Complemet of set A. Ve Diagram Type III Premise : If A, the B Premise : Not B is true. Coclusio: Not A is true.. THE STRAIGHT LINE (a) Gradiet A B A B Gradiet of AB = m = y y A Equatio of a straight lie Eample: (A) = 7 + = 3 (B) = + = (A B) = (A B) = = 3 (A B ) = 7 (A B) = (A B) = = 9 (A B) = Gradiet Form: y = m + c m = gradiet c = y-itercept. MATHEMATICAL REASONING (a) Statemet A mathematical setece which is either true or false but ot both. Implicatio If a, the b a atecedet b cosequet Itercept Form: y a b a = itercept b = y itercept p if ad oly if q ca be writte i two implicatios: If p, the q If q, the p Argumet Three types of argumet: Type I Premise : All A are B Premise : C is A Coclusio: C is B Type II Premise : If A, the B Premise : A is true Coclusio: B is true. y-it ercept Gradiet of straight lie m = -itercept = b a. STATISTICS (a) Class, Modal Class, Class Iterval Size, Midpoit, Cumulative frequecy, Ogive Eample : The table below shows the time take by 8 studets to type a documet. Time (mi) - -9 Frequecy 7

- -9 3-3 3-39 - -9 For the class : Lower limit = mi Upper limit = mi 9

5 For the class : Lower limit = mi Upper limit = mi 9 Lower boudary = 9. mi Upper boudary =. mi Class iterval size = Upper boudary lower boudary =. 9. = mi Modal class = 9 mi Midpoit of modal class = 9 = 7 To draw a ogive, a table of upper boudary ad cumulative frequecy has to be costructed. Time Upper Cumulative Frequecy (mi) boudary frequecy TRIGONOMETRY si o = Opposite AB hypoteuse AC cos o = adjacet BC hypoteuse AC ta o = opposite AB adjacet BC Acroym: Add Sugar To Coffee Trigoometric Graphs. y = si From the ogive : Media = 9. mi First quartile =. mi Third quartile = 3 mi Iterquartile rage = 3. = 9. mi. Histogram, Frequecy Polygo Eample: The table shows the marks obtaied by a group of studets i a test.. y = cos 3. y = ta Marks 3 3 Frequecy 8 8. ANGLE OF ELEVATION AND DEPRESSION (a) Agle of Elevatio

6 The agle of elevatio is the agle betweee the horizotal lie draw from the eye of a observer ad the lie joiig the eye of the observer to a object which is higher tha the observer. The agle of elevatio of B from A is BAC (a) Covert umber i base to a umber i base, or 8. Method: Repeated divisio. Eample: Agle of Depressio 3 = The agle of depressio is the agle betwee the horizotal lie from the eye of the observer a the lie joiig the eye of the observer to a object which is lower tha the observer. The agle of depressio of B from A is BAC. 9. LINES AND PLANES (a) Agle Betwee a Lie ad a Plae 3 = 8 Covert umber i base,, 8 to umber i base. Method: By usig place value Eample: (a) = 3 = = 7 = = + + = 9 I the diagram, (a) BC is the ormal lie to the plae PQRS. AB is the orthogoal projectio of the lie AC to the plae PQRS. The agle betwee the lie AC ad the plae PQRS is BAC Agle Betwee Two Plaes Covert betwee umbers i base, ad 8. Method: Number i base m Number i base Number i base. Eample: Covert to umber i base. 3 = = Therefore, = (d) Covert umber i base two to umber i base eight ad vice versa. Usig a coversio table I the diagram, (a) The plae PQRS ad the plae TURS itersects at the lie RS. MN ad KN are ay two lies draw o each plae which are perpedicular to RS ad itersect at the poit N. The agle betwee the plae PQRS ad the plae TURS is MNK. FORM NOTES. NUMBER BASES Eample : Base Base = 3 8

8 =. GRAPHS OF FUNCTIONS (a) Liear Graph y = m + c Reflectio Descriptio: Reflectio i the lie Eample: Reflectio i the

Eample: A clockwise rotatio of 9 o about the cetre (, ).

. TRANSFORMATION (a) Traslastio h Descriptio: Traslastio k Eample : Traslastio 3 (e) Eample : Elargemet of scale

7 8 =. GRAPHS OF FUNCTIONS (a) Liear Graph y = m + c Reflectio Descriptio: Reflectio i the lie Eample: Reflectio i the lie y =. Quadratic Graph y = a + b + c Rotatio Descriptio: Directio rotatio of agle about the cetre. Eample: A clockwise rotatio of 9 o about the cetre (, ). Cubic Graph y = a 3 + c (d) Reciprocal Graph a y (d) Elargemet Descriptio: Elargemet of scale factor, with the cetre.. TRANSFORMATION (a) Traslastio h Descriptio: Traslastio k Eample : Traslastio 3 (e) Eample : Elargemet of scale factor with the cetre at the origi. Area of image Area of object k = scale factor k Combied Trasformtios Trasformatio V followed by trasformatio W is writte as WV. 3. MATRICES (a) a c a c b d b d a ka k b kb 7

8 a c b e d g f ae bg h ce dg af bh cf dh a b (d) If M =, the c d M - d b = ad bc c a (e) If a + by = h c + dy = k a b h c d y k d b h y ad bc c a k (f) a c Matri has o iverse if ad bc = b d. VARIATIONS (a) Direct Variatio If y varies directly as, Writt i mathematical form: y, Writte i equatio form: y = k, k is a costat. Iverse Variatio If y varies iversely as, Writte i mathematical form: Writte i equatio form: y k y, k is a costat. Joit Variatio If y varies directly as ad iversely as z, Writte i mathematical form: Writte i equatio form: costat. y, z k y, k is a z. GRADIENT AND AREA UNDER A GRAPH (a) Distace-Time Graph Gradiet = Rate of chage of speed = v u t = acceleratio Distace = Area below speed-time graph. PROBABILITY (a) Defiitio of Probability Probability that evet A happe, A ( ) PA ( ) S ( ) S = sample space Complemetary Evet P(A) = P(A) Probability of Combied Evets (i) P(A or B) = P(A B) (ii) P(A ad B) = P(A B) 7. BEARING Bearig Bearig of poit B from A is the agle measured clockwise from the orth directio at A to the lie joiig B to A. Bearig is writte i 3 digits. Eample : Bearig B from A is o 8. THE EARTH AS A SPHERE (a) Nautical Miles autical mile is the legth of the arc o a great circle which subteds a agle of at the cetre of the earth. Distace Betwee Two Poits o a Great Circle. Gradiet = distace time = speed Distace = autical miles = agle betwee the parallels of latitude measured alog a meridia of logitude. Total distace Average speed = Total time Speed-Time Graph 8

9 (ii) the elevatio of the combied solid o the vertical plae parallel to GPS as viewed from D. = agle betwee the meridias of logitude measured alog the equator. (a) Distace Betwee Two Poits o The Parallel of Latitude. Distace = cos o = agle of the parallel of latitude. (i) Pla (d) (e) Shortest Distace The shortest distace betwee two poits o the surface of the earth is the distace betwee the two poits measured alog a great circle. Kot kot = autical mile per hour. (ii) C-elevatio 9. PLAN AND ELEVATION (a) The diagram shows a solid right prism with rectagular base FGPN o a horizotal table. The surface EFGHJK is the uiform cross sectio. The rectagular surface EKLM is a slatig plae. Rectagle JHQR is a horizotal plae. The edges EF, KJ ad HG are vertical. Draw to full scale, the pla of the solid. D-elevatio A solid i the form of a cuboid is joied to the solid i (a) at the plae PQRLMN to form a combied solid as show i the diagram. The square base FGSW is a horizotal plae. Draw to full scale (i) the elevatio of the combied solid o the vertical plae parallel to FG as viewed from C, 9

SS3 QUESTIONS FOR 2018 MATHSCHAMP. 3. How many vertices has a hexagonal prism? A. 6 B. 8 C. 10 D. 12

SS3 QUESTIONS FOR 2018 MATHSCHAMP. 3. How many vertices has a hexagonal prism? A. 6 B. 8 C. 10 D. 12 SS3 QUESTIONS FOR 8 MATHSCHAMP. P ad Q are two matrices such that their dimesios are 3 by 4 ad 4 by 3 respectively. What is the dimesio of the product PQ? 3 by 3 4 by 4 3 by 4 4 by 3. What is the smallest