F.3 Mathematics Final Term Exam Syllabus

Size: px

Start display at page:

Download "F.3 Mathematics Final Term Exam Syllabus"

Judith Harmon
6 years ago
Views:

1 F.3 Mathematics Final Term Exam Syllabus The Final Term Exam cnsists f: 20 Multiple Chice Questins (40 Marks), 10 Extended Questins (60 Marks) = Ttal 100 Marks Allcated time: 90 minutes Yu will need: black r blue pen (fr all answers except drawings/cnstructins), pencil, straight-edge, cmpass, prtractr (fr checking, nt fr cnstructin), ne sheet f tracing paper. The fllwing is a summary f frmulae and cncepts yu need t knw. This is in n way cmprehensive and it is yur respnsibility t g thrugh all class ntes, wrksheets and hmewrk t ensure full cverage f the exam syllabus. Chapter 1: Mre abut Factrisatin f Plynmials 1.1 Factrisatin Using Identities Difference f Tw Squares Identity: (+)( ) Perfect Square Identities: (+) +2+ and ( ) Factrisatin using Crss Methd 1.3 Factrisatin Using the Difference and Sum f Tw Cubes Identities Sum f Tw Cubes Identity: + (+)( + ) Difference f Tw Cubes Identity: ( )( ++ ) Chapter 2: Law f Indices + IGCSE Standard Frm (nte: Standard Frm = Scientific Ntatin) 2.1 Zer Index and Negative Integral Indices Zer Index: =1, where 0 Negative Integral Indices: =, where 0 and is a psitive integer. Laws f Integral Indices (where and are integers and, 0): = = ( ) = () = = 2.2 Scientific Ntatin A psitive number expressed in scientific ntatin is in the frm f 10, where 1 <10 and is an integer. [IGCSE] Adding, subtracting, multiplying and dividing tw numbers in standard frm. 2.3 Ntatin fr Different Numeral Systems Decimal system: 236 in expanded frm is (10,10,1 are place values.) Binary system: 1101 in expanded frm is (2,2,1 are place values, and 0 is a place hlder.) Hexadecimal system: 4 in expanded frm is (16,16,1 are place values.) 2.4 Inter-cnversin f Numbers f Different Numeral Systems T cnvert frm Binary r Hexadecimal t Decimal, evaluate the expanded frm. Example: 1101 = =8+4+1=13 Example: 4 = = =3146 T cnvert frm Decimal, use lng divisin. Examples: cnverting 21 int binary and 3108 int hexadecimal T cnvert between binary and hexadecimal, cnvert t decimal first. pg. 1

2 Chapter 12: Crdinate Gemetry f Straight Lines 12.1 Distance between Tw Pints The distance frmula is used t measure the distance between pints (, ) and (, ) and is given by: =( ) +( ) 12.2 Slpe f a Straight Line The slpe frmula used t measure the slpe f, where (, ) and (, ), is given by: = Nte 1: the slpe can be psitive (slping up frm left t right), negative (slping dwn frm left t right), zer (hrizntal line) r undefined (vertical line). Nte 2: Even thugh a slpe f 5 is steeper than a slpe f 3, the value f 5 is smaller than 3. The slpe and inclinatin (see Chapter 5) are related as fllws: h = 12.3 Parallel Lines and Perpendicular Lines If tw lines with slpes and are parallel, then = If tw lines with slpes and are perpendicular, then = Pint f Divisin A pint f divisin is a pint that cuts a line segment int tw parts. Mid-pint Frmula If (,) is the mid-pint f line segment jining (, ) and (, ), then we use the mid-pint frmula: = +,= Sectin Frmula If (,) is a pint n the line segment jining (, ) and (, ) such that :=: then we use the mid-pint frmula: = +,= Nte: if yu are required t find the rati f :, let :=1: s use either = Applicatins f the Analytic Apprach in Gemetry r = + 1+ Chapter 10: Area and Vlume (III) 10.1 Pyramids Vlume f a pyramid= base area height Vlume f a frustum=vlume f the larger pyramid vlume f the smaller pyramid Ttal surface area f a pyramid=ttal area f all lateral faces+area f the base Nte 1: make sure yu knw where the height is! (Yu may need t use Pythagras) Nte 2: make sure yu knw what the terms slant height, right pyramid and regular pyramid mean! Nte 3: if yu are asked t find the surface area f a frustum, dn t frget the flat surface n the tp 10.2 Circular Cnes The distance between any pint f the circumference f the base and the vertex f the right circular cne is the slant height l. It is given by the Pythagrean frmula l = +h where is the radius f the base area and h is the height f the cne. Vlume f a circular cne= h Vlume f a frustum=vlume f the larger cne vlume f the smaller cne Ttal surface area f a pyramid=curved surface area+area f the base=l+ Nte: if yu are asked t find the surface area f a frustum, dn t frget the flat surface n the tp 10.3 Spheres Vlume f a sphere = pg. 2

3 Surface area f a sphere=4 Nte: if yu are asked t find the surface area f a hemisphere r quarter-sphere, dn t frget t include the flat surfaces (if necessary) 10.4 The Dimensins f Length, Area and Vlume Linear measurements apply t lengths f line segments r curves. Dimensin = 1 Quadratic measurements apply t areas f planes r curved figures. Dimensin = 2 Cubic measurements apply t vlumes f slids r space. Dimensin = 3 Hint: cunt the number f linear measurements required t calculate. Example 1: =2 nly ne measurement () is required therefre is linear Example 2: =l+ tw measurements ( and l in the first term; is used twice in the secnd term) are required therefre is quadratic. Here, the TSA is the sum f tw ther areas. Example 3: = h three measurements ( is used twice, h nce) are required therefre is cubic Similar Plane Figures and Similar Slids Recall that similarity means tw bjects that have the same shape. Fr tw similar 2-D figures: if and are their areas, l and l are tw crrespnding linear measurements, then = l l Fr tw similar 3-D slids: if and are tw crrespnding surface areas, and l and l are tw crrespnding linear measurements, then = l l if and are their vlumes, and l and l are tw crrespnding linear measurements, then = l l IGCSE: Transfrmatin [IGCSE] Reflectin Objects can be reflected abut a mirrr line. The mirrr line can be a vertical line, hrizntal line, r a diagnal line (examples: =, =, =+1) If yu are asked t describe this transfrmatin, yu must specify the mirrr line Example: Reflectin abut = 4. [IGCSE] Rtatin Objects are typically rtated abut 90 clckwise, 180 r 90 cunter-clckwise (anti-clckwise) If yu are asked t describe this transfrmatin, yu must specify the rtatin and the centre f rtatin Example: Rtatin 90 CCW abut (0,2). [IGCSE] Translatin Objects are translated hrizntally (with respect t -axis) and vertically (with respect t -axis) If yu are asked t describe this transfrmatin, yu must specify the hrizntal and vertical cmpnents f the translatin Example: Translatin by 4 units left and 3 units up r Translatin by 4 3 ) [IGCSE] Enlargement Objects are enlarged (scale factr > 1) r reduced (scale factr between 0 and 1). If yu are asked t describe this transfrmatin, yu must specify the scale factr and the centre f enlargement Example: Enlargement with scale factr 2 and centre at (1, 1). Chapter 3: Percentages 3.0 Pre-Learning Simple Interest: with principal (), interest rate (%) and perid f time (), The final amunt is =1+ The simple interest is = = 3.1 Cmpund Interest Cmpund Interest: with principal (), interest rate (%) and perid f time (), The final amunt is =1+ pg. 3

The cmpund interest is = =1+ 3.2 Increasing at a Cnstant Rate: with quantity (), rate (%) and perid f time (), The new value is = 1+ The grwth factr is 1+ 3.

4 The cmpund interest is = = Increasing at a Cnstant Rate: with quantity (), rate (%) and perid f time (), The new value is = 1+ The grwth factr is Decreasing at a Cnstant Rate: with quantity (), rate (%) and perid f time (), The new value is = 1 The decay factr is Mre abut Percentage Changes Successive Percentage Change Example: if a quantity underges an increase f % fllwed by a decrease f % then: The final value = (1+%)(1 %) Percentage Change = 100% 3.5 Taxatin Valuatin Rates =rateable value rates percentage change. T find the rates fr each quarter, divide by 4. Salaries Tax If there is an allwance (r deductin), then the net chargeable incme is fund by subtracting it frm the incme. The Salaries Tax is calculated by splitting up the net chargeable incme int tax bands and multiplying each f them by the crrespnding fixed tax rate. The tax rates increase and are called prgressive rates. Chapter 4: Linear Inequalities in One Unknwn + IGCSE Inequalities 4.1 Basic Knwledge f Inequalities T find the slutin f an inequality, make the unknwn a subject. Represent the slutin f inequalities using the number line. Example: >7 Basic Prperties f Inequalities Transitive Prperty: If > and > then > Additive Prperty: If > then +>+ (and > ) Multiplicative Prperty: If > and >0 then > (and > ) Als, if <0 then < (and < ); that is, the inequality sign is reversed. Reciprcal Prperty: If >>0, then < ; that is, the inequality sign is reversed. 4.2 Linear Inequalities in One Unknwn [IGCSE] Represent slutins f inequalities with tw variables n the crdinate axes. (This means yu need t knw hw t find the equatin f straight lines.) Example: Shade the regin that satisfies >1,<3, 4, >1 means the crdinates lie t the right f the line =1, s the regin is t the right f this line =1 =4 = =3 4 means the crdinates lie belw and n the line =4, s the regin is belw this line means the crdinates lie abve and n the line =, s the regin is abve the line <3 means the crdinates lie t the left f the line =3, s the regin is t the left this line pg. 4

Chapter 9: Mre abut 3-D figures 9.0 Key Pints frm Mathematics in previus years (see NCM 3B, pp. 72-73) Reflectinal Symmetry Rtatinal Symmetry 2-D Representatin f Slids Euler s Frmula: + =2 9.

5 Chapter 9: Mre abut 3-D figures 9.0 Key Pints frm Mathematics in previus years (see NCM 3B, pp ) Reflectinal Symmetry Rtatinal Symmetry 2-D Representatin f Slids Euler s Frmula: + =2 9.1 Symmetry f 3-D Figures Reflectinal Symmetry Rtatinal Symmetry Summary f reflectinal and rtatinal symmetries f a cube and tetrahedrn: CUBE TETRAHEDRON 3 lateral planes 6 diagnal planes: each plane uses tw ppsing edges Ttal: 9 planes f reflectin REFLECTIONAL SYMMETRY each plane uses an edge and the midpint f the ppsite edge Ttal: 6 planes f reflectin ROTATIONAL SYMMETRY 3 axes thrugh the centre f ppsing faces (rder 4) 4 axes thrugh ppsing vertices (rder 3) 6 axes thrugh ppsing edges (rder 2) Ttal: 13 axes f rtatin 4 axes thrugh a vertex and the ppsing face (rder 3) 3 axes thrugh ppsing faces (rder 2) Ttal: 7 axes f rtatin 9.2 Nets f 3-D Figures Yu will need t identify cinciding vertices and/r edges. 9.3 Further Knwledge n 2-D Representatins f 3-D Objects 9.4 Pints, Straight Lines and Planes f 3-D Figures Distance between a Pint and a Straight Line If is a pint that is sme distance frm line l, then the shrtest distance between and l is where is a pint alng l and is perpendicular t l. is the prjectin f n l. Distance between a Pint and a Plane If is a pint that is utside plane, then the shrtest distance between P and is where is a pint n and is perpendicular t every line n which passes thrugh. is the prjectin f n. Relatinship between Tw Straight Lines If tw lines l and l are parallel, then the distance between them is cnstant. If tw lines l and l are nt parallel, then the angle between the lines is the acute angle at the pint f intersectin. pg. 5

Relatinship between a Straight Line and a Plane A straight line l and plane intersect at the pint f intersectin.

If =90, then the prjectin f line l is just the pint f intersectin. Relatinship between Tw Planes When tw planes intersect, they intersect at a straight line called the line f intersectin.

T find the angle between tw planes: Identify the line f intersectin l Find a pint n l such that tw lines ne n each plane that ends at is perpendicular t l.

6 Relatinship between a Straight Line and a Plane A straight line l and plane intersect at the pint f intersectin. If a pint n is a prjectin f a pint n l, then the line segment is the prjectin f line segment. The acute angle between and is the angle between the line l and plane. If =90, then the prjectin f line l is just the pint f intersectin. Relatinship between Tw Planes When tw planes intersect, they intersect at a straight line called the line f intersectin. The angle between the tw planes is the acute angle between tw lines ne n each plane that are perpendicular t the line f intersectin. T find the angle between tw planes: Identify the line f intersectin l Find a pint n l such that tw lines ne n each plane that ends at is perpendicular t l. The angle between the planes is the acute angle between the tw lines. 9.5 Knwledge n Regular Plyhedra A regular plyhedrn, r Platnic slid, is a 3-D bject where each face is frmed by identical regular plygns and the same number f faces meet at each vertex. There are 5 regular plyhedral: Chapter 11: Applicatins in Trignmetry 11.0 Use the mnemnic SOHCAHTOA t remember the trig. ratis: sin=, cs=, tan= 11.1 Gradients Gradients (in this curriculum) is a psitive quantity t indicate hw steep a slpe is. It is calculated as: vertical rise gradient= hrizntal run Gradients are typically expressed as a fractin r as a rati in the frm 1: ( is an integer) The angle between the slpe and the hrizntal is the inclinatin. The relatinship between gradient and inclinatin is given by: gradient=tan When calculating gradients n a map, pay attentin t the scale. Yu will need t make sure bth the vertical rise and hrizntal run measurements have the same units. pg. 6

7 11.2 Angles f Elevatin and Depressin When lking at an bject that is lcated abve the hrizntal, the angle f elevatin is the acute angle frmed between the line f sight (direct line between yur eyes and the bject) and the hrizntal. When lking at an bject that is lcated belw the hrizntal, the angle f elevatin is the acute angle frmed between the line f sight and the hrizntal. Bearings are used t indicate the relative psitins f bjects n a hrizntal plane. (The angles f elevatin and depressin are used t indicate relative psitins f bjects n a vertical plane.) Reduced bearing (r cmpass bearing) is made up f 3 parts: r, an acute angle, r. Examples: 35, 89, 1 Whle circle bearing (r true bearing) is the angle measurement taken in a clck-wise directin frm nrth. The integral part f the angle is written in 3 digits. Examples: 135,001,359.9 Chapter 5: Intrductin t Prbability + IGCSE Prbabilities 5.1 The Meaning f Prbability The prbability f an event happening is given by ()= ( ) The prbability f an event is within the range f 0 (impssible) t 1 (certain): 0 () Mre Abut Prbability Use a tree diagram r tabulatin methd t describe prbability scenaris. Remember t list the utcmes if asked! [IGCSE] It als helps t write the prbability values n the branches f the tree diagram r in each cell f the table. Gemetric Prbability: If an event happens in a certain regin f the figure, then ()= 5.3 Experimental Prbability Experimental Prbability: If an event happens ver a number f trials, then Experimental Prbability= Theretical Prbability: This is the prbability f event happening using deductive reasning. Example: the theretical prbability f getting a 1 n a fair dice is. Fr a large number f trials, experimental prbability theretical prbability. 5.4 Expected values Expected number f ccurrences: If the prbability f an event happening is, then we expect that after trials this event will ccur times. Expected value f a variable: If a variable has n pssible values,,,, and the crrespnding prbabilities are,,, respectively, then the expected value f the variable =,,, [IGCSE] Cmplementary prbability: The prbability f an event happening () is cmplementary t the prbability f the event nt happening ( ), thus ( )=1 () [IGCSE] Tw events are said t be Mutually Exclusive if they d nt have shared utcmes. If tw events and are Mutually Exclusive, we can apply the Additive Law: ( r )=()+() If tw events are nt mutually exclusive, we need t take away the duplicates. (D nt cunt the same utcme mre than nce). [IGCSE] Tw events are said t be Independent if, after ne event, the prbability f the ther is unaffected. If tw events and are Independent, we can apply the Multiplicative Law: ( and )=() () If tw events are nt independent, then we need t first wrk ut the prbability f the secnd event after the first event has ccurred befre multiplying the tw prbabilities tgether. Chapter 7: Mre n Deductive Gemetry 7.0 Key References frm Mathematics in previus years (see NCM 3A, pp ) Angles in Triangles: [ sum f ], [ext. f ] Cngruent Triangles [crr. s, s],[crr. sides, s], [SSS],[ASA],[AAS], [SAS],[RHS] Similar Triangles [crr. s,~ s],[crr. sides,~ s], [],[3 sides prprtinal], [rati f 2 sides,inc. ] Issceles Triangles [base s,iss. ], [axis f symmetry f iss. bisects vert. ], [axis f symmetry f iss. bisects the base], [axis f symmetry f iss. base], [sides pp.eq. s] pg. 7

8 7.1 Using the Deductive Apprach t Slve Gemetric Prblems n Triangles First, find ut what the end gal is. Example: if yu need t prve ne line is a perpendicular bisectr f anther, then yu need t prve a right angle exists and that tw lengths are equal. Next, lk fr the given infrmatin. Tgether with any cmmn side r angle, can yu find a pair f cngruent r similar triangles? Hint: yu may need t add auxiliary (extra) lines. 7.2 Special Lines in Triangles Angle Bisectr Perpendicular Median Altitude Bisectr A straight line which bisects an interir angle f a triangle. A line segment jining a vertex t the mid-pint f the ppsite side. A straight line which is perpendicular t and bisects ne side f a triangle A line segment jining a vertex t the ppsite side. This line segment and the ppsite side are perpendicular t each ther. 7.3 Relatins between Lines in a Triangle Triangle Inequality: the sum f any tw sides must be greater than the third Cncurrent lines: three lines are cncurrent if they intersect at the same pint Centres f Triangles: In-centre Circumcentre Centrid Orthcentre The intersectin f angle bisectrs. The intersectin f perpendicular The intersectin f medians. The intersectin f altitudes. bisectrs. It is always inside the triangle. It can be inside (acute), n the hyptenuse (rightangled) r utside the triangle (btuse). It is always inside the triangle. It can be inside (acute), n the right-angled vertex (right-angled) r utside the triangle (btuse). The crrespnding inscribed circle is the largest circle that fits inside the triangle. The crrespnding circumcircle is the smallest circle that enclses the triangle. Chapter 8: Quadrilaterals 8.0 Key References frm Mathematics in previus years (see NCM 3B, pp. 2-3) Angles Relating t Parallel Lines. Use these if // is given r deduced [crr. s, AB//CD] [alt. s, AB//CD] [int. s, AB//CD] Cnditins fr Parallel Lines. Use these if and are given r deduced t prve // [crr. s equal] [alt. s equal] [int. s supp.] pg. 8

8.1 Definitins (and Prperties) f Special Quadrilaterals Trapezium: a quadrilateral with nly ne pair f parallel ppsite sides Parallelgram: a quadrilateral with tw pairs f parallel ppsite sides Rhmbus:

9 8.1 Definitins (and Prperties) f Special Quadrilaterals Trapezium: a quadrilateral with nly ne pair f parallel ppsite sides Parallelgram: a quadrilateral with tw pairs f parallel ppsite sides Rhmbus: a quadrilateral with fur equal sides Rectangle: a quadrilateral with fur equal angles Square: a quadrilateral with all sides equal and all angles equal 8.2 Parallelgrams Prperties f Parallelgrams. If parallelgram is given r deduced The ppsite sides f a parallelgram are equal [pp.sides f //gram] The ppsite angles f a parallelgram are equal [pp. s f //gram] The diagnals f a parallelgram bisect each ther [diagnals f //gram] Tests fr Parallelgrams. If yu need t prve a quadrilateral is a parallelgram Bth pairs f ppsite sides f a quadrilateral are equal [pp.sides equal] Bth pairs f ppsite angles f a quadrilateral are equal [pp. s equal] The diagnals f a quadrilateral bisect each ther [diags.bisect each ther] One pair f ppsite sides f a quadrilateral are equal and parallel [2 sides equal and //] 8.3 Rhmbuses, Rectangles and Squares Prperties f Rhmbuses All prperties f a parallelgram Diagnals are perpendicular t each ther [prperty f rhmbus] Each interir angle is bisected by the diagnal Nte: A quadrilateral with diagnals perpendicular t each ther is nt necessarily a rhmbus [eg. kite], but a parallelgram with diagnals perpendicular t each ther must be a rhmbus. Prperties f Rectangles All prperties f a parallelgram All angles are right angles Diagnals are equal [prperty f rectangle] Diagnals bisect each ther int 4 equal segments Prperties f Squares All prperties f a rectangle All prperties f a rhmbus [prperty f square] Angle between a side and the diagnal is 45º 8.4 Prfs Related t Parallelgrams In the flwchart belw yu can see that prperties f certain quadrilaterals als apply t ther quadrilaterals. Example: all prperties f a rhmbus apply t a square. 8.5 Mid-pint Therem and Intercept Therem Mid-pint Therem The straight line jining the mid-pints f tw sides f a triangle is parallel t the third side and its length is half the third side s length. [mid pt.therem] Intercept Therem When three r mre parallel straight lines make equal intercepts n a given transversal, these lines will als make equal intercepts n any ther transversal. [intercept therem] pg. 9

10 NOT IN F.3 FINAL EXAM: IGCSE Transfrmatin Negative Enlargement Objects are flipped if the scale factr is less than 0. IGCSE Quadratic Inequalities [IGCSE] Find the slutin f quadratic inequalities and represent them n a number line. Example: 9 > 0 Rearranging gives > 9. If we let = 9, then we have = 3 and = 3. When we test, say = 0, we see that 0 < 9 and therefre = 0 is nt a slutin. Thus, < 3 and > 3. pg. 10

Geometry Strand Students will use visualization and spatial reasoning to analyze characteristics and properties of geometric shapes.

Geometry Strand Students will use visualization and spatial reasoning to analyze characteristics and properties of geometric shapes. 2005 Gemetry Standards Algebra Strand Nte: The algebraic skills and cncepts within the Algebra prcess and cntent perfrmance indicatrs must me maintained and applied as students are asked t investigate,