Worked solutions. 1 Algebra and functions 1: Manipulating algebraic expressions. Prior knowledge 1 page 2. Exercise 1.2A page 8. Exercise 1.
|
|
- Annabelle Chrystal Powers
- 5 years ago
- Views:
Transcription
1 WORKED SOLUTIONS Worked solutions Algera and functions : Manipulating algeraic epressions Prior knowledge page a (a + ) a + c( d) c cd c e ( f g + eh) e f e g + e h d i( + j) + 7(k j) i + ij + 7k j e l(l + ) (l + l ) l + l l l l l f m(m n) (m + n) m mn m n a ( + )( + ) + + ( )( + ) + c ( )( ) + d ( ) + 9 e ( + )( + ) + + f ( + )( + )( + ) a a a a a a a 9 c d (c ) c e (d) d f ( e f ) ef ef a c 7 d e f Eercise.A a + + a + a page HarperCollinsPulishers 7 Edecel A-level Mathematics Year and AS 979 ( )( + ) ( )( + ) + ( + ) (n) + (n + ) n + n + n + n + n + (n + n + ) Since is a multiple of, (n + n + )must e an even numer. ( + )( + ) ( + )( ) ( + ) + ( + ) ( ) + ( ) Should e: ( + ) + ( + ) ( ) ( ) Should e: Should e: Let p n + where n is an integer. Consequentl p is an odd numer. p (n + ) n + n + (n + n) + Since (n + n) is even, (n + n) + is odd. p(n +)(m + ) p(mn + n + m + ) mnp + pn + mp + p (mnp + pn + mp + p), an even numer Eercise.A page a ( + ) ()() () + ()() () + ()() () + ()() () ( + ) ()() () + ()() () + ()() () + ()() () + ()() ()
2 ALGEBRA AND FUNCTIONS : MANIPULATING ALGEBRAIC EXPRESSIONS a ( ) ()() ( ) + ()() ( ) + ()() ( ) + ()() ( ) + ()() ( ) + ()() ( ) ( ) ( ) ()() ( ) + ()() ( ) + ()() ( ) + ()() ( ) + ()() ( ) + + ( + )( ) ( + )( + + ) + + ()() ( ) ( + ) ()( ) ( ) + ()( ) ( ) + ()( ) ( ) + ()( ) ( ) ( ) ()( ) ( ) + ()( ) ( ) + ()( ) ( ) + ()( ) ( ) ( + ) ()()() (c) + ()()() (c) 9 c + c 9 c + c c + c (c )(c + ) c or c Eercise.B page a c a c d The are the coefficients for a cuic epansion from Pascal s triangle. C, C, C, C, C, C All the rows prior to the inde required have to e written in Pascal s triangle: a laorious and error-prone eercise. C 9! 79!! 7 Eercise.A page a ( + ) C + C () + C () + C () + C ()! + + (!!!! )(!!!! ) +!! ( )+ ( )!!!! ( + ) C + C () + C () + C ()! 7! 9!!!!!!! +!!!( ) a ( + ) )(! 7 ( + ) ( + ) ( ) )( (! ) a + C + C + C + C + +!!!!!!!!! +!!! ( ) C () + C () ( ) a ( + ) ( + ) + C () ( ) + C () ( ) + C ( )! 7!!!!! +! ( 9!! )! + ( )!! 9 +! (!! ) )(! )(! HarperCollinsPulishers 7 Edecel A-level Mathematics Year and AS 979
3 WORKED SOLUTIONS 7 7 ( + ) + 7 ( 77 )( 7 )! 7 7 C + C () () + C() () + C + C +!! 7!!!! +!! +!!!!! !! 9 ( ) C 7 7 C +! +!! 9 7 9! 7!! C() + C () + C() () + C! 79!!!!! + +!!! +! 7!! Eercise.A a ( + ) ( + ) c ( ) d + z + z doesn t factorise. a ( )( ) ( + )( + ) c ( + )( ) d ( 7)( + 7) 7( ) Should e: 7( ) page a ( ) ( )( ) c ( + ) d ( ) e (9 + 7z)(9 7z) f ( 7 + ) the quadratic cannot e factorised. + ( )( + ) or ( )( + ) (7 + )( ) 7 ( + )( + ) ( + )( + ) ( + )( + ) ( + )( + ) ( + )( + ) ( + )( + ) ( + )( + ) ( + )( + ) ( + )( + ) ( + )( + ) ( + )( + ) ( + )( + ) p(q r ) r(p q) pq pr pr qr p (q r r) rq qr qr p or p q r r r + r q 9 ( ) + ( + ) Eercise.A page a + + f() + ( ) + so ( ) is a factor of f(). + + ( )( + )( + ) a + ( + ) ( ) + + ( + ) ( + ) + + ( + )( ) + ( + ) a f() f() so ( ) is a factor ( )( )( + ) HarperCollinsPulishers 7 Edecel A-level Mathematics Year and AS 979
4 ALGEBRA AND FUNCTIONS : MANIPULATING ALGEBRAIC EXPRESSIONS f() f( ) so ( + ) is a factor ( + )( + + ) g() + + g( ) so ( + ) is a factor ( + )( + )( )( ) c f() + f() + so ( ) is a factor. + ( )( + 9 ) g() + 9 g( ) ( ) + ( ) 9( ) so ( + ) is a factor. + ( )( + )( + )( ) a (A + B + C)( ) + D : D : (C)( ) + D C Equating coefficients: : A : A + B B ( )( ) (A + B + C)( ) + D : D : (C)( ) + D C 7 Equating coefficients: : A : A + B B ( + + 7)( ) + c + + (A + B + C)( + ) + D : + D : (C)() C Equating coefficients: : A : A + B D B + + ( + )( + ) a f() + f() + f() f( ) 7 9 c f() f( ) a + (A + B + C + D)( ) + E : + E E : (D)( ) + D Equating coefficients: : A : A + B B : B + C C + ( + + )( ) (A + B + C + D)( + ) + E : 7 E : (D)() 9 D Equating coefficients: : A : A + B B : B + C E 9 C ( 7 + )( + ) 9 f() f() + + so ( ) is a factor ( )( + + 9) g() g() so ( ) is a factor. g() ( )( ) f() ( + )( )( ) HarperCollinsPulishers 7 Edecel A-level Mathematics Year and AS 979
5 WORKED SOLUTIONS Eercise.A page Eercise.7A page a a 9 c c d 7d e e f f a ± c 7 7 a a c c or c 7 d d e 9e f f g or f g a ( ) c ± ( ) Should e: 7 7 Area π a π or π 9 a 9 or a c c c c 9 Volume π 9 π and z z z z 7 or a a a e + e f f a 7c d d a g h + h gh j k k j a ( a) ( a)( a) HarperCollinsPulishers 7 Edecel A-level Mathematics Year and AS 979 a a + a a + a 7 ( a + )( a ) a ( a + a) ( a + a)( a + a) a+ a + a + a a + a + a 9 a a ( a a)( a + a) a ( a)( + a) Eercise.A page 7 a + c a c + a c + + a c a c + 9
6 ALGEBRA AND FUNCTIONS : MANIPULATING ALGEBRAIC EXPRESSIONS a + + c Eam-stle questions page 9 a a + a f() + + f() so ( ) is a factor of f(). f( ) so ( + ) is not a factor of f(). C 9 + ( 7) ( 7)( + ) ( 7)( + )( ) 7 HarperCollinsPulishers 7 Edecel A-level Mathematics Year and AS ( ) a+ a ( a+ ) ( a+ )+ ( a ) a a + a + + a a + a + which is rational if a and are rational. 9 a f() 9 + f() 9 + hence ( ) is a factor of f(). + ( ) ( )( + ) g() + g( ) + + hence ( + ) is a factor of g(). ( + ) + + f() 9 + ( )( + ) ( ) a ( ) The inde on the racket should e and the contents of the racket should e. The coefficient of < (it is ) (A + B + C)( ) + D : D D 79
7 WORKED SOLUTIONS : (C)( ) + 79 C Equating coefficients: : A : A + B B ( + + )( ) ± a a a ( ) a For : a a a 7 a ( ) C () + C () ( ) + C () ( ) + C () ( ) + C ( )! + +!!!!! ( )!!! ( ) +!!! ( ) +!!! ( ) c. (.) ( + )( ) ( + )( 7 ) So one numer is too high. As the numers are smmetrical, the should e a. 7 ( a+ ) ( a ) ( a+ ) ( a ) ( a ) a ( + ) a a a a a + a a + a a a + a which is irrational unless is a square numer. a HarperCollinsPulishers 7 Edecel A-level Mathematics Year and AS f() f() so ( ) is a factor of f() ( ) ( )( + 9 7) g() g( ) so ( + ) is a factor of g(). 9 ( + ) f() ( )( + )( )( + ) (A + B + C + D + E)( + ) + F : + + F F : (E)() E Equating coefficients: : A : 9 A + B 7
8 ALGEBRA AND FUNCTIONS : EQUATIONS AND INEQUALITIES B : B + C C : C + D D ( )( + ) a + C + C + C + C! +! (! )+ (! )+ ( )!!!! + + +!!!! ( ) + Total numer of alls is (7 ) + (9 ) + P( RR) a + C + C + C + C + +!! +! +!...!!!!!!!! c It is onl an approimation ecause the epansion isn t complete When + 7 is divided + to get an epression for the area of the cross-section (width depth), then there is a remainder of. + Consequentl, this is not a possile correct epression for the volume of the cuoid. When + is divided + to get an epression for the area of the cross-section (width depth) then there is no remainder. Consequentl, this is a possile correct epression for the volume of the cuoid. Algera and functions : Equations and inequalities Prior knowledge page ( + )( ) or + ( ) ( ) or a,, c ac ( ) ()()( ) so two real roots. ± ±. or. + Add and. 7, + + Sustitute into ( ) + ( ) < 7 < 7 < c + or a HarperCollinsPulishers 7 Edecel A-level Mathematics Year and AS 979
9 WORKED SOLUTIONS Eercise.A page a where. Intercepts and turning point at (, ). No -intercept, -intercept and turning point at (, ). c + where. -intercepts at (, ) and (, ), -intercept at (, ) and turning point at, a Line of smmetr at, -intercept at (, ), -intercept and turning point at (, ) Line of smmetr at.7. -intercepts at (, ) and (, ), -intercept at (, ) and turning point at (.7,.) HarperCollinsPulishers 7 Edecel A-level Mathematics Year and AS 979 9
10 ALGEBRA AND FUNCTIONS : EQUATIONS AND INEQUALITIES c Line of smmetr at.. -intercepts at (, ) and (, ), -intercept at (, ) and turning point at (.,.) Line of smmetr at. (appro.). -intercepts at (.7, ) and (, ), -intercept at (, ) and turning point at appro. (., ). ( + )( ) Epand the rackets. 7 This equation has the approimate -intercepts and the -intercept at the turning point when.,. in the equation. 9 Rearranging Solutions for 9 can e found where the graphs intersect, hence and. HarperCollinsPulishers 7 Edecel A-level Mathematics Year and AS 979
11 WORKED SOLUTIONS Eercise.A page a + + ac () ()()() ac < so no real roots. + + ac () ()()() ac > so two distinct real roots. c + + ac () ()()() ac so two equal real roots. + ac ( ) ()()() ac > so has two distinct real roots. a ac ( ) ()()( ) 9 ac > so two distinct real roots. + 7 ac ( ) ()()(7) ac < so no real roots. c + ac ( ) ()()() ac so two equal real roots. + (k + ) + (k ) ac so two equal real roots. (k + ) ()()(k ) Epand and simplif. k k + or k + k Both quadratic equations have the solutions k or k. For k, and for k,. a + ac () ()( )() ac > so two distinct real roots. 9 + ac ( ) ()()(9) ac so two equal real roots. c + ac ( ) ()()() 9 ac < so no real roots. + c a ac < () ()()( c) < + c < c < ac () ()()( c) + c c c ac > () ()()( c) > + c > c > Eercise.A a + ( + ) ( ) [( ) ] c ( + ) + 7 ( + ) 9 + ( ) + ( ) a ( ) ( ) ( ) c ( + ) + ( ) + + page HarperCollinsPulishers 7 Edecel A-level Mathematics Year and AS 979
12 ALGEBRA AND FUNCTIONS : EQUATIONS AND INEQUALITIES + 7 ( ) ( ) At the turning point has a minimum value, i.e. when. When,, the coordinates of the turning point. [( + ) ] ( + + ) a + + ( + ) ( ) + ( ) ut can go no further. c ( + ) 7 ( ) ( ) ( ) > Consequentl ac < ecause the curve of this equation will not intercept the -ais and so there are no real roots. ( )[( + ) ] ( )( + + ) ( )( + + ) + + Eercise.A page a i 9 ( + )( ) or ± ()()( 9) ii or iii 9 or iv or i + + ( + )( + ) or ± 9 ii or iii ( + ) or iv or c i 7 Cannot e factorised. ± ( 7) ii ± iii ( ) ± iv ± d i ( ) or ± ii or iii ( ) 9 or iv or Factorisation: can onl e used on equations that factorise sometimes spotting factors can e difficult can solve: + + cannot solve: 7 Quadratic formula: can e used to solve an equation with real roots including ones that don t factorise cumersome and consequentl eas to make a mistake can solve: 7 cannot solve: + 7 HarperCollinsPulishers 7 Edecel A-level Mathematics Year and AS 979
13 WORKED SOLUTIONS Completing the square: can e used to solve an equation with real roots including ones that don t factorise can e cumersome manipulations if is odd and a > ; consequentl eas to make a mistake can solve: 7 cannot solve: + 7 Calculator: eas if ou know how to use our calculator does not help ou to understand the underling methods can solve: + + and 7 cannot solve: + 7 a ( ) ± Completed the square. The equation was alread in form of a completed square. ± ± Does not factorise so used the quadratic formula. Alternativel, could have completed the square ut chose not to as a >. + ( )( ) ( 7)( ) ± ± 7 or 7 a + + c c a + + a a c ( a ) a a ( a ) ( a ) ( a ) a c a ac + a a ac + a + ± ac a a ± ac a c + ( )( + ) or Equation factorises so eas to do this. Alternativel, could have completed the square ut chose not to as a >. Eercise.A page a + + Sutract from. and d + 7 ( )( 7) or 7 Equation factorises so eas to do this. Alternativel, could have completed the square ut chose not to as a >. a Cannot e factorised if ac is not a square numer. ac ()()( ) not a square numer so does not factorise. HarperCollinsPulishers 7 Edecel A-level Mathematics Year and AS Multipl. + Add and. and c + Add and. and 9
14 ALGEBRA AND FUNCTIONS : EQUATIONS AND INEQUALITIES + Multipl. + Add and. Sustitute into. If there are n unknowns then ou need n distinct equations involving the n unknowns. + Sutract from. As neither the coefficient of nor is the same in oth equations, when one equation is sutracted from the other an unknown is not eliminated. + Add and. 9 and + ( ) Sutract from. and. ears old. 7 z z Multipl. z Add and. z Multipl. z Add and. Sustitute into : z Sustitute into : Eercise.B page a + + From. Sustitute into. and + From. Sustitute into. ( ) and c + From. + Sustitute into. + and 9 + Rearrange. Sustitute into. + ( ) Epand and simplif. Sustitute into. Solve. HarperCollinsPulishers 7 Edecel A-level Mathematics Year and AS 979
15 WORKED SOLUTIONS + 9 From. 9 Sustitute into. 9 and The graphs intersect at the point (, ). + From. + Sustitute into. + + and c c + From. Sustitute into. 7 7 c p and c 7p The simultaneous equations onl work if the cost of a small ag of sweets is 7p and the cost of a ar of chocolate is p. Although two equations can e formed and solved simultaneousl from the given figures, p is too little for a ar of chocolate so the student must e correct: one of the calculations is wrong. z z From. + Sustitute into. ( + ) z z From. z + Sustitute into. ( + ), z and Eercise.A page 7 a + Add and. + ( 7)( + ) 7 or or 7 + From. Sustitute into. + ( ) + ( )( ) or or c + ( + )( ) or or a Sustitution or elimination: for elimination the second equation would need to e multiplied so that can susequentl e eliminated. Sustitution onl: neither addition nor sutraction of the equations will eliminate a variale. c Sustitution onl: neither addition nor sutraction of the equations will eliminate a variale. a + Multipl. HarperCollinsPulishers 7 Edecel A-level Mathematics Year and AS 979
16 ALGEBRA AND FUNCTIONS : EQUATIONS AND INEQUALITIES + Add and. + 9 ( + )( ) or 7 or + From. Sustitute into. ( ) ± or c From. + Sustitute into. ( + ) + ± ( ) ± ± and ± 7 From. 7 Sustitute into. ( 7) ( + )( ) or 7 or 9 So the coordinates of the points of intersection of this line and this curve are (, 7 ) and (, ). 9 + From. + Sustitute into. + ( + ) + ±. or.. or. So the coordinates of the points of intersection of this line and the circle are (.,.) and (.,.). + From. + Sustitute into. + ( + ) ( + ) and The line is a tangent to the circle. The intersect once at (, ). Eercise.7A page a > > > c + < < < d ( ) 7 7 a > > HarperCollinsPulishers 7 Edecel A-level Mathematics Year and AS 979
17 WORKED SOLUTIONS c + < < > < d ( ) 7 7 a ( ) + + and > > > < < {,,, } 7 < < < or < + < < + < < < < < {, 9,, 7,,,,,,,,,,,,,, 7} c ( ) and < ( + ) < 7 < < 7 r πr 7 r. So. r miles 7 7 HarperCollinsPulishers 7 Edecel A-level Mathematics Year and AS 979 d cannot e oth less than and greater than or 7 equal to. 9 7 or + where {,,,,, } a cannot e oth less than and greater than in this wa. Can e rewritten as < or > cannot e oth less than or greater than in this wa. Can e rewritten as < or > 7
18 ALGEBRA AND FUNCTIONS : EQUATIONS AND INEQUALITIES c cannot e oth less than and greater than., and 7 a c d + j < 7 j < j < 7 + j > 7 j > 7 7 < j < Eercise.A page a + > + > ( + )( ) > < or > + ( )( ) c 9 ( + )( ) < or > d < ( ) < < < a + 7 < < ( + )( + ) < < < + > ( + )( + ) > < < c > d > ( )( + ) > < or > ( ) < < a > ( )( + ) > < < and + > > < {,,,,, } + 9 ( )( + ) or < < So < {all integers less than.} c ( + ) 9 + ± d and < < {,,, } < < ( ) < < < or > > < So < {all integers less than.} r πr r.9 So.9 r miles a + < < ( )( ) < < < {,,, } < < HarperCollinsPulishers 7 Edecel A-level Mathematics Year and AS 979
19 WORKED SOLUTIONS ( + )( ) < < < {,,,, } c > ( + ) > > ( + ) > Inequalit true for all values of. d + > > + > ( )( + ) > < or > {all integers ecept,, } a + < ( + )( ) < < < and + < ( + )( ) < < < So < <, i.e. {,,, } + < or + < So < <, i.e. {,,,,, } c < and > < < and < or > So < <, i.e. { } d + > or + < < or > or < < {all integers ecluding,,, } Eam-stle questions page + Line of smmetr at. HarperCollinsPulishers 7 Edecel A-level Mathematics Year and AS 979 -intercepts at (, ) and (, ), -intercept at (, ) and turning point at (.,.) (appro.). 7 ac 9 ()()( 7) > so two distinct real roots ± 9 ( ) ±.79 or.79 > <.79 or >.79 + Multipl. + Add and. and ( )( + ) or So or 7 7 -intercepts at (, ) and (, ) and -intercept at (, ); a < ( )( ) + + Satisfies the intercepts and a <. + + ac 9 ()()() 9 < so no real roots. The statement + + for all values of is correct. The discriminant is <, consequentl the equation has no real roots and so it does not intercept the -ais. As a > all of the curve will e aove the -ais. 9
20 ALGEBRA AND FUNCTIONS : EQUATIONS AND INEQUALITIES ( ) 7 ( ) At the turning point will have a minimum value i.e. when 7. When 7, ( ) So coordinates of turning point are ( 7, ) + Multipl and multipl. + Sutract from. and So the coordinates of the point of intersection of these two lines are (, ). 7 Rearrange. 7+ Sustitute into ( )( ) ( 7, ) and (, ) a 9 > + < No solution (as no overlap of inequalities). > > ( + )( ) > < or > a + + > ( + )( + 7) > < 7 or > and > > So >, i.e. {all integers greater than } + + > or > ( + )( + 7) > < 7 or > or > > So < 7 or > {all integers less than 7 or greater than } + > > ± 7 ±.7 or.77 <.7 or >.77 a + a ac < ()(a)( ) < + a < a < a < ac ()(a)( ) + a a a c ac > ()(a)( ) > + a > a > a > HarperCollinsPulishers 7 Edecel A-level Mathematics Year and AS 979
21 WORKED SOLUTIONS 7 a + + c a + + c a a + 9 c + a a a + c a 9 a a + 9 a c a a a + 9 a c a a a c + ± 9 a a ± 9 a c a Perimeter: ( + ) + ( + ) ( + )( + ) + + Rearrange. 9 Sustitute into ( 9)( + ) 9 or or So width or + So length or πr r. πr > r >.. < r. + 7z 7z From z 7 Sustitute into and rearrange. From Sustitute into and solve. 7 + ( )( ) or or z or 7 7 One set of solutions is z,, 7 Another set of solutions is, z, 7 HarperCollinsPulishers 7 Edecel A-level Mathematics Year and AS 979
22 ALGEBRA AND FUNCTIONS : SKETCHING CURVES Algera and functions : Sketching curves Prior knowledge p 9 a ( 7)( + ) so or 7 ( + 7)( 7) so 7 or 7 c ( ) so d ( ) so or 7 e ( )( + ) so or f ( )( ) so or a ( ) 7 ± 7 ± 7 ( ) ± ± or c ( + ) 7 + ± 7 ± 7 a (, 7), minimum as a > (, ), minimum as a > c (, 7), minimum as a > a a,, c 9 ac ( 9) so one real root. a,, c ac ( ) so no real roots. c a, 7, c ac 9 ( ) so two real roots. Eercise.A p a 7 (, ) (, 9) Roots at and so factors are ( ) and ( ). ( )( ) + a 9 7 (/, 9/) The sketch is not correct. ( 7)( + ) So should cut -ais at (, ) and (7, ). a > so graph should e U-shaped. When, so -intercept at (, ). ( ) So turning point at (, ). HarperCollinsPulishers 7 Edecel A-level Mathematics Year and AS 979
23 WORKED SOLUTIONS a 7 a.... ( /, 9/) a m s c Roots at and so factors are ( + ) and ( ), ut graph shape indicates a <, so ( + )( ) + + Eercise.A p a The sketch is not correct. Point of inflection should e at (, ) and curve should e a reflection in the -ais of the curve shown. The graphs are reflections of each other in the line C ( ) onl HarperCollinsPulishers 7 Edecel A-level Mathematics Year and AS 979
24 ALGEBRA AND FUNCTIONS : SKETCHING CURVES Eercise.B p a a a ( ).. a ( )( + ) ( 7) HarperCollinsPulishers 7 Edecel A-level Mathematics Year and AS
25 WORKED SOLUTIONS ( ) Using the quadratic formula to solve. or Eercise.A p 9 a The curve is the wrong shape: it should not e a paraola. The curve should e a U-shape sitting on the -ais at (, ) and intercepting the -ais at (, ). The do not generate the same curve ecause the are different functions. ( + ) ( + )( + )( + )( + ) ( ) ( )( )( )( ) ( ) (+) a B ( ) and C ( ) Eercise.B p 7 a HarperCollinsPulishers 7 Edecel A-level Mathematics Year and AS 979 7
26 ALGEBRA AND FUNCTIONS : SKETCHING CURVES a ( )( + )( + ) a Eercise.A p 7 a Asmptotes at and Asmptotes at and c In oth graphs the asmptotes are in the same locations. The ke difference is that the curves of the graph of are much closer to the origin than those of. a -intercept at (, ); asmptotes at and. -intercept at (,.7); asmptotes at and. a + ( + ) a iv ii c iv d i a Asmptotes at and ; -intercept at (, ). ( ) ( + ) HarperCollinsPulishers 7 Edecel A-level Mathematics Year and AS 979
27 WORKED SOLUTIONS Asmptotes at and ; -intercept at (, ) No. The asmptotes are in the correct locations at and. However, the graph drawn is for a reciprocal with a positive numerator whereas the given equation has a negative numerator. Eercise.B p 7 a Asmptotes at and Asmptotes at and a Asmptotes at and ; intercept at (,.). a 7 Asmptotes at and ; -intercept at (, ). 7 ( ) + ( ) a iii i c ii d i a Asmptotes at and ; -intercept at (, ) c In oth graphs the asmptotes are in the same locations. The ke differences are (i) that the curves of the graph of are much closer to the origin than those of and (ii) the graph of is a reflection in the -ais. HarperCollinsPulishers 7 Edecel A-level Mathematics Year and AS 979 Asmptotes at and ; -intercept at (, )
28 ALGEBRA AND FUNCTIONS : SKETCHING CURVES Yes, the graph is for the given function. Correct shape with asmptotes at and ; -intercept at (, 7 9 ). Eercise.A p ( + ) ( ) rearranged is ( + )( ) ( ) ( + ) ( ) ( ). There are two solutions to There are three points of intersection. There are no solutions to ecause the graphs do not intersect. ecause the graphs intersect twice ( + ) rearranged is ( + ) ( + ). There are no solutions to ( + ) ecause the graphs do not intersect. Also, there are no real solutions to the fourth root of a variale with a value of. 9 7 There are two solutions to ( )( ) ecause the graphs intersect twice. 7 At points of intersection, the equations are equal. HarperCollinsPulishers 7 Edecel A-level Mathematics Year and AS ( ) ( ) ( )( + ), or Sustituting into one of the original equations:, or 9
29 WORKED SOLUTIONS 7 There are two points of intersection. There are two points of intersection. 7 Eercise.A p m (g) m V (cm ) Using the graph, when m, V.7. The volume of the ring is.7 cm. a C, iii A, ii c B, i V r V kr k π π 7 V π r π 7 7π When r 7, V V V a k k When, () V (cm ) d t d kt k. Constant speed is mph. d d (miles) 7 9 t t (hours) Using the graph, when t., d 7. The distance travelled is 7 miles. m V m kv 9. k 9.. When V.7, m g HarperCollinsPulishers 7 Edecel A-level Mathematics Year and AS r r (cm) a Using the graph, when r, V cm. Using the graph, when V π, r cm. c Sustituting r into the equation gives V. Likewise sustituting V π into the equation gives r. a k From the graph, when,. k k c 9
30 ALGEBRA AND FUNCTIONS : SKETCHING CURVES Eercise.7A p 9 a f( + ): one solution. 9 7 f() + : asmptotes at and. f() + : no solutions. c f( ): one solution. 7 9 d f( + ): one solution. 9 7 a f( + ): asmptotes at and. 7 c f( ): asmptotes at and. HarperCollinsPulishers 7 Edecel A-level Mathematics Year and AS 979 d f() : asmptotes at and. f( ): Q at (, ); one solution. f() + : Q at (, ); one solution.
31 WORKED SOLUTIONS a f( + ): asmptotes at and. 7 f ( + ) ( + ) c (, ) f( + ): P at (, ); asmptote at. f() + : P at (, ); asmptote at. f( + ): f() + has one solution (one point of intersection). f() + f( + ) a, f( ) where : asmptotes at 9 and 9. Ais intercepts at (, ), (, ) and (, ). 7 a f( + ) ( + )( + ) : two solutions. f() + ( + )( ) + : two solutions. c f ( ) ( ) : two solutions. d f() ( + )( ) : two solutions. 9 9 c, d f() + : asmptotes at 9 and 9. a, f( + ) + : no solutions. f() : no solutions. f( + ) f()+ 9 9 HarperCollinsPulishers 7 Edecel A-level Mathematics Year and AS 979
32 ALGEBRA AND FUNCTIONS : SKETCHING CURVES c If f() then f( + ) + and f( + ) f() + + f( ) ( ) ( ) + ( ) + ( ) Eercise.A p 9 a f(): one solution. 7 f(): one solution. 7 c f(): one solution. 7 f : one solution. d 7 a f(): asmptotes at and. f(): asmptotes at and. c f( ): asmptotes at and. HarperCollinsPulishers 7 Edecel A-level Mathematics Year and AS 979
33 WORKED SOLUTIONS d f(): asmptotes at and. f( ): P(, ), R(, ), Q(, ). f( ) has two solutions. f(): P(, ), R(, ), Q(, ). f() has two solutions. f( ) f(): has no solutions (the do not intersect). f( ) f() a, f() where : asmptotes at,, and. Ais intercepts at (, ), ( 9, ), (, ), (9, ) and (, ). 9 9 c, d f() where : asmptotes at 9 and 9. Ais intercepts at (, ), (, ) and (, ). a f(): asmptotes at and. 7 f ( + ) c (, ) f( ) and f(): in oth cases asmptote at. For f( ), P is at (, ). For f(), P is at (, ) a f() ( + )( ) : two solutions. 7 HarperCollinsPulishers 7 Edecel A-level Mathematics Year and AS 979
34 ALGEBRA AND FUNCTIONS : SKETCHING CURVES f() ( + )( ) : two solutions. 7 c f ( + )( ) : two solutions. c If f(), f( + ) + and f( + ) ()( + ) 9 f() + + f Eam-stle questions p 9 a f( + ) f( ) c f() + d f() The cale will touch the rollercoaster at two points in the given interval (two points of intersection) f d ( + )( ) : two solutions. a,, d f() + : no solutions. f() ()( + ): no solutions. 7 r p r kp k k r p The echange rate is reals to. r p 7 f() f() Using the graph: a r 9 r p p HarperCollinsPulishers 7 Edecel A-level Mathematics Year and AS 979
35 WORKED SOLUTIONS a f( + 9) f() k π A πl When l 7, A 9π A 7 9 / c f() The shape is a circle 7 a Asmptotes at and. There are two points of intersection d f ( ) and + Asmptotes at and. There are two points of intersection The rait and mice populations are onl the same when the are oth first introduced to the island. A l A kl. k a f() + f() c f( ) d f( ) HarperCollinsPulishers 7 Edecel A-level Mathematics Year and AS 979
36 ALGEBRA AND FUNCTIONS : SKETCHING CURVES 9 a d a f() + 9 c i f() + a where a >. ii f( a) where a > 9 or a < HarperCollinsPulishers 7 Edecel A-level Mathematics Year and AS 979
37 WORKED SOLUTIONS Coordinate geometr : Equations of straight lines Prior Knowledge (, ), (, ) a A:, B:, C:, D:, E:, F:, A, E D, F c B, C Eercise.A p The lines not in the form a + + c, where a, and c are integers are: c ecause the right-hand side is not zero d ecause the right-hand side is not zero e ecause c is not an integer f ecause a, and c are not integers. a Rearrange +. + where a, and c Rearrange. + where a, and c a Multipl :, and rearrange. where a, and c Multipl : +, and rearrange. + where a, and c c Multipl : + Multipl : +, and rearrange. + where a, and c Rearrange: + Rearrange and divide : + Gradient and intercept ; coordinates (, ) Equation written down incorrectl. It should e. p Not all epressions in the equation have een multiplied. It should e. Although the values are wrong this step is actuall correct. With the correct values it should e +. Although the values are wrong this step is actuall correct. With the correct values it should e +. ( ) where a, and c 7 When the line intercepts the ais,. Sustituting : + So the coordinates of the intercept are (, ). When the line intercepts the ais,. Sustituting : + So the coordinates of the intercept are (, ). Eercise.A p a m m + c m d m + + HarperCollinsPulishers 7 Edecel A-level Mathematics Year and AS 979 7
38 COORDINATE GEOMETRY : EQUATIONS OF STRAIGHT LINES a m + m + + m + m + + First find the point of intersection m and (, ) (, ) m + Find the equation of the first line. m and (, ) (, ) m Find the equation of the second line. m and (, ) (, ) m + + Find the point of intersection Point of intersection is (, 7). 7 m and (, ) (, ) m When ou sustitute in, will equal if the line goes through the origin. ( ) ( ) The line does not go through the origin. Eercise.A p a m (, ) (, ) (, ) ( 7, ) m 7 m m m (, ) ( 9, ) (, ) (, ) m 9 m m c m (, ) (, ) (, ) (, 9) m 9 m m HarperCollinsPulishers 7 Edecel A-level Mathematics Year and AS 979
39 WORKED SOLUTIONS a m (, ) ( a, a) (, ) ( a, a) m a a a a m a a m m (, ) ( a, a) (, ) ( a, a) m a a a a m a a m Should e m coordinate incorrect in (, ). It should e (, ). Numerator and denominator confused. Should e: m m m a m (, ) (, ) (, ) (, ) m m 9 m Lie on a straight line with a gradient of. m (, ) (, ) (, ) ( 9, ) m 9 m m Do not lie on a straight line with a gradient of. c m (, ) (, 7) (, ) (, ) m 7 m 9 m Lie on a straight line with a gradient of. d m (, ) ( 7, ) (, ) (, ) m 7 m m Lie on a straight line with a gradient of. m (, ), ( ) (, ) (, ) m m m m (, ), (, ) (, ) m 9 m 9 m The gradient, m, is the same etween oth of the pairs of points, so we can conclude that all the points lie on a straight line. Ascent: m (, ) (, ) (, ) (, ) HarperCollinsPulishers 7 Edecel A-level Mathematics Year and AS 979 9
40 COORDINATE GEOMETRY : EQUATIONS OF STRAIGHT LINES m m m Descent: m (, ) (, ) (, ) (, ) m m m The descent is the steepest part ecause >. 7 m (, ), (, ), m m m Eercise.A p a (, ) (, ) (, ) ( 7, ) 7 + (, ) (, ) (, ) ( 9, ) 9 9 a (, ) (, ) (, ) (, 9) (, ) (, ) (, ) ( 7, ) Let (, ) (, ) HarperCollinsPulishers 7 Edecel A-level Mathematics Year and AS 979 Let (, ) (, ) and and and confused. The net line should e: With this correction, the remainder of the calculation should e: ( ) ( ) + + So + 9 intercept when, so when (, ) (, ) (, ) (, )
41 WORKED SOLUTIONS + + intercept when so (, ) (, ) (, ) (, 7) 7 7 7( ) Gradient 7 and -intercept (, ) Line A: (, ) ( 7, ) (, ) (, ) m Line B: (, ) (, ) (, ) ( 7, ) m HarperCollinsPulishers 7 Edecel A-level Mathematics Year and AS 979 Line B is steeper ecause >. (The negatives can e ignored ecause the just indicate the direction of the line.) 7 a (, ), ( ) ( ) (, ), (, ), 7 (, ), intercept when + 7 so 7 + intercept when + so (, ), 7 (, ),
42 COORDINATE GEOMETRY : EQUATIONS OF STRAIGHT LINES 7 7 ( ) Rearranging: + 9 Line A: (, ) (, ) (, ) (, ) 9 9 Line B: 7 (, ) (, ) (, ) (, ) At the point of intersection: 7 7 Sustituting into either equation gives. So point of intersection is (, ). Point A: 9 7, Point B: 7 +, (, ) ( 7, ) (, ), ( + ) ( ) + 7 ( ) When, so this line does not pass through the point (, ). Eercise.A p a m in oth equations so the lines are parallel. m in oth equations so the lines are parallel. Line A: (, ) ( 7, ) (, ) (, ) m m 7 m Line B: (, ) (, ) (, ) (, ) m m m m for oth, so the lines are parallel. From 7, m 7. The intercept of is when so is at (, ). (, ) (, ) m 7( ) 7 HarperCollinsPulishers 7 Edecel A-level Mathematics Year and AS 979
43 WORKED SOLUTIONS a Arrange the equations in the form m + c: +, m is not equal so the lines are not parallel. Arrange the equations in the form m + c: +, Line A: m in oth equations so the lines are parallel. (, ) (, ) (, ) (, ) m m m Line B: (, ) (, ) (, ) (, ) m m m m for oth, so the lines are parallel. From + + 7, 7, so m. The intercept of + is when so is at (, ). (, ) (, ) m ( ) + 7 a Arrange the equations in the form m + c: 7, + m in oth equations so the lines are parallel. Arrange the equations in the form m + c: 7, m is not equal so the lines are not parallel. Arrange the equations in the form m + c: +, +, m +,,, m,, m +,,, m So + and are parallel lines as m for oth lines. And + and + are parallel lines as m for oth lines. Eercise.B p 7 a m and m so m m so the lines are not perpendicular. m and m so m m so the lines are perpendicular. m for the given line ut the gradient of a line perpendicular to this will e m. Let (, ) (, ). m (, ) incorrectl sustituted. Correcting this and using correct value of m gives: So a m and m so m m so the lines are perpendicular. m and m so m m so the lines are not perpendicular. Line A: (, ) (, ) (, ) ( 9, ) m m 9 m Line B: (, ) (, ) (, ) (, ) m m m m and m so m m so the lines are perpendicular. Arrange the equations in the form m + c: +, +, m +,,, m,, m +,,, m HarperCollinsPulishers 7 Edecel A-level Mathematics Year and AS 979
44 COORDINATE GEOMETRY : EQUATIONS OF STRAIGHT LINES So + and + are perpendicular lines as mm. And + and + are perpendicular lines as mm. And and + are perpendicular lines as mm. And and + are perpendicular lines as m m. From 7, 7, 7, m mm So m The intercept of + is when so is at (, ). (, ) (, ) m 7 a m and m so m m perpendicular. m 9 and m so mm 9 not perpendicular. Descrie as line A: (, ) (, ) (, ) (, ) m m m m Descrie as line B: (, ) (, ) (, ) (, ) m m m m Descrie as line C: (, ) (, ) (, ) (, ) so the lines are so the lines are m m m m Descrie as line D: (, ) (, ) (, ) (, ) m m m m So A is parallel to C, B is parallel to D, A is perpendicular to B and D, and C is perpendicular to B and D. So the quadrilateral can onl e a square or a rectangle. Eercise.A p (, ) (, ) (, ) (, ) a False: m and m so m m so the lines are not perpendicular. False: c and c so the lines do not share the same intercept. c False: negative gradient means the oject is travelling ack to the start. d True: speed is the gradient and the gradients are the same (the difference in the signs just indicates direction) so the are travelling at the same speed. HarperCollinsPulishers 7 Edecel A-level Mathematics Year and AS 979
45 WORKED SOLUTIONS e False: m m so the lines are not parallel. f False: positive gradient means the oject is travelling awa from the start. We have een given two pairs of coordinates for each printer. The cost of the printer is actuall the start of the costs when, i.e. the intercept. We have also een given the coordinates when. Anna s printer: (, ) (, ) (, ) (, 9) Bhavini s printer: (, ) ( 7,. ) (, ) (, 7. ) m is the cost per ear to run the printer and is in Anna s case and in Bhavini s case. c is the cost to u the printer and is in Anna s case and 7. in Bhavini s case. The cost of one ink cartridge for Anna s printer is. The cost of one ink cartridge for Bhavini s printer is. Nothing is significant aout the intercepts. Although ou can mathematicall work out the intercept in each case, this would actuall e a negative value which would not make sense as ou cannot have a negative numer of ears. Tai firm A: (, ) (, ) (, ) (, ) When,. When 7,. Tai firm B: (, ) (, ) (, ) (, ) + When,. When 7,. Tai firm C: (, ) (, ) (, ) (, ) + When,. When 7, 9. Tai firm A is the cheapest on a -mile journe. Tai firm C is the cheapest on a 7-mile journe. (, ) ( 7., ) (, ) ( 7,. ) HarperCollinsPulishers 7 Edecel A-level Mathematics Year and AS 979
46 COORDINATE GEOMETRY : EQUATIONS OF STRAIGHT LINES Eam-stle questions p m, (, ) (, ) m ( ) + a Line A: (, ) ( 7, ) (, ) (, ) m m 7 m m Line B: (, ) (, ) (, ) (, ) m m m Line A is steeper ecause >. m mm so m (, ) (, ) m + a Runner A starts miles awa from runner B s home. The two runners meet 9 miles awa from runner B s home so runner A will have run miles. Runner A: (, ) (, ) (, ) ( 9, ) 9 + Runner B: (, ) (, ) (, ) ( 9, ) 9 9 c Runner B runs faster. Gradient speed and >. a m mm so m (, ) (, ) ( ) m + + When, intercept (, ) When, intercept (, ) c Area of triangle h units Line AB: (, ) (, ) (, ) (, ) m m m Line BC: (, ) (, ) (, ) (, ) HarperCollinsPulishers 7 Edecel A-level Mathematics Year and AS 979
47 WORKED SOLUTIONS m m m mm So AB and BC are perpendicular to each other and ABC is a right-angled triangle. 7 Pa as ou go: (, ) (, ) (, ) (, ) For this option: rides: so 7 rides: 7 so 9 For other option (ook of tickets): Cost of rides + so pa as ou go option is cheaper. Cost of 7 rides + ( ) so ook of tickets option is cheaper. Line A: (, ) (, ) (, ) (, a) m a a a m, (, ) (, ) m m m Gradient perpendicular to this: m (, ) (, ) m( ) ( ) ( )+ + Solving gives 9 7 Sustituting this value ack into the original equation gives 7 Point of intersection (, ), 9 ( 7 7 ) + + Rearranging gives m So m m( ) 9 7 a Line A: + 7 (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) HarperCollinsPulishers 7 Edecel A-level Mathematics Year and AS 979 7
48 COORDINATE GEOMETRY : CIRCLES Line B: (, ) (, ) (, ) (, ) 9 ( ) 9 The two equations are the inverses of each other. At the point of intersection: 9 + ( ) 9 Rearranging gives Sustituting ack into one of the original equations gives. So the point of intersection is (, ). c The transformation linking the two lines is a reflection in the line. 9 7 Coordinate geometr : Circles Prior knowledge p 7 Gradient 9 ( ) Gradient of first line Perpendicular gradient Equation of perpendicular line through (, ) is given ( ). + 9 Sustitute into ( ) When 9, 9 Hence point of intersection (9, ). Complete square on +. + ( 7) 9 + ( 7) Eercise.A p 7 9 d The point of intersection is the onl point where the temperature in degrees Celsius and the temperature in degrees Fahrenheit have the same numerical value, i.e. at (, ). a Centre (, ), radius Centre (9, ), radius c Centre (, ), radius d Centre (, ), radius 7 a ( + ) + ( 9) 9 and ( + ) + ( + ) 9 and c ( ) + and + 9 d ( ) + ( ) and + + a ( ) + ( + ) ( + ) + ( ) 9 c ( + ) + ( ) a ( 9) + ( + 7) 9 ( 9) + ( + 7) Centre (9, 7), radius ( + ) + 9 ( + ) + Centre (, ), radius HarperCollinsPulishers 7 Edecel A-level Mathematics Year and AS 979
49 WORKED SOLUTIONS c ( + ) + ( + 9) + 79 ( + ) + ( + 9) 7 Centre (, 9), radius d ( + ) + ( ) 9 + d ( ) + ( ) r ( ) + (7 ) ( + ) + ( ) Centre (, ), radius 9 r ( ) + () The circle has the equation ( + ) + ( 7). ( 9) + ( ) 7 d ( ) + ( ) d ( ) + (7 ) d () + () The circle has the equation ( ) + ( ). The locations on the -ais will lie on the circumference of the circle. When, ( ) + ( ) ( ) + 9 ( ) ± ± or The circle cuts the -ais at (, ) and (, ). Rewrite the equation of the circle completing the square. ( + p) p + ( + ) 9 9 ( + p) + ( + ) p Since r, p p p p Since p is a positive constant, p. The centre of the circle has the coordinates ( p, ). Since p, the centre has the coordinates (, ). The centre is units from the origin. Eercise.B p d ( ) + ( ) d ( 7 7) + ( ( )) d ( ) + () d HarperCollinsPulishers 7 Edecel A-level Mathematics Year and AS 979 a The centre is the midpoint etween D and E. Centre (, ) (, ) The radius is the distance etween the centre and a point on the circumference. Let the centre (, ) (, ) and D(, 9) (, ). d ( ) + ( ) r ( ) + ( 9) r () + ( ) r c The circle has the equation ( ) + ( ). Centre of circle + (, ) (9, ) r (9 ( )) + ( ) r () + ( ) 9 The circle has the equation ( 9) + ( ) 9. a The student has correctl worked out that AB 9, ut the equation of the circle needs the square of the radius, not the square of the diameter. The diameter is 9 7, so the radius is 7. ( +.) + ( 7) ( 7 ) a + 9 +, (, ) d ( ) + ( ) r (9 ) + ( ( )) r () + () r c ( ) + ( + ) d Sustitute the coordinates of H into the equation of the circle. ( ) + ( + ) ( ) + () Since oth sides of the equation are satisfied, H lies on C. a The centre of the circle is the midpoint of ST 9+ + (, ) (, 7). The midpoint of UV will also e the centre of the circle, (, 7). Hence + p + q, (, 7) + p p + q 7 q 9
50 COORDINATE GEOMETRY : CIRCLES r ( 9 ( )) + ( 7) r ( 7) + ( ) The circle has the equation ( + ) + ( 7). 7 The equation of the circle can e rewritten as ( + ) + ( ), so (, ) is the centre of the circle. Sustitute + into ( + ) + ( + ) Epand and simplif Solve. ( + )( ) or M and N are the two intersection points, it doesn t matter which is which. When, +, so M (, ) When, +, so N (, ) The midpoint of MN + + (, ) (, ) Since the midpoint of MN is the centre of the circle, MN is a diameter of the circle. Eercise.A p 7 a d ( ) + ( ) DE ( ) + ( ) DE ( ) + () EF ( ) + ( 9) EF ( ) + ( ) DF ( ) + ( 9) DF ( ) + () Since +, DE + EF DF. Gradient of DE Gradient of EF 9 Since m m, DE and EF are perpendicular. c DF is a diameter of the circle. Centre + + (, 9 ) (, ) For the radius: d ( ) + ( ) r ( ) + ( ) r () + () The equation of the circle is ( ) + ( ). a Prove using Pthagoras theorem (could compare gradients). PQ ( ) + ( ) ( ) + ( ) PR ( ) + ( ) ( ) + ( ) QR ( ) + ( ) ( ) + () Since PR + QR PQ, Pthagoras theorem is satisfied and the triangle PQR is right-angled. Centre + + (, ) (, ) For the radius: d ( ) + ( ) r ( ) + ( ) r ( ) + ( ) Area of triangle h Area of circle πr π π Green area (π ) m c π. m.., so tins are required. Since AB is a diameter, ACB 9 and the lengths of the sides satisf Pthagoras theorem. AC + BC AB ( 9) + ( ) ( + ) ( )( ) or When, d + 9 Radius 9 9 When, d + 7 Radius 7 7 a Sustitute and into ( + ) + ( ) r. ( + ) + ( ) r () + ( ) r r Sustitute and q into ( + ) + ( ) ( + ) + (q ) ( ) + (q ) (q ) 9 q ± q ± q or Since q >, q HarperCollinsPulishers 7 Edecel A-level Mathematics Year and AS 979
51 WORKED SOLUTIONS c Gradient of AB Gradient of AC ( ) Gradient of BC ( ) d m m so AC and BC are perpendicular. Since the angle in a semicircle is a right angle, AB is a diameter of the circle. a Gradient of L ( ) Gradient of L ) 9 m m so L and L are perpendicular. Because L and L are perpendicular and intersect at point W on the circumference, SV is a diameter. Centre + (, ) (, ) For the radius: d ( ) + ( ) r ( ) + ( ) r ( ) + () The equation of C is given ( ) + ( ) Epanding and simplifing: + + a PR PQ + QR PR ( (t + 7)) + ((t + 7) ) ( t) + (t + ) t + t + 7 PQ ( ) + ((t + 7) (t + )) ( ) + ( ) QR ( (t + 7)) + ((t + ) ) ( t) + (t + ) t + t + Hence t + t + 7 t + t + + t t Hence the coordinates of P and R are (, ) and (, ). Centre of circle + + (, ) (, ) For the radius: d ( ) + ( ) r ( ) + ( ) r (7) + ( ) The equation of C is given ( ) + ( ). JL JK + KL JL ( 7) + ( ) ( ) + ( ) JK ( ( )) + ( (p )) ( ) + ( p) p p + 9 KL ( 7) + ((p ) ) ( ) + (p ) p p + 9 Hence p p p p + 9 p p + p p + (p )(p ) p or Eercise.A p a 9+ 7, (, ) Gradient of DE ( ) 9 7 c Gradient of FG d Equation is given ( ). +. e When, ( ) + as required. f For the radius: d ( ) + ( ) r (9 ) + ( ) r () + () The equation is given ( ) + ( ). Gradient of JK 7 7 Gradient of isector 7 Midpoint + 7+ (, ) 9 (, ) Equation of isector is given 9 7 ( ) When, 9 7 ( ) Centre of circle (9, ) For the radius: d ( ) + ( ) r (9 ) + ( 7) r () + () 7 The equation is given ( 9) + ( ) 7. a Midpoint + +, (, ) + HarperCollinsPulishers 7 Edecel A-level Mathematics Year and AS 979
52 COORDINATE GEOMETRY : CIRCLES + Coordinates of V are (, ). Gradient of UV Gradient of isector Equation of isector is given ( ). When, ( ) Centre of circle (, ) For the radius: d ( ) + ( ) r ( ) + ( ) r ( ) + ( 9) The equation is given ( ) + ( ). d 7 d + m a Equation of circle can e rewritten as ( + ) + ( + ) Hence centre of circle (, ) When, ( + ) + ( + ) ( + ) + ( + ) + ± or Length of chord ( ) c Height of triangle ( ) Base of triangle PQ Area of triangle h a Midpoint of AB + + (, 7 ) (, ) 7 Gradient of AB Gradient of isector Equation of isector is given ( + ) + 7 Midpoint of BC + 7 (, ) (, ) Gradient of BC 7 ( ) Gradient of isector Equation of isector is given ( ). + 9 c 9 ecause gradients are perpendicular. d Gradient AB from a Gradient BC from Hence AB is perpendicular to BC and angle ABC 9. e AC is a diameter. 7 For each circle, let the first point A, the second B and the third C. a A(, ), B(, ) and C(, ) Midpoint of AB + + (, ) 9 (, ) Gradient of AB 7 Gradient of isector 7 Equation of isector is given 9 7 ( ) Midpoint of AC + + (, ) (, ) Gradient of AC Gradient of isector Equation of isector is given ( ) For point of intersection, When, Centre of circle (, ) For the radius: d ( ) + ( ) r ( ) + ( ) r () + ( ) The equation is given ( ) + ( ). A(, ), B(, ) and C(, ) Midpoint of AB + + (, ) (, ) Gradient of AB Gradient of isector Equation of isector is given ( ) + 7 Midpoint of AC + (, ) (, ) Gradient of AC ( ) Gradient of isector HarperCollinsPulishers 7 Edecel A-level Mathematics Year and AS 979
53 WORKED SOLUTIONS Equation of isector is given ( ( )) + For point of intersection, When, + Centre of circle (, ) For the radius: d ( ) + ( ) r ( ) + ( ) r ( ) + () The equation is given ( + ) + ( ). c A(9, ), B(7, ) and C(, ) Midpoint of AB (, ) (, ) Gradient of AB 9 7 Gradient of isector Equation of isector is given ( ) + Midpoint of AC 9 + (, ) (, ) Gradient of AC ( ) 9 Gradient of isector Equation of isector is given ( ) + For point of intersection, + + When, + Centre of circle (, ) For the radius: d ( ) + ( ) r ( 7) + ( ) r ( 7) + () Equation of isector is given + ( + ) For point of intersection, + When, + Centre of circle (, ) For the radius: d ( ) + ( ) r ( + ) + ( + ) r ( 7) + ( 9) The equation is given ( + ) + ( + ). a Circle A: Circle B: + + Sutract from. 9 7 Hence the line on which the circles intersect is given 7. Sustitute 7 into the equation for Circle A. + ( 7) + ( 7) ( )( ) or When, 7 When, 7 Points of intersection are (, ) and (, ). Length of chord: d ( ) + ( ) d ( ) + ( ( )) d () + () The length of the chord is is The equation is given ( ) + ( ). Eercise.A p d A(, ), B( 7, 9) and C(, ) Midpoint of AB 7 (, 9) (, ) Gradient of radius Gradient of AB ) ( ) Gradient of the tangent The equation of the tangent is given Gradient of isector ( 7) Equation of isector is given + ( + ) Midpoint of AC (, ) (, ) Gradient of AC ( ) a + ( ) + 9 Since oth sides of the equation agree, T lies on ( ) Gradient of isector the circle. HarperCollinsPulishers 7 Edecel A-level Mathematics Year and AS 979
54 COORDINATE GEOMETRY : CIRCLES Centre of circle (, ) Gradient of radius ( ) Gradient of the tangent The equation of the tangent is given + ( ) + Hence + a Centre (, ) Radius c Let centre X, point P (, 7) and point where tangent meets circle T. The length of PT is required, where PT + XT PX For PX: d ( ) + ( ) PX ( ) + ( 7) PX ( ) + ( ) PT + XT PX PT + PT PT 9 m For A: Gradient of radius 7 7 Gradient of the tangent The equation of the tangent is given 7 ( ) Hence ( ) + 7 For B: Gradient of radius 7 7 Gradient of the tangent The equation of the tangent is given 7 ( ) Hence ( ) + 7 Put tangents equal. ( ) + 7 ( ) + 7 ( ) ( ) ( ) + 7 (this could also e deduced smmetr since AB is horizontal) When 7, (7 ) + 7 ( ) + 7 AXBQ is a kite. Split into congruent triangles AXQ and BXQ. Base of triangle AXQ 9 Perpendicular height of triangle AXQ Area of triangle AXQ 7 Area of AXBQ 7 7 Rewrite the equation of the circle as ( + ) + ( ) Centre (, ), radius 9 Let centre X and point where tangent meets circle T. PT + XT PX For PX: d ( ) + ( ) PX (7 ( )) + ( ) PX () + () PT + XT PX PT + 9 PT 9 PT 9 7 a Rewrite the equation of the circle as ( ) + ( ) Centre (, ), radius Gradient of radius Gradient of the tangent The equation of the tangent is given ( ) When, q + q 9 q 9 q Use d ( ) + ( ) For PT: PT ( ) + ( ) PT ( ) + () For PX: PX ( ) + ( ) PX ( ) + () TX is the radius, so TX Hence PT + TX PX ecause +. 9 Hence the coordinates of Q are (7, 9 ). HarperCollinsPulishers 7 Edecel A-level Mathematics Year and AS 979
55 WORKED SOLUTIONS c Since triangle is right-angled, area TX PT. TX PT Area 7 a Gradient of tangent Gradient of radius Equation of line with gradient through (, ) is given ( ) Find point of intersection of and Hence the point (7, 9) lies on the circumference. For the radius: d ( ) + ( ) r (7 ) + (9 ) r () + ( ) The equation of the circle is given ( ) + ( ) Gradient of tangent Gradient of radius Equation of line with gradient through (, ) is given ( + ) ( + ) + Find point of intersection of ( + ) + and +. ( + ) + + ( + ) Hence the point (.,.) lies on the circumference. For the radius: d ( ) + ( ) r (.) + (.) r (.) + (.) 7. The equation of the circle is given ( + ) + ( ) 7. Eercise.B p a Sustitute into +. + ( ) ( ) The equation has repeated roots so is a tangent of the circle. Sustitute 7 into +. (7 ) ( 7) The equation has repeated roots so 7 + is a tangent of the circle. Sustitute into ( ) + ( ) ( ) The equation has repeated roots so is a tangent of the circle. Sustitute + c into +. + ( + c) ( + c) c + c c + (c ) + (c c ) For ac : (c ) (c c ) c c + 7 c + c + c 9c + c + c (c )(c + ) c or Sustitute + c into ( + c) ( + c) + + c + c + c + + ( c) + (c c + ) + ( c) + (c c + ) HarperCollinsPulishers 7 Edecel A-level Mathematics Year and AS 979
56 COORDINATE GEOMETRY : CIRCLES For ac : ( c) (c c + ) c + c c + c c + c 99 c c + (c )(c ) c or The tangents are + and +. Sustitute m + into ( ) + ( ). ( ) + (m + ) ( ) + (m + ) + + m + m + 9 ( + m ) + (m ) + For ac : (m ) ( + m ) 7m m m m m 7 m + m + (7m + )(m + ) m 7 or For the tangent 7 + : For the tangent + : + + Eam-stle questions p a The equation of the circle is given ( a) + ( ) r, so centre (9, ). Gemma has copied the signs from the equation ut needed to change them oth. Olivia has stated the length of the radius, not the diameter. Radius Diameter Radius Equation is given ( 7) + ( + ) Epand On the -ais,. ( + ) + ( 7) ( + ) + ± ± or Coordinates are (, ) and (, ). On the -ais, ( + ) + ( 7) () + ( 7) ( 7) 7 ± 7 ± or Coordinates are (, ) and (, ). Dilgusha has not halved the coefficients of and when completing the square. She has added rather than sutracting. When square-rooting, which shouldn t have een possile in this situation, she has squarerooted + and made it negative. Correct solution: Complete the square on and. ( + ) 9 + ( 7) 9 + ( + ) + ( 7) Centre (, 7) Radius a d ( ) + ( ) r ( ) + ( ) r () + ( 9) Radius cm The circle has the equation ( ) + ( ) c Gradient of radius Gradient of the tangent The equation of the tangent is given ( ) 7 + d d ( ) + ( ) d (9 ) + ( ) d (7) + () Distance 7 cm 7 m e Unlikel athlete will remain at same point whilst turning. Hammer s motion will not e horizontal ecause it will e affected gravit. HarperCollinsPulishers 7 Edecel A-level Mathematics Year and AS 979
57 WORKED SOLUTIONS a Midpoint of AB 9+, (, ) For the radius, d ( ) + ( ) r ( 9) + ( ) r () + ( ) The circle has the equation ( ) + ( ). Gradient of radius 9 Gradient of the tangent The equation of the tangent is given ( 9) Centre of circle, X ( 9, ) Radius 9 7 PX ( ( 9)) + ( ) PX (7) + (7) 7 PT PX XT PT PT 9 a r ( 7) + ( ) r () + () 7 Equation is given ( 7) + ( ) 7. Midpoint of XA 7+ + (, ) (9, 7) Gradient of XA 7 Gradient of the tangent The equation of the tangent is given 7 ( 9) or +. c Let P e the midpoint of XA. XM XA AM ecause XM and XA are radii and since MN is the isector of XA it is a line of smmetr for triangle MAX (hence XM AM). MP MX XP MP 7 MP 7 MN 9 a ( + ) + ( ) 7 ( + ) + (7 ) ( + ) + ( ) ( + 9)( ) 9 or HarperCollinsPulishers 7 Edecel A-level Mathematics Year and AS 979 When 9, 7 ( 9) When, 7 Since the -coordinate of A is less than the -coordinate of B, A is ( 9, ) and B is (, ). c Tangent at A: Gradient of radius 9 ( ) Gradient of the tangent (Note that since the centre of the circle lies on + 7, AB is a diameter, so the tangent is perpendicular to + 7.) The equation of the tangent is given ( + 9) Hence + Tangent at P: Gradient of radius ( ) 7 Gradient of the tangent 7 The equation of the tangent is given 7 ( + ) Hence 7 ( + ) + From which, + 7 ( + ) When, + 7 Coordinates of T are (, 7 ). a Equation is given ( ) + ( ) 9 The centre is (, ). The radius is. Method : Similar triangles PTX and PAO: XP OP TX OA Let length of XP a a + ( +a) a + ( + a) a + ( + a) a a + a a a a a (a )(a + ) a (can t e ) Method : Since m at T, ( ) + (m ) m m ( + m ) + ( m) + 7
Number Plane Graphs and Coordinate Geometry
Numer Plane Graphs and Coordinate Geometr Now this is m kind of paraola! Chapter Contents :0 The paraola PS, PS, PS Investigation: The graphs of paraolas :0 Paraolas of the form = a + + c PS Fun Spot:
More informationGraphs and polynomials
1 1A The inomial theorem 1B Polnomials 1C Division of polnomials 1D Linear graphs 1E Quadratic graphs 1F Cuic graphs 1G Quartic graphs Graphs and polnomials AreAS of STud Graphs of polnomial functions
More informationHigher. Polynomials and Quadratics. Polynomials and Quadratics 1
Higher Mathematics Polnomials and Quadratics Contents Polnomials and Quadratics 1 1 Quadratics EF 1 The Discriminant EF Completing the Square EF Sketching Paraolas EF 7 5 Determining the Equation of a
More informationGraphs and polynomials
5_6_56_MQVMM - _t Page Frida, Novemer 8, 5 :5 AM MQ Maths Methods / Final Pages / 8//5 Graphs and polnomials VCEcoverage Areas of stud Units & Functions and graphs Algera In this chapter A The inomial
More information1 k. cos tan? Higher Maths Non Calculator Practice Practice Paper A. 1. A sequence is defined by the recurrence relation u 2u 1, u 3.
Higher Maths Non Calculator Practice Practice Paper A. A sequence is defined b the recurrence relation u u, u. n n What is the value of u?. The line with equation k 9 is parallel to the line with gradient
More informationUNCORRECTED SAMPLE PAGES. 3Quadratics. Chapter 3. Objectives
Chapter 3 3Quadratics Objectives To recognise and sketch the graphs of quadratic polnomials. To find the ke features of the graph of a quadratic polnomial: ais intercepts, turning point and ais of smmetr.
More informationy intercept Gradient Facts Lines that have the same gradient are PARALLEL
CORE Summar Notes Linear Graphs and Equations = m + c gradient = increase in increase in intercept Gradient Facts Lines that have the same gradient are PARALLEL If lines are PERPENDICULAR then m m = or
More informationMay 27, QUADRATICS.notebook. Apr 26 17:43. Apr 26 18:27. Apr 26 18:40. Apr 28 10:22. Apr 28 10:34. Apr 28 10:33. Starter
1. Factorise: 2 - - 6 2. Solve for : 2( + 1) = - 1 3. Factorise: 2-25 To solve quadratic equations.. Factorise: 2 2-8 5. State the gradient of the line: + 12 = 2 Apr 26 17:3 Apr 26 18:27 Solving Quadratic
More information5Higher-degree ONLINE PAGE PROOFS. polynomials
5Higher-degree polnomials 5. Kick off with CAS 5.2 Quartic polnomials 5.3 Families of polnomials 5.4 Numerical approimations to roots of polnomial equations 5.5 Review 5. Kick off with CAS Quartic transformations
More informationMATH 115: Final Exam Review. Can you find the distance between two points and the midpoint of a line segment? (1.1)
MATH : Final Eam Review Can ou find the distance between two points and the midpoint of a line segment? (.) () Consider the points A (,) and ( 6, ) B. (a) Find the distance between A and B. (b) Find the
More informationMaths A Level Summer Assignment & Transition Work
Maths A Level Summer Assignment & Transition Work The summer assignment element should take no longer than hours to complete. Your summer assignment for each course must be submitted in the relevant first
More informationGrade 12 Mathematics. unimaths.co.za. Revision Questions. (Including Solutions)
Grade 12 Mathematics Revision Questions (Including Solutions) unimaths.co.za Get read for universit mathematics b downloading free lessons taken from Unimaths Intro Workbook. Visit unimaths.co.za for more
More informationEdexcel New GCE A Level Maths workbook
Edexcel New GCE A Level Maths workbook Straight line graphs Parallel and Perpendicular lines. Edited by: K V Kumaran kumarmaths.weebly.com Straight line graphs A LEVEL LINKS Scheme of work: a. Straight-line
More informationFigure 5.1 shows some scaffolding in which some of the horizontal pieces are 2 m long and others are 1 m. All the vertical pieces are 2 m.
A place for everthing, and everthing in its place. samuel smiles (8 904) Coordinate geometr Figure. shows some scaffolding in which some of the horizontal pieces are m long and others are m. All the vertical
More informationMathematics. Mathematics 2. hsn.uk.net. Higher HSN22000
hsn.uk.net Higher Mathematics UNIT Mathematics HSN000 This document was produced speciall for the HSN.uk.net website, and we require that an copies or derivative works attribute the work to Higher Still
More informationJUST IN TIME MATERIAL GRADE 11 KZN DEPARTMENT OF EDUCATION CURRICULUM GRADES DIRECTORATE TERM
JUST IN TIME MATERIAL GRADE 11 KZN DEPARTMENT OF EDUCATION CURRICULUM GRADES 10 1 DIRECTORATE TERM 1 017 This document has been compiled by the FET Mathematics Subject Advisors together with Lead Teachers.
More informationPolynomial Degree and Finite Differences
CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson, you Learn the terminology associated with polynomials Use the finite differences method to determine the degree of a polynomial
More informationMathematics. Mathematics 2. hsn.uk.net. Higher HSN22000
Higher Mathematics UNIT Mathematics HSN000 This document was produced speciall for the HSN.uk.net website, and we require that an copies or derivative works attribute the work to Higher Still Notes. For
More informationHigher. Polynomials and Quadratics. Polynomials and Quadratics 1
Higher Mathematics Contents 1 1 Quadratics EF 1 The Discriminant EF 3 3 Completing the Square EF 4 4 Sketching Parabolas EF 7 5 Determining the Equation of a Parabola RC 9 6 Solving Quadratic Inequalities
More informationSt Peter the Apostle High. Mathematics Dept.
St Peter the postle High Mathematics Dept. Higher Prelim Revision Paper I - Non~calculator Time allowed - hour 0 minutes FORMULE LIST Circle: The equation g f c 0 represents a circle centre ( g, f ) and
More informationTHOMAS WHITHAM SIXTH FORM
THOMAS WHITHAM SIXTH FORM Algebra Foundation & Higher Tier Units & thomaswhitham.pbworks.com Algebra () Collection of like terms. Simplif each of the following epressions a) a a a b) m m m c) d) d d 6d
More informationab is shifted horizontally by h units. ab is shifted vertically by k units.
Algera II Notes Unit Eight: Eponential and Logarithmic Functions Sllaus Ojective: 8. The student will graph logarithmic and eponential functions including ase e. Eponential Function: a, 0, Graph of an
More informationUNCORRECTED. To recognise the rules of a number of common algebraic relations: y = x 1 y 2 = x
5A galler of graphs Objectives To recognise the rules of a number of common algebraic relations: = = = (rectangular hperbola) + = (circle). To be able to sketch the graphs of these relations. To be able
More informationMathematics. Polynomials and Quadratics. hsn.uk.net. Higher. Contents. Polynomials and Quadratics 1. CfE Edition
Higher Mathematics Contents 1 1 Quadratics EF 1 The Discriminant EF 3 3 Completing the Square EF 4 4 Sketching Parabolas EF 7 5 Determining the Equation of a Parabola RC 9 6 Solving Quadratic Inequalities
More informationAQA Level 2 Further mathematics Number & algebra. Section 3: Functions and their graphs
AQA Level Further mathematics Number & algebra Section : Functions and their graphs Notes and Eamples These notes contain subsections on: The language of functions Gradients The equation of a straight
More informationZETA MATHS. Higher Mathematics Revision Checklist
ZETA MATHS Higher Mathematics Revision Checklist Contents: Epressions & Functions Page Logarithmic & Eponential Functions Addition Formulae. 3 Wave Function.... 4 Graphs of Functions. 5 Sets of Functions
More information( ) ( ) or ( ) ( ) Review Exercise 1. 3 a 80 Use. 1 a. bc = b c 8 = 2 = 4. b 8. Use = 16 = First find 8 = 1+ = 21 8 = =
Review Eercise a Use m m a a, so a a a Use c c 6 5 ( a ) 5 a First find Use a 5 m n m n m a m ( a ) or ( a) 5 5 65 m n m a n m a m a a n m or m n (Use a a a ) cancelling y 6 ecause n n ( 5) ( 5)( 5) (
More informationwe make slices perpendicular to the x-axis. If the slices are thin enough, they resemble x cylinders or discs. The formula for the x
Math Learning Centre Solids of Revolution When we rotate a curve around a defined ais, the -D shape created is called a solid of revolution. In the same wa that we can find the area under a curve calculating
More informationChapter 1 Coordinates, points and lines
Cambridge Universit Press 978--36-6000-7 Cambridge International AS and A Level Mathematics: Pure Mathematics Coursebook Hugh Neill, Douglas Quadling, Julian Gilbe Ecerpt Chapter Coordinates, points and
More informationThe CENTRE for EDUCATION in MATHEMATICS and COMPUTING cemc.uwaterloo.ca Euclid Contest. Wednesday, April 15, 2015
The CENTRE for EDUCATION in MATHEMATICS and COMPUTING cemc.uwaterloo.ca 015 Euclid Contest Wednesday, April 15, 015 (in North America and South America) Thursday, April 16, 015 (outside of North America
More informationPure Core 1. Revision Notes
Pure Core Revision Notes Ma 06 Pure Core Algebra... Indices... Rules of indices... Surds... 4 Simplifing surds... 4 Rationalising the denominator... 4 Quadratic functions... 5 Completing the square....
More informationDEPARTMENT OF MATHEMATICS
DEPARTMENT OF MATHEMATICS AS level Mathematics Core mathematics 1 C1 2015-2016 Name: Page C1 workbook contents Indices and Surds Simultaneous equations Quadratics Inequalities Graphs Arithmetic series
More informationCircles - Edexcel Past Exam Questions. (a) the coordinates of A, (b) the radius of C,
- Edecel Past Eam Questions 1. The circle C, with centre at the point A, has equation 2 + 2 10 + 9 = 0. Find (a) the coordinates of A, (b) the radius of C, (2) (2) (c) the coordinates of the points at
More informationEdexcel New GCE A Level Maths workbook Circle.
Edexcel New GCE A Level Maths workbook Circle. Edited by: K V Kumaran kumarmaths.weebly.com 1 Finding the Midpoint of a Line To work out the midpoint of line we need to find the halfway point Midpoint
More informationMathematics. Polynomials and Quadratics. hsn.uk.net. Higher. Contents. Polynomials and Quadratics 52 HSN22100
Higher Mathematics UNIT OUTCOME 1 Polnomials and Quadratics Contents Polnomials and Quadratics 5 1 Quadratics 5 The Discriminant 54 Completing the Square 55 4 Sketching Parabolas 57 5 Determining the Equation
More informationCollege Algebra Final, 7/2/10
NAME College Algebra Final, 7//10 1. Factor the polnomial p() = 3 5 13 4 + 13 3 + 9 16 + 4 completel, then sketch a graph of it. Make sure to plot the - and -intercepts. (10 points) Solution: B the rational
More informationSTUDY KNOWHOW PROGRAM STUDY AND LEARNING CENTRE. Functions & Graphs
STUDY KNOWHOW PROGRAM STUDY AND LEARNING CENTRE Functions & Graphs Contents Functions and Relations... 1 Interval Notation... 3 Graphs: Linear Functions... 5 Lines and Gradients... 7 Graphs: Quadratic
More informationModule 3, Section 4 Analytic Geometry II
Principles of Mathematics 11 Section, Introduction 01 Introduction, Section Analtic Geometr II As the lesson titles show, this section etends what ou have learned about Analtic Geometr to several related
More informationb UVW is a right-angled triangle, therefore VW is the diameter of the circle. Centre of circle = Midpoint of VW = (8 2) + ( 2 6) = 100
Circles 6F a U(, 8), V(7, 7) and W(, ) UV = ( x x ) ( y y ) = (7 ) (7 8) = 8 VW = ( 7) ( 7) = 64 UW = ( ) ( 8) = 8 Use Pythagoras' theorem to show UV UW = VW 8 8 = 64 = VW Therefore, UVW is a right-angled
More informationSample Problems For Grade 9 Mathematics. Grade. 1. If x 3
Sample roblems For 9 Mathematics DIRECTIONS: This section provides sample mathematics problems for the 9 test forms. These problems are based on material included in the New York Cit curriculum for 8.
More informationCoordinate geometry. + bx + c. Vertical asymptote. Sketch graphs of hyperbolas (including asymptotic behaviour) from the general
A Sketch graphs of = a m b n c where m = or and n = or B Reciprocal graphs C Graphs of circles and ellipses D Graphs of hperbolas E Partial fractions F Sketch graphs using partial fractions Coordinate
More informationThe Not-Formula Book for C1
Not The Not-Formula Book for C1 Everything you need to know for Core 1 that won t be in the formula book Examination Board: AQA Brief This document is intended as an aid for revision. Although it includes
More informationCircles, Mixed Exercise 6
Circles, Mixed Exercise 6 a QR is the diameter of the circle so the centre, C, is the midpoint of QR ( 5) 0 Midpoint = +, + = (, 6) C(, 6) b Radius = of diameter = of QR = of ( x x ) + ( y y ) = of ( 5
More informationMath 030 Review for Final Exam Revised Fall 2010 RH/ DM 1
Math 00 Review for Final Eam Revised Fall 010 RH/ DM 1 1. Solve the equations: (-1) (7) (-) (-1) () 1 1 1 1 f. 1 g. h. 1 11 i. 9. Solve the following equations for the given variable: 1 Solve for. D ab
More informationA-Level Notes CORE 1
A-Level Notes CORE 1 Basic algebra Glossary Coefficient For example, in the expression x³ 3x² x + 4, the coefficient of x³ is, the coefficient of x² is 3, and the coefficient of x is 1. (The final 4 is
More informationHigher Tier - Algebra revision
Higher Tier - Algebra revision Contents: Indices Epanding single brackets Epanding double brackets Substitution Solving equations Solving equations from angle probs Finding nth term of a sequence Simultaneous
More informationFind the dimensions of rectangle ABCD of maximum area.
QURTIS (hapter 1) 47 7 Infinitel man rectangles ma e inscried within the right angled triangle shown alongside. One of them is illustrated. a Let = cm and = cm. Use similar triangles to find in terms of.
More informationFurther Mathematics Summer work booklet
Further Mathematics Summer work booklet Further Mathematics tasks 1 Skills You Should Have Below is the list of the skills you should be confident with before starting the A-Level Further Maths course:
More informationName Please print your name as it appears on the class roster.
Berkele Cit College Practice Problems Math 1 Precalculus - Final Eam Preparation Name Please print our name as it appears on the class roster. SHORT ANSWER. Write the word or phrase that best completes
More informationApplications. 12 The Shapes of Algebra. 1. a. Write an equation that relates the coordinates x and y for points on the circle.
Applications 1. a. Write an equation that relates the coordinates and for points on the circle. 1 8 (, ) 1 8 O 8 1 8 1 (13, 0) b. Find the missing coordinates for each of these points on the circle. If
More information2016 SEC 4 ADDITIONAL MATHEMATICS CW & HW
FEB EXAM 06 SEC 4 ADDITIONAL MATHEMATICS CW & HW Find the values of k for which the line y 6 is a tangent to the curve k 7 y. Find also the coordinates of the point at which this tangent touches the curve.
More informationCore Mathematics 2 Coordinate Geometry
Core Mathematics 2 Coordinate Geometry Edited by: K V Kumaran Email: kvkumaran@gmail.com Core Mathematics 2 Coordinate Geometry 1 Coordinate geometry in the (x, y) plane Coordinate geometry of the circle
More informationNEXT-GENERATION Advanced Algebra and Functions
NEXT-GENERATIN Advanced Algebra and Functions Sample Questions The College Board The College Board is a mission-driven not-for-profit organization that connects students to college success and opportunit.
More informationc) domain {x R, x 3}, range {y R}
Answers Chapter 1 Functions 1.1 Functions, Domain, and Range 1. a) Yes, no vertical line will pass through more than one point. b) No, an vertical line between = 6 and = 6 will pass through two points..
More informationLearning Goals. College of Charleston Department of Mathematics Math 101: College Algebra Final Exam Review Problems 1
College of Charleston Department of Mathematics Math 0: College Algebra Final Eam Review Problems Learning Goals (AL-) Arithmetic of Real and Comple Numbers: I can classif numbers as natural, integer,
More information8 Differential Calculus 1 Introduction
8 Differential Calculus Introduction The ideas that are the basis for calculus have been with us for a ver long time. Between 5 BC and 5 BC, Greek mathematicians were working on problems that would find
More informationDiagnostic Tests Study Guide
California State Universit, Sacramento Department of Mathematics and Statistics Diagnostic Tests Stud Guide Descriptions Stud Guides Sample Tests & Answers Table of Contents: Introduction Elementar Algebra
More informationQuadratics NOTES.notebook November 02, 2017
1) Find y where y = 2-1 and a) = 2 b) = -1 c) = 0 2) Epand the brackets and simplify: (m + 4)(2m - 3) To find the equation of quadratic graphs using substitution of a point. 3) Fully factorise 4y 2-5y
More informationChapter 4 Analytic Trigonometry
Analtic Trigonometr Chapter Analtic Trigonometr Inverse Trigonometric Functions The trigonometric functions act as an operator on the variable (angle, resulting in an output value Suppose this process
More informationCopyrighted by Gabriel Tang B.Ed., B.Sc. Page 111.
Algera Chapter : Polnomial and Rational Functions Chapter : Polnomial and Rational Functions - Polnomial Functions and Their Graphs Polnomial Functions: - a function that consists of a polnomial epression
More informationInternational Examinations. Advanced Level Mathematics Pure Mathematics 1 Hugh Neill and Douglas Quadling
International Eaminations Advanced Level Mathematics Pure Mathematics Hugh Neill and Douglas Quadling PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE The Pitt Building, Trumpington Street,
More informationEquations and Inequalities
Equations and Inequalities Figure 1 CHAPTER OUTLINE 1 The Rectangular Coordinate Systems and Graphs Linear Equations in One Variable Models and Applications Comple Numbers Quadratic Equations 6 Other Types
More informationN5 R1.1 Linear Equations - Revision
N5 R Linear Equations - Revision This revision pack covers the skills at Unit Assessment and eam level for Linear Equations so ou can evaluate our learning of this outcome. It is important that ou prepare
More informationTime: 1 hour 30 minutes
Paper Reference(s) 6664/01 Edexcel GCE Core Mathematics C Silver Level S4 Time: 1 hour 0 minutes Materials required for examination Mathematical Formulae (Green) Items included with question papers Nil
More information1. Find the domain of the following functions. Write your answer using interval notation. (9 pts.)
MATH- Sample Eam Spring 7. Find the domain of the following functions. Write your answer using interval notation. (9 pts.) a. 9 f ( ) b. g ( ) 9 8 8. Write the equation of the circle in standard form given
More informationNATIONAL QUALIFICATIONS
H Mathematics Higher Paper Practice Paper A Time allowed hour minutes NATIONAL QUALIFICATIONS Read carefull Calculators ma NOT be used in this paper. Section A Questions ( marks) Instructions for completion
More informationParabolas. Example. y = ax 2 + bx + c where a = 1, b = 0, c = 0. = x 2 + 6x [expanding] \ y = x 2 + 6x + 11 and so is of the form
Parabolas NCEA evel Achievement Standard 9157 (Mathematics and Statistics.) Appl graphical methods in solving problems Methods include: graphs at curriculum level 7, their features and their equations
More informationCopyrighted by Gabriel Tang B.Ed., B.Sc. Page 1.
Chapter : Linear and Quadratic Functions Chapter : Linear and Quadratic Functions -: Points and Lines Sstem of Linear Equations: - two or more linear equations on the same coordinate grid. Solution of
More information6. COORDINATE GEOMETRY
6. CRDINATE GEMETRY Unit 6. : To Find the distance between two points A(, ) and B(, ) : AB = Eg. Given two points A(,3) and B(4,7) ( ) ( ). [BACK T BASICS] E. P(4,5) and Q(3,) Distance of AB = (4 ) (7
More informationAdvanced Algebra Scope and Sequence First Semester. Second Semester
Last update: April 03 Advanced Algebra Scope and Sequence 03-4 First Semester Unit Name Unit : Review of Basic Concepts and Polynomials Unit : Rational and Radical Epressions Sections in Book 0308 SLOs
More informationWorking Out Your Grade
Working Out Your Grade Please note: these files are matched to the most recent version of our book. Don t worry you can still use the files with older versions of the book, but the answer references will
More information5 Find an equation of the circle in which AB is a diameter in each case. a A (1, 2) B (3, 2) b A ( 7, 2) B (1, 8) c A (1, 1) B (4, 0)
C2 CRDINATE GEMETRY Worksheet A 1 Write down an equation of the circle with the given centre and radius in each case. a centre (0, 0) radius 5 b centre (1, 3) radius 2 c centre (4, 6) radius 1 1 d centre
More informationMath Intermediate Algebra
Math 095 - Intermediate Algebra Final Eam Review Objective 1: Determine whether a relation is a function. Given a graphical, tabular, or algebraic representation for a function, evaluate the function and
More informationMaths Higher Prelim Content
Maths Higher Prelim Content Straight Line Gradient of a line A(x 1, y 1 ), B(x 2, y 2 ), Gradient of AB m AB = y 2 y1 x 2 x 1 m = tanθ where θ is the angle the line makes with the positive direction of
More informationSolutions to the Math 1051 Sample Final Exam (from Spring 2003) Page 1
Solutions to the Math 0 Sample Final Eam (from Spring 00) Page Part : Multiple Choice Questions. Here ou work out the problems and then select the answer that matches our answer. No partial credit is given
More information1Number ONLINE PAGE PROOFS. systems: real and complex. 1.1 Kick off with CAS
1Numer systems: real and complex 1.1 Kick off with CAS 1. Review of set notation 1.3 Properties of surds 1. The set of complex numers 1.5 Multiplication and division of complex numers 1.6 Representing
More informationGlossary. Also available at BigIdeasMath.com: multi-language glossary vocabulary flash cards. An equation that contains an absolute value expression
Glossar This student friendl glossar is designed to be a reference for ke vocabular, properties, and mathematical terms. Several of the entries include a short eample to aid our understanding of important
More informationQUADRATIC GRAPHS ALGEBRA 2. Dr Adrian Jannetta MIMA CMath FRAS INU0114/514 (MATHS 1) Quadratic Graphs 1/ 16 Adrian Jannetta
QUADRATIC GRAPHS ALGEBRA 2 INU0114/514 (MATHS 1) Dr Adrian Jannetta MIMA CMath FRAS Quadratic Graphs 1/ 16 Adrian Jannetta Objectives Be able to sketch the graph of a quadratic function Recognise the shape
More information1. (A) Factor to get (2x+3)(2x 10) = 0, so the two roots are 3/2 and 5, which sum to 7/2.
Solutions 00 53 rd AMC 1 A 1. (A) Factor to get (x+3)(x 10) = 0, so the two roots are 3/ and 5, which sum to 7/.. (A) Let x be the number she was given. Her calculations produce so x 9 3 = 43, x 9 = 19
More informationP1 Chapter 4 :: Graphs & Transformations
P1 Chapter 4 :: Graphs & Transformations jfrost@tiffin.kingston.sch.uk www.drfrostmaths.com @DrFrostMaths Last modified: 14 th September 2017 Use of DrFrostMaths for practice Register for free at: www.drfrostmaths.com/homework
More informationMEI Core 1. Basic Algebra. Section 1: Basic algebraic manipulation and solving simple equations. Manipulating algebraic expressions
MEI Core Basic Algebra Section : Basic algebraic manipulation and solving simple equations Notes and Examples These notes contain subsections on Manipulating algebraic expressions Collecting like terms
More informationMathematics Paper 1 (Non-Calculator)
H National Qualifications CFE Higher Mathematics - Specimen Paper F Duration hour and 0 minutes Mathematics Paper (Non-Calculator) Total marks 60 Attempt ALL questions. You ma NOT use a calculator. Full
More informationPure Core 2. Revision Notes
Pure Core Revision Notes June 06 Pure Core Algebra... Polynomials... Factorising... Standard results... Long division... Remainder theorem... 4 Factor theorem... 5 Choosing a suitable factor... 6 Cubic
More informationAnswers to Homework Book 9
Answers to Homework ook Checking answers Eercise.H (page ) a) 0 7 0 = 7, 7 b) (7 ) 0 = 0 (or, better, (7 ) 0 = 00), c) 0 = 00, d) 0 = 0, 0 e) =, 7 f) 0 0 = 00, g) 70 0 = 00, h) 0 0 0 = 00, 0 a) 0 7 =,
More informationDiagnostic Assessment Number and Quantitative Reasoning
Number and Quantitative Reasoning Select the best answer.. Which list contains the first four multiples of 3? A 3, 30, 300, 3000 B 3, 6, 9, 22 C 3, 4, 5, 6 D 3, 26, 39, 52 2. Which pair of numbers has
More informationPerth Academy Mathematics Department Intermediate 2 Unit 3 Revision Pack. Contents:
Perth Academ Mathematics Department Intermediate Unit Revision Pack Contents: Algebraic Operations: Fractions Fractions Formulae Surds Indices Quadratic Functions: Y = a Y = a + b Y = ( + a) + b Turning
More informationA marks are for accuracy and are not given unless the relevant M mark has been given (M0 A1 is impossible!).
NOTES 1) In the marking scheme there are three types of marks: M marks are for method A marks are for accuracy and are not given unless the relevant M mark has been given (M0 is impossible!). B marks are
More informationCircle. Paper 1 Section A. Each correct answer in this section is worth two marks. 5. A circle has equation. 4. The point P( 2, 4) lies on the circle
PSf Circle Paper 1 Section A Each correct answer in this section is worth two marks. 1. A circle has equation ( 3) 2 + ( + 4) 2 = 20. Find the gradient of the tangent to the circle at the point (1, 0).
More information2017 Canadian Team Mathematics Contest
The CENTRE for EDUCATION in MATHEMATICS and COMPUTING cemc.uwaterloo.ca 017 Canadian Team Mathematics Contest April 017 Solutions 017 University of Waterloo 017 CTMC Solutions Page Individual Problems
More informationFurther factorising, simplifying, completing the square and algebraic proof
Further factorising, simplifying, completing the square and algebraic proof 8 CHAPTER 8. Further factorising Quadratic epressions of the form b c were factorised in Section 8. by finding two numbers whose
More informationObjectives To solve equations by completing the square To rewrite functions by completing the square
4-6 Completing the Square Content Standard Reviews A.REI.4. Solve quadratic equations y... completing the square... Ojectives To solve equations y completing the square To rewrite functions y completing
More informationSolutions to O Level Add Math paper
Solutions to O Level Add Math paper 4. Bab food is heated in a microwave to a temperature of C. It subsequentl cools in such a wa that its temperature, T C, t minutes after removal from the microwave,
More informationSkills Practice Skills Practice for Lesson 9.1
Skills Practice Skills Practice for Lesson.1 Name Date Meeting Friends The Distance Formula Vocabular Define the term in our own words. 1. Distance Formula Problem Set Archaeologists map the location of
More informationThe Coordinate Plane. Circles and Polygons on the Coordinate Plane. LESSON 13.1 Skills Practice. Problem Set
LESSON.1 Skills Practice Name Date The Coordinate Plane Circles and Polgons on the Coordinate Plane Problem Set Use the given information to show that each statement is true. Justif our answers b using
More informationy = f(x + 4) a) Example: A repeating X by using two linear equations y = ±x. b) Example: y = f(x - 3). The translation is
Answers Chapter Function Transformations. Horizontal and Vertical Translations, pages to. a h, k h, k - c h -, k d h 7, k - e h -, k. a A (-,, B (-,, C (-,, D (,, E (, A (-, -, B (-,, C (,, D (, -, E (,
More informationCourse 15 Numbers and Their Properties
Course Numbers and Their Properties KEY Module: Objective: Rules for Eponents and Radicals To practice appling rules for eponents when the eponents are rational numbers Name: Date: Fill in the blanks.
More informationDifferentiation and applications
FS O PA G E PR O U N C O R R EC TE D Differentiation and applications. Kick off with CAS. Limits, continuit and differentiabilit. Derivatives of power functions.4 C oordinate geometr applications of differentiation.5
More informationHIGHER SCHOOL CERTIFICATE EXAMINATION MATHEMATICS 2/3 UNIT (COMMON) Time allowed Three hours (Plus 5 minutes reading time)
N E W S O U T H W A L E S HIGHER SCHOOL CERTIFICATE EXAMINATION 996 MATHEMATICS /3 UNIT (COMMON) Time allowed Three hours (Plus minutes reading time) DIRECTIONS TO CANDIDATES Attempt ALL questions. ALL
More informationabc Mathematics Pure Core General Certificate of Education SPECIMEN UNITS AND MARK SCHEMES
abc General Certificate of Education Mathematics Pure Core SPECIMEN UNITS AND MARK SCHEMES ADVANCED SUBSIDIARY MATHEMATICS (56) ADVANCED SUBSIDIARY PURE MATHEMATICS (566) ADVANCED SUBSIDIARY FURTHER MATHEMATICS
More informationMath 154A Elementary Algebra Fall 2014 Final Exam Study Guide
Math A Elementar Algebra Fall 0 Final Eam Stud Guide The eam is on Tuesda, December 6 th from 6:00pm 8:0pm. You are allowed a scientific calculator and a " b 6" inde card for notes. On our inde card be
More information