Higher Mathematics Booklet CONTENTS
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1 Higher Mathematics Bklet CONTENTS Frmula List Item Pages The Straight Line Hmewrk The Straight Line Hmewrk Functins Hmewrk 3 Functins Hmewrk 4 Recurrence Relatins Hmewrk 5 Differentiatin Hmewrk 6 Differentiatin Hmewrk 7 Plynmials Hmewrk 8 Quadratics Hmewrk 9 Integratin Hmewrk 0 Integratin Hmewrk Further Trignmetry Hmewrk Further Trignmetry Hmewrk 3 The Circle Hmewrk 4 The Circle Hmewrk 5 Vectrs Hmewrk 6 Vectrs Hmewrk 7 Further Calculus Hmewrk 8 Further Calculus Hmewrk 9 Lgarithms and Epnentials Hmewrk 0 Lgarithms and Epnentials Hmewrk The Wave Functin Hmewrk The Wave Functin Hmewrk 3 Revising Unit (including A/B questins) Revising Unit (including A/B questins) Revising Unit 3 (including A/B questins) Unit Practice Assessments (NABS) Unit Practice Assessments (NABS) Unit 3 Practice Assessments (NABS) Higher Eam Technique Mdules Past Paper Questin Map Tpic Checklist The fllwing items will be issued thrughut the year in separate bklets. Please keep them safe and return them t yu teacher at the end f the year. Higher Objective Test Bklet Additinal Practice Papers Revisin Bklets
2 Higher Mathematics Bklet DETAILS Hmewrk All questins in the hmewrk shuld be attempted. The questins in the b cntain imprtant revisin f previus tpics. Care f bklet Please keep these bklets in gd cnditin and return at the end f the year Please d nt mark the cpies with pen
3 The Straight Line Hmewrk. Find the angles which the lines jining the fllwing pairs f pints make with the psitive directin f the -aes (OX) (a) (0, 0), (, ) (b) (0, 0), ( 3, ) (c) (-3, 6), (5, 4) (6). Find the equatins f the lines thrugh the fllwing pints, with the given gradients. Give yur answers in the frm y b m( a) (a) (, 3), 4 (b) (0, 4), - (c) (-4, ), ½ 3. Find the equatin f the line thrugh (-3, ) and parallel t the line 3y Find the equatin f the line cnnecting pints (, -4) and (3, -6). () 5. P is the pint (4, 0), Q is (0, -3) and R is (-5, -). Given that PQRS is a parallelgram, btain the equatin f RS. 6. Prve that the pints A(-, ), B(-, 0) and C(7, -8) are cllinear. 7. Slve the equatin ( 9) 9( ) Slve the fllwing simultaneus equatins 4 y 8 3y TOTAL 7
4 The Straight Line Hmewrk. Find the equatin f the median AD f triangle ABC where the crdinates f A, B and C are (-, 3), (-3, -4) and (5, ) respectively.. Find the equatin f the altitude PR f triangle OPQ where the crdinates f O, P and Q are (-, 3), (-3, -4) and (5, ) respectively. 3. Find the equatin f the perpendicular bisectr f the line jining A(, -) and B(8, 3). 4. A triangle ABC has vertices A( -4, -3), B(-, ) and C(6, -3) y B 0 A C (a) Shw that the triangle ABC is right angled at B. (b) The medians AD and BE intersect at M. (i) Find the equatins f AD and BE. (6) (ii) Find the crdinates f M () 5. Slve 5 ( 4) 0 6. Simplify each epressin (a) b b (b) 48 f 6 f 0 ( 6h 5 ) 3 (c) 4 TOTAL 6
5 Functins Hmewrk 3. Cnvert t radians (simplify where pssible) (a) 60 (b) 5 (c) 330. Epress 60 in radians crrect t decimal places. () 3. Cnvert t degrees 4 9 (a) radians (b) radians (c) radians Sketch the graph f y ) sin(3 60, f ( ), g ( ) and h( ) g( f ( )) (a) Find a frmula fr h(). () (b) Fr what values f is h undefined? () 6. f ( ) and g ( ) 3 Evaluate g ( f ()) and f ( g( )) 7. Epress the fllwing as eact values (a) sin5 (b) tan50 () 8. Slve cs3, fr TOTAL 4
6 Functins Hmewrk 4. The fllwing sketch shws the graph f f(). - Make separate sketches f the fllwing graphs: (a) y f () (b) y f ( ) (c) y 3 f ( ) -6. g (a) y g() ( ). Make separate sketches f the fllwing graphs: (b) y 3 (c) ( 4) y 3. (a) Shw that the functin f ( ) 8 3 can be written in the frm f ( ) a( b) c, where a, b and c are cnstants. (b) Hence, r therwise, find the crdinates f the turning pint f the functin f. () 4. (a) Epress 7 in the frm a and b. a ( b) and write dwn the values f (b) State the minimum value f 7 and justify yur answer. () 5. Find the crdinates f the pints f intersectin f the curve y 3 5 and the line y Evaluate 64 () TOTAL
7 Recurrence Relatins Hmewrk 5. (a)write dwn the net 3 terms f the sequence given by the recurrence relatin T 0.9T, T 40 () n n 0 (b) What is the limit f this sequence? (). A car designer has calculated that water escapes frm an engine s cling system at a rate f 0% per mnth. The system was initially filled with 36 litres f clant and each mnth she adds 4 litres f clant t the system. (a) Find a recurrence relatin t describe this. () (b) Calculate the vlume f clant in the system after 6 mnths. () (c) If the clant drps belw 8 litres the engine will verheat. Is the engine in danger f verheating? (eplain fully) () 3. A recurrence relatin is defined by U n au n b fr sme cnstants a and b. (a) If U 90, U3 430 and U 4 90 calculate the values f a and b. (b) What is the initial value, U 0, f this sequence? () 4. (a) Given that f ( ) and g ( ) 3, find functins k( ) f ( g( )) and h( ) g( f ( )). (b) If f ( g( p)) f ( g( p)) 6 fr sme number p, find the value(s) f p. (6) TOTAL
8 Differentiatin Hmewrk 6. Differentiate the fllwing functins with respect t. (a) f ( ) 3 6 (b) f ( ) ( 3). Differentiate each f the fllwing functins with respect t the relevant variable 5 3 (a) f ( ) ( ) (b) f ( t) t ( t t ) (c) g( p) (6) Calculate the rate f change f f ( ) 3 at. () 4. Find the equatin f the tangent t the curve with equatin y 4 3 at the at the pint where. 5. The gradient f the tangent t the curve y a b is equal t 30 at the pint pint (3, ). Find the values f a and b. 6. Triangle PQR has vertices (, 3), (-3, -) and (3, 0) respectively. (a) Find the equatins f the perpendicular bisectrs f sides RQ and PR. (b) Find the crdinates f the pint T, the pint f intersectin f these tw lines. (6) TOTAL 5
9 Differentiatin Hmewrk 7. Differentiate the fllwing functins with respect t (a) y (b) ( 3) y (6) 3 '. Fr the fllwing graph f f (), sketch the graph f f ( ). y (, 0 3 () 3 3. Fr what values f is the functin h ( ) 3 decreasing? 4. The diagram shws a sketch f the parabla y 6 and the line y. y y 6 y 0 (a) Find the gradient f the tangent t the parabla at the pint (0, 0). () (b) Hence r therwise calculate the size f the angle between the line y and the tangent t the parabla at the pint (0, 0). 5. Sketch the graph f the curve y ( 4), indicating all imprtant pints. (5) 6. Sketch the graph f y ) 3cs( 30, TOTAL 5
10 Plynmials Hmewrk 8 3. (a) Shw that is a rt f the equatin () (b) Hence find the ther rts. 3. Given that 4 is a factr f h( ) 5 a 80, find (a) The value f a () (b) Hence slve the equatin 3 5 a 80 0 when a takes this value. () When a bis divided by the remainder is 0, and when divided by the remainder is. Find a and b. (6) 4. Find the equatin f the perpendicular bisectr f CD where the crdinates f C and D are (4,3) and (,-3) respectively Find the equatin f the tangent t the curve y 3, at the pint. TOTAL
11 Quadratics Hmewrk 9. Decide the nature f the rts in the fllwing equatin: (). Prve that y 6 is a tangent t y 5 and find the pint f cntact. 3. Find p if the rts f ( p ) p ( p ) 0are equal. 4. Epress 7 in the frm ( a) b (a) Hence state the minimum value f 7. (b) State the maimum value f and the 7 crrespnding value f. (5) 5. A statistician claims that purchasing 5 new street cleaners will reduce the amunt f litter by 65% after cleaning. Hwever, after testing it is fund that a further 400kg f litter are drpped between cleaning times. What is the minimum weight f litter fund at any pint after cleaning? (5) 6. Differentiate 3 4 () TOTAL
12 Integratin Hmewrk 0. Integrate the fllwing: d a) b) ( 4 3) d (5) 3 r. Evaluate dr r 9 k 3. If d, find the value f k The functin g, defined n a suitable dmain, is given by g ( ). (a) Describe any restrictin n the dmain f g. () (b) Find an epressin fr h() where h( ) g( g( )), giving yur answer as a fractin in its simplest frm. 5. Find the rate f change f the functin f ( t) 5t(3t 4), when t TOTAL 0
13 Integratin Hmewrk. Find the area between the curve limits and 4. y 6 5 and the ais, between the. Find the area between the curves y and y 3 (5) dy 3. Given that 4 3 d and that y when find y in terms f. 4. y y f () Sketch the graph f (a) f ( ) (b) f () () 5. Find the equatin f the curve frm questin 4 in the frm y k( a)( b)( c) 6. Given that 3 y 5 dy, find. d TOTAL
14 Further Trignmetry Hmewrk. Sketch the graph f the functin f ( ) sin( 30). (). If tan 3, find the eact value f sin. () 3. (a)write cs cs sin sin in the frm cs( A B) (b) Hence slve the equatin cs cs sin sin, 6 6 fr 0 () 4. The diagram shws the crss-sectin f an adjustable ramp which is made frm tw right-angled triangles, PQR and RQS. Angle RQS = a and PQR = b. Find the eact value f sin(a+b) (a) Given that (-,0) is a pint f intersectin f the curve y 3 3 n the -ais. Find the ther pints f intersectin n this ais. (b) State the pint f intersectin n the y-ais. (c) Find the crdinates f the statinary pint(s) and justify the(ir) nature. (d) Sketch the graph f the functin. (9) TOTAL
15 Further Trignmetry Hmewrk 3. Slve the fllwing equatins: a) sin sin 0 ( 0 ) b) 4cs 6cs 0 ( 0 360). (0). The diagram shws tw curves y cs and y sin where Find the -crdinate f the pint f intersectin at A. (5) 3. A rectangular sheet f cardbard is t be flded t frm a clsed cubid with a length f cm, breadth cm and a vlume f 7cm 36 (a) Shw that the height is. 6 (b) Shw that the ttal surface area A is given by the frmula A( ) 4 cm (c) Find the dimensins f the cubid s that the ttal surface area is minimum. (7) TOTAL
16 The Circle Hmewrk 4. Find the equatins f the fllwing circles with: (a) centre (0,0) and radius 3 (b) centre (-,3) and radius 5 (). State the centre and radius f each f the fllwing circles (a) y 4 8y 6 0 (b) y 6y 0 (c) y 6 4y Prve whether pint (-, ) lies inside r utside the circle y 4 5y 8 0 () 4. The end pints f the diameter f a circle are A(-6, 4) and B(, -). Find the equatin f the circle. 5. Find the value f k if pint (-, k) lies n the circle y 5y ABC has vertices A(-, -6), B(3, 4) and C(-3, 7). Find the: (a) Equatin f the altitude frm A (b) Equatin f the median frm C (c) Crdinates f the pint f intersectin between the altitude and median abve. () TOTAL 5
17 The Circle Hmewrk 5. Prve that T(5, ) lies n the circle ( ) ( y ) 5, and find the equatin f the tangent at T. (5). Shw that y is a tangent t the circle y 8 y 9 0 and find the pint f cntact. 3. The diagram shws tw identically sized circles that have line AB as a cmmn tangent with T as the pint f cntact. The equatin f AB is = -. One f the circles has equatin y y 7 0. Find the equatin f the ther circle. A T B 4. Calculate the length f the tangent frm A(,-4) t the circle with equatin ( 5) ( y 4) 5. (6) 5. If f and g are functins defined n set R by f ( ) 3 and g ( ) 4 3 (a) Find g ( f ( )) (b) Find f ( f ( )) TOTAL
18 Vectrs Hmewrk 6. R divides EF in the rati 3:. Find the crdinates f R given that the crdinates f E and F are (5, 0, 0) and (0, 0, -5) respectively.. In this diagram AB u and DA v. DC AB, and DC is parallel t AB. Write dwn, in terms f u and v, the vectrs: (a) DC (b) BC (c) BM (d) DM (5) 3. A(,3,-), B(5,4,) and C(-,6,3) are vertices f a parallelgram ABCD. Find the crdinates f the pint D. 4. A, B, and C are the pints (3, -4, 4), (5,, 0) and (8,, -6) respectively. Shw that A, B and C are cllinear and find the rati in which C divides AB. (5) 3 5. Find the value f k if ( 3) is a factr f 5 k Evaluate 3 d TOTAL 4
19 Vectrs Hmewrk 7. The psitin vectrs f the pints P and Q are p 3i 4j and q 3i 4j 5k respectively. Calculate the size f angle POQ, where O is the rigin. (9). (a) Shw that ( a b).( a b) a b. (b) Hence evaluate this epressin when 3 a and 0 5 b 3. (5) p 3. If a 3 and b 6 4 Find the value f p given that a and b are perpendicular. 4. Fr what real values f c will the equatin 3 5 c 0 have real rts. () u 5. Evaluate du 4 u TOTAL 3
20 Further Calculus Hmewrk 8. Differentiate (a) cs (b) 4p 3sin p (). Find 3 (a) (5 sin ) d (b) 4 ( cs y ) dy 5y 3. Let 4 f ( ) ( 3). Find '( ) f. () 4. Find the area between the curve y 4sin and the ais frm. 3 and Find the equatin f the tangent t the curve y, at the pint. ( ) (6) 6. If tan( A) 3 4 and tan( B ) withut a calculatr that 7, where A and B are acute angles, shw A B. 4 cs A 7. Prve that tan A cs A () TOTAL 3
21 Further Calculus Hmewrk 9. Find f '( ) given that f ( ) sin( 3) (). Differentiate the fllwing (a) f ( ) cs (b) f ( ) (6) sin 3. Find the derivative with respect t f :- f ( ) ( 3 ) cs( 3 ) 4. Find ( 3) d and hence find the eact value f ( 3) d 0 5. Integrate 6 sin( ) d () 6. (a) Shw that has a rt between - and -. (b) Find this rt crrect t decimal place. 7. Find the equatin f the tangent t the circle y 5 at the pint (-4, 3). TOTAL 5
22 Lgarithms and Epnentials Hmewrk 0. Make sketches t shw y lg 3 and y lg 3( ) (). Find a) lg 3 7 b) lg 9 c) lg Simplify lg 3 lg Slve fr :- lg( 3) lg 0 5. Find, if The diagram shws a sketch f part f the graph f y lg 5. a) Make a cpy f the graph f y lg 5. On yur cpy, sketch the graph f y lg 5 () b) Find the crdinates f the pint where it crsses the -ais. () 7. Slve tan, given 0 80 TOTAL 3
23 Lgarithms and Epnentials Hmewrk. A medical technician btains this print-ut f a wave generated by an scillscpe. The technician knws that the equatin f the first branch f the graph fr 0 3 shuld be f the frm y ae k. (a) Find the values f a and k. (b) Find the equatin f the secnd branch f the curve (i.e. fr 3 6). (). A lake is plluted by a cleaning agent frm a dye-wrks. Every day, the pllutant is brken dwn by natural prcesses, (rain, the fish in the lake etc.) The frmula fr the percentage f the pllutant is :- 0. 0d P(d)=00 e where P(d)=the percentage f pllutant after d days. (a) What percentage f the pllutant will remain in the lake after 0 days? () (b) Hw lng will it take fr half the pllutant t be remved frm the lake? 3. A scientific eperiment is carried ut t find a cnnectin between tw variables P and t. Results are recrded, and when the graph f lg 0 P against lg 0 t is pltted, the best fitting line is drawn in as shwn in the diagram. It is thught that the relatinship between P and t is f the frm P kt n (a) Shw that lg 0 P nlg 0 t lg 0 k (b) Find k and n. (5) 4. Find the pints f intersectin f the line 3y and the circle y 0 4y 9 0. (5) TOTAL
24 The Wave Functin Hmewrk. Epress 3sin 3cs in the frm k cs( ) where k 0 and Epress 3 cs sin in the frm k sin( ) where k 0 and Epress 5cs8 sin8 in the frm r sin(8 ) where r 0 and (a)write dwn the maimum value f f ( ) 3cs 4 sin. (5) (b) Find the value f fr which this maimum ccurs, fr () 3 5. The straight line OA is a tangent t the curve y at pint A(,) O A (a) Find the equatin f the line OA. (b) Calculate the finite area bunded by the curve, y ais and the line OA. (6) TOTAL 4
25 The Wave Functin Hmewrk 3. Slve this equatin algebraically fr 0. 3sin cs. (a) Find the minimum value f 7 3sin 4cs. (5) 6 (b) Hence r therwise find the maimum value f f ( ) 7 3sin 4cs and the value f at which this ccurs. () 3. The epressin 0sin0T 40cs0T represents the displacement f a wave after T secnds and can be written in the frm 0 5 sin(0t 63.4 ). a) Write dwn the amplitude and perid f the wave. () b) Use yur values f R and t sketch the graph f Rsin( 0T ) against T fr 0 T 36, shwing clearly the pints where the graph cuts the -ais and any statinary pints. (6) 3 4. Fr what values f is functin f ( ) 4 9 decreasing? TOTAL 3
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