hsn.uk.net Higher Mathematics UNIT Specimen NAB Assessment HSN50 This dcument was prduced speciall fr the HSN.uk.net website, and we require that an cpies r derivative wrks attribute the wrk t Higher Still Ntes. Fr mre details abut the cpright n these ntes, please see http://creativecmmns.rg/licenses/b-nc-sa/.5/sctland/
Higher Mathematics Unit Specimen NAB Assessment UNIT Specimen NAB Assessment Outcme. Shw that ( + ) is a factr f f ( ) = + 8 and hence factrise full f ( ).. Use the discriminant t determine the nature f the rts f the equatin + = 0. Outcme. Find 6 d, # where 0.. The curve = ( ) is shwn in the diagram belw. = ( ) O Calculate the shaded area enclsed between the curve and the -ais between = 0 and =. 5 5. The diagram shws the line with equatin = and the curve with equatin = 5. = 5 O = Write dwn the integral which represents the shaded area. D nt carr ut the integratin. hsn.uk.net Page HSN50
Higher Mathematics Unit Specimen NAB Assessment Outcme 6. Slve the equatin sin = fr 0 <. 7. The diagram belw shws tw right-angled triangles. 9 (a) Write dwn the values f sin and cs. (b) B epanding cs( + ) shw that the eact value f cs( + ) is 6 65. 8. (a) Epress sin5 cs cs5 sin + in the frm sin( a b ) (b) Use ur answer frm part (a) t slve the equatin Outcme +. sin5 cs + cs5 sin = fr 0 < < 60. 9. (a) A circle has radius 7 units and centre (, ). Write dwn the equatin f the circle. (b) A circle has equatin 0 + 0 + 6 = 0. Write dwn its radius and the crdinates f its centre. 0. Shw that the straight line = is a tangent t the circle with equatin + + 6 + + 8 = 0. 5. The pint P( 0,5 ) lies n the circle with centre (,0), as shwn in the diagram belw. O P( 0,5) Find the equatin f the tangent t the circle at P. hsn.uk.net Page HSN50
Higher Mathematics Unit Specimen NAB Assessment Marking Instructins Pass Marks Outcme Outcme Outcme Outcme 6 8 7 0 Outcme Plnmials and Quadratics. 8. 8 8 0 Since f ( ) = 0, ( + ) is a factr. ( ) = ( + )( + ) f b = ( + )( )( ) = ( ) = 0 Since b ac ac > 0, the rts are real and distinct. Outcme Integratin 6. d = ( 6 ) $ d # 6 = + c = + c. $ ( ) = ( ) 0 $ 0 d d 5 % & = ' 5 ( ) * 0 5 = ( ( ) ( ) ) 0 = 5 ( r ) 0 0 Knw t evaluate f ( ) Cmplete evaluatin and cnclusin Quadratic factr Factrise quadratic Use the discriminant Calculate discriminant and state nature f rts Epress in standard frm Integrate term with negative pwer Cnstant f integratin Knw t integrate with limits Use crrect limits Integrate Prcess limits Cmplete prcess 5 hsn.uk.net Page HSN50
Higher Mathematics Unit Specimen NAB Assessment 5. 5 = 7 = 0 ( 7) = 0 = 0 r = 7 $ d 0 7 Shaded area is (( ) ( 5 ) ) square units. Outcme Trignmetr 6. sin = S A T C 7. (a) = r = 8 r 8 AC 9 5 DF 0 6 + ( ) = sin = = + = +, - = + =,. 9 5 5 5 sin = = and cs = 0 = (b) ( ) cs + = cs cs sin sin 5 5 5 = 0 6 65 65 = = 6 65 6 Strateg t find intersectin Slve quadratic Use $ ( upper ) with limits frm quadratic lwer d Rearrange t standard frm One slutin Secnd slutin Calculate remaining sides sin and cs Use cmpund angle frmula Substitute values 8. (a) sin5 cs + cs5 sin = sin( 5 + ) Use cmpund angle frmula (b) sin( 5 + ) = 80 5 + = 60 r 80 60 = 5 r 05 a S T A C 80 + a 60 a ( ) a = sin = 60 a Substitute sin( 5 + ) Prcess sin One slutin Secnd slutin hsn.uk.net Page HSN50
Higher Mathematics Unit Specimen NAB Assessment Outcme Circles 9. (a) ( ) + ( + ) = 9 Centre 0. (b) The centre is ( 5, ) The radius is ( 5) + ( ) = 7 + + 6 + + 8 = 0 + ( ) + 6 + ( ) + 8 = 0 b ac = = 6 6 = 0 5 + 0 + 5 = 0 + + = 0 Since the discriminant is zer, the line is a tangent t the circle. 5 0 5. mpc = = 0 + S m = since the radius and tangent are tgt 5 perpendicular. + 5 5 = 0 5 = ( 0) 5 Square f radius State centre Knw hw t calculate radius Prcess radius Strateg fr finding intersectin Epress in standard frm Knw t calculate discriminant Calculate discriminant Cnclusin Knw hw t find gradient f radius Prcess gradient f radius Gradient f tangent Equatin f tangent 5 hsn.uk.net Page 5 HSN50
Practice Assessment () Unit - Mathematics (H) Outcme Marks. Shw that ( + ) is a factr f g ( ) = + + 6, and epress g () in full factrised frm. (). Use the discriminant t determine the nature f the rts f the equatin + 5 + = 0. () Outcme. Find d. (). Calculate the shaded area shwn in the diagram. = ( ) (5) 5. The diagram shws the line with equatin = + 5 and the curve with equatin = 5 +. Write dwn the integral which represents the shaded area. D nt carr ut the integratin. = + 5 = 5 + - 5 ()
Outcme 6. Slve algebraicall the equatin sin = fr 0 " <. () 7. The diagram belw shws tw right-angled triangles PQR and SRT. S Q SQ = QR = PR = RT = P R T (a) Write dwn the values f cs and sin. () (b) B epanding sin ( + ) shw that the eact value f sin ( + ) is 8 80. () 8. (a) Epress cs cs0 sin sin 0 in the frm cs ( A + B). () (b) Hence slve the equatin cs cs0 sin sin 0 = fr 0 < < 60. () Outcme 9. (a) A circle f radius 6 units has as its centre the pint C(,-). Write dwn the equatin f this circle. () (b) A circle has equatin + + = 0. Write dwn the crdinates f its centre and calculate its radius. () 0. Shw that the line with equatin = 5 is a tangent t the circle with equatin + + 6 0 = 0. (5). A circle has as its centre the pint C(-,), as shwn in the diagram. The pint P(-9,) lies n the circumference f the circle. Find the equatin f the tangent at P. P(-9,) C(-,) ()
Practice Assessment () Unit - Mathematics (H) Outcme Marks. Shw that ( ) is a factr f f ( ) = +, and epress f () in full factrised frm. (). Use the discriminant t determine the nature f the rts f the equatin 6 + = 0. () Outcme & #. Find $ + d % ". (). Calculate the shaded area shwn in the diagram. = + (5) 5. The diagram shws the curves with equatins = + and = +. Write dwn the integral which represents the shaded area. D nt carr ut the integratin. = + = + ()
Outcme 6. Slve algebraicall the equatin tan = fr 0 " <. () 7. The diagram shws tw right-angled triangles EFG and EHG. H FEG = and HEG =. 5 E G 7 F (a) Write dwn the values f sin and cs. () (b) B epanding cs( + ) shw that the eact value f cs( + ) is 5. () 8. (a) Epress sin cs 0 cs sin 0 in the frm sin ( A B). () (b) Hence slve the equatin sin cs 0 cs sin 0 = 9 fr 0 < < 80. () Outcme 9. (a) A circle has radius 0 units and centre (5,-). Write dwn the equatin f the circle. () (b) A circle has equatin + + 0 + = 0. Write dwn its radius and the crdinates f its centre. () 0. Shw that the line with equatin = 0 is a tangent t the circle with equatin + 6 + 6 + 0 = 0. (5). A circle has AB as a diameter, as shwn in the diagram. A and B have crdinates (-,5) and (0,8) respectivel. Find the equatin f the tangent at B. B A ()
Practice Assessment () Unit - Mathematics (H) Outcme Marks. Shw that ( + ) is a factr f f ( ) = + 5 +, and epress f () in full factrised frm. (). Use the discriminant t determine the nature f the rts f the equatin 5 + = 0. () Outcme. Find d. 5 (). Calculate the shaded area shwn in the diagram. = 6 + 8 (5) 5. The diagram shws the curve with equatin = 8 + 8 and the line + = 8. Write dwn the integral which represents the shaded area. D nt carr ut the integratin. = 8 + 8 + = 8 ()
Outcme 6. Slve algebraicall the equatin cs = fr 0 " <. () 7. The diagram shws tw right-angled triangles ABC and ABD. D DAB = and CAB =. C A 5 B (a) Write dwn the values f cs and sin. () (b) B epanding cs( ) shw that the eact value f cs( ) is 8. () 8. (a) Epress sin cs5 + cs sin5 in the frm sin (A + B). () (b) Hence slve the equatin sin cs5 + cs sin5 = 7 fr 0 < < 80. () Outcme 9. (a) A circle has a radius f unit and centre (-,6). Write dwn the equatin f this circle. () (b) A circle has equatin + 6 + 5 = 0. Write dwn its radius and the crdinates f its centre. () 0. Shw that the line with equatin = 7 is a tangent t the circle with equatin + + 8 + 5 = 0. (5). A circle has as its centre the pint C(5,). The pint P(9,) lies n its circumference. Find the equatin f the tangent at P. ()
Unit - Practice Assessments Answers Practice Assessment Outcme :. prf, g ( ) = ( + )( + )( ). b " ac = real, distinct and irratinal ' & # Outcme :. + C $ ' + C ' % ". 6 units 5 d 0 0 5. + " " ( + 5) d # " Outcme : 6. { } 8. (a), 6 cs( + 0) 7. (a) (b) { 5 5, 5 5 } cs =, sin = (b) prf 0 8 Outcme : 9. (a) ( ) + ( + ) = 6 (b) C(,-), r = 0. rt, =, a tangent. " = ( + 9) = + Practice Assessment Outcme :. prf, f ( ) = ( )( + )( + ). b " ac = " nt real ' & # Outcme :. + + C $ ' + + C ' % ". 9 units 5. + " " ( " + ) d # " 7 Outcme : 6. { } 8. (a) d 0 0, sin( 0) 7. (a) (b) { 6, 7 6 } 7 0 sin =, cs = (b) prf 5 5 Outcme : 9. (a) ( 5) + ( + ) = 00 (b) C(,-5), r = 5 0. rt, = 5, a tangent. 8 = ( 0) " = + 8 Practice Assessment Outcme :. prf, g ( ) = ( + )( )( ). b " ac = real, distinct and ratinal ' & # Outcme :. + C $ ' + C ' % " 7. 6 units 5. " " ( " 8 + 8) d # 7 " 7 Outcme : 6. { } 0 8. (a) 7 8 d 0 7. (a) 8, 8 sin( + 5) (b) { 5, 05 5 } 5 cs =, sin = (b) prf Outcme : 9. (a) ( + ) + ( 6) = (b) C(,0), r = 0. rt, =, a tangent. = ( 9) " = +