Standrewspaisley
Higher MathematicsGCC Straight Line[SQA][SQA]1.2. Find the size of the angle a that the linejoining the points A (0, 1) and B (3 3, 2)makes with the positive direction of thex -axis.yO B (3 3, 2)a 3xA (0, 1)[SQA]3.[SQA]4. Find the equation of the line through the point (3, 5) which is parallel to the linewith equation 3x 2y 5 0.hsn.uk.netPage 1Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes2
Higher Mathematics[SQA]5.[SQA]6.[SQA]7. Triangle PQR has vertex P on the x -axis,as shown in the diagram.yQ(4, 6)Q and R are the points (4, 6) and (8, 2)6 x – 7 y 18 0respectively.The equation of PQ is 6x 7y 18 0.(a) State the coordinates of P.OP(b) Find the equation of the altitude ofthe triangle from P.TxR(8, –2)(c) The altitude from P meets the lineQR at T. Find the coordinates of T.hsn.uk.netPage 2134Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes
Higher Mathematics[SQA]8.[SQA]9.hsn.uk.netPage 3Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes
Higher Mathematics10. Triangle ABC has vertices A (4, 0) ,B (4, 16) and C (18, 20) , as shown inthe diagram opposite.yCPMedians AP and CR intersect at thepoint T (6, 12) .BTQRO[SQA]xA(a) Find the equation of median BQ.3(b) Verify that T lies on BQ.1(c) Find the ratio in which T divides BQ.211. Triangle ABC has vertices A (2, 2) ,B (12, 2) and C (8, 6) .yC (8, 6)(a) Write down the equation of l1 ,the perpendicular bisector ofAB.(b) Find the equation of l2 , theperpendicular bisector of AC.1A (2, 2)OB (12, 2)x4(c) Find the point of intersection oflines l1 and l2 .1(d) Hence find the equation of thecircle passing through A, B andC.2[END OF QUESTIONS]hsn.uk.netPage 4Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes
Higher MathematicsGCC Straight CNCNContentG1G8, G7, G1Page 1Answerproof(i) H (4, 72 ), (ii) proofU1 OC11995 P2 Q1Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes
Higher Mathematics[SQA]2. Find the size of the angle a that the linejoining the points A (0, 1) and B (3 3, 2)makes with the positive direction of thex -axis.yO B (3 3, 2)a 3xA (0, 1)PartMarks3LevelCCalc.NCContentG2 1 ss: know how to find gradient orequ. 2 pd: process 3 ic: interpret exact value[SQA] 1U1 OC12000 P1 Q32 ( 1) 3 3 0 2 tan a gradient stated or implied by 33 a nswerU1 OC11992 P1 Q134. Find the equation of the line through the point (3, 5) which is parallel to the linewith equation 3x 2y 5 0.PartMarks2hsn.uk.netLevelCCalc.CNContentG3, G2Page 2AnswerU1 OC11991 P1 Q1Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes2
Higher entG3, G3AnswerMarks62LevelCCCalc.NCNCContentG3, G5, G8G8AnswerU1 OC11998 P1 Q16.Part(a)(b)hsn.uk.netPage 3U1 OC11992 P1 Q2Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes
Higher Mathematics[SQA]7. Triangle PQR has vertex P on the x -axis,as shown in the diagram.yQ(4, 6)Q and R are the points (4, 6) and (8, 2)6 x – 7 y 18 0respectively.The equation of PQ is 6x 7y 18 0.(a) State the coordinates of P.OP(b) Find the equation of the altitude ofthe triangle from P.T1xR(8, –2)(c) The altitude from P meets the lineQR at T. Find the coordinates of T.Part(a)(b)(c)Marks134 1 ic:LevelCCCCalc.CNCNCNContentG4G7G84AnswerP( 3, 0)y 12 ( x 3)T (5, 4) 2 pd: find gradient (of QR) 3 ss: know and use m1 m2 1 4 ic: state equ. of altitude 2 mQR 2 3 malt. 12 4 y 0 12 ( x 3) 5 6 7 8 5 6 7 8ic:ss:pd:pd:state equ. of line (QR)prepare to solve sim. equ.solve for xsolve for yhsn.uk.netU1 OC12009 P1 Q21 1 P ( 3, 0)interpret x-interceptPage 43y 2 2( x 8)x 2y 3 and 2x y 14x 5y 4Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes
Higher evelCCCCalc.CNCNCNContentG5, G3G1G1Page 5AnswerU1 OC11994 P2 Q2Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes
Higher NContentG8Page 6AnswerU1 OC11995 P1 Q6Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes
Higher Mathematics10. Triangle ABC has vertices A (4, 0) ,B (4, 16) and C (18, 20) , as shown inthe diagram opposite.Medians AP and CR intersect at thepoint T (6, 12) .yCPBTQROxA(a) Find the equation of median BQ.3(b) Verify that T lies on BQ.1(c) Find the ratio in which T divides tG7A6G24Answery 16 25 ( x ( 4))proof2:1U3 OC12010 P1 Q21 1 ss: know and find midpoint of AC 2 pd: calculate gradient of BQ 3 ic: state equation 1 (11, 10)6 2 15or equiv3 y 16 25 ( x ( 4))or y 10 25 ( x 11) 4 ic: 4 2(6) 5(12) 12 60 72substitute in for T and complete 5 ss: valid method for finding theratio6 ic: complete to simplified ratiohsn.uk.netPage 7 5 e.g. vectorapproach 10 5BT , TQ 4 2 6 2 : 1Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes
Higher Mathematics[SQA]11. Triangle ABC has vertices A (2, 2) ,B (12, 2) and C (8, 6) .yC (8, 6)(a) Write down the equation of l1 ,the perpendicular bisector ofAB.1A (2, 2)(b) Find the equation of l2 , theperpendicular bisector of AC.B (12, 2)xO4(c) Find the point of intersection oflines l1 and l2 .1(d) Hence find the equation of thecircle passing through A, B NCNCNContentG3, G7G7G8G8, G9, G10Answerx 73x 2y 23(7, 1)( x 7)2 (y 1)2 26 1 ic:state equation of a vertical line 1 x 7 2 3 4 5process coord. of a midpointfind gradient of ACstate gradient of perpendicularstate equation of straight line 2 3 4 5pd:ss:ic:ic:U2 OC42001 P2 Q7midpoint (5, 4)mAC 23m 32y 4 32 ( x 5) 6 pd: find pt of intersection 6 x 7, y 1 7 ss: use standard form of circle equ. 8 ic: find radius and complete 7 ( x 7)2 (y 1)2 8 ( x 7)2 (y 1)2 26or 7 x2 y2 14x 2y c 0 8 c 24[END OF QUESTIONS]hsn.uk.netPage 8Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes
Higher MathematicsGCC Quadratics and PolynomialsPaper 1 Section AEach correct answer in this section is worth two marks.1. Which of the following diagramsshows a parabola with equationy ax2 bx c, where2. The diagram shows the graph of acubic.y a 0(1, 2) b2 4ac 0?y –1A.O2xxOyWhat is the equation of this cubic?B.xOyxOA.y x ( x 1)( x 2)B.y x ( x 1)( x 2)C.y x ( x 1)( x 2)D.y x ( x 1)( x 2)C.yOxD.3. If f ( x ) ( x 3)( x 5) , for whatvalues of x is the graph of y f ( x )above the x -axis?hsn.uk.netPage 1A. 5 x 3B. 3 x 5C.x 5, x 3D.x 3, x 5Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes
Higher Mathematics4. What is the solution of x2 4x 0,where x is a real number?A. 4 x 0B.x 4, x 0C.0 x 4D.x 0, x 47. A function f is given byf ( x ) 2x2 x 9.Which of the following describes thenature of the roots of f ( x ) 0?A. No real rootsB. Equal rootsC. Real distinct rootsD. Rational distinct roots5. Solve 6 x x2 0.A. 3 x 2B.x 3, x 2C. 2 x 3D.x 2, x 36. The discriminant of a quadraticequation is 23.Here are two statements about thisquadratic equation:I. the roots are real;8. The roots of the equationkx2 3x 2 0 are equal.II. the roots are rational.What is the value of k?Which of the following is true?A. 98B. 89C. only statement II is correctC.89D. both statements are correctD.98A. neither statement is correctB. only statement I is correcthsn.uk.netPage 2Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes
Higher Mathematics11. A function f is defined on the set ofreal numbers byf ( x ) x3 x2 x 3.9. The diagram shows the graph withequation y k( x 1)2 ( x t) .yWhat is the remainder when f ( x ) isdivided by ( x 1) ?A. 010B. 2C. 3O15xD. 4What are the values of k and t?ktA. 2 5B. 25C.2 5D.2510. A parabola intersects the axes atx 2, x 1 and y 6, asshown in the diagram.y6–2–1OxWhat is the equation of theparabola?A.y 6( x 1)( x 2)B.y 6( x 1)( x 2)C.y 3( x 1)( x 2)D.y 3( x 1)( x 2)hsn.uk.netPage 3Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes
Higher Mathematics[END OF PAPER 1 SECTION A]Paper 1 Section B[SQA]12. (a) Express f ( x ) x2 4x 5 in the form f ( x ) ( x a)2 b.2(b) On the same diagram sketch:(i) the graph of y f ( x ) ;(ii) the graph of y 10 f ( x ) .4(c) Find the range of values of x for which 10 f ( x ) is positive.[SQA][SQA]13. Find the values of x for which the function f ( x ) 2x3 3x2 36x is increasing.14. Given that k is a real number, show that the roots of the equation kx2 3x 3 kare always real numbers.[SQA]15. For what value of k does the equation x2 5x (k 6) 0 have equal roots?[SQA]16.[SQA]17. Find the values of k for which the equation 2x2 4x k 0 has real roots.18. (a) (i) Show that ( x 1) is a factor of f ( x ) 2x3 x2 8x 5.(ii) Hence factorise f ( x ) fully.(b) Solve 2x3 x2 8x 5 0.1453251(c) The line with equation y 2x 3 is a tangent to the curve with equationy 2x3 x2 6x 2 at the point G.Find the coordinates of G.5(d) This tangent meets the curve again at the point H.Write down the coordinates of H.hsn.uk.netPage 41Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes
Higher Mathematics[SQA]19. Factorise fully 2x3 5x2 4x 3.[SQA]20.[SQA]21. One root of the equation 2x3 3x2 px 30 0 is 3.4Find the value of p and the other roots.4[SQA]22.[SQA]23. Express x4 x in its fully factorised form.4[SQA]24.[SQA]25. Express x3 4x2 7x 10 in its fully factorised form.[SQA]26.hsn.uk.netPage 5Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes4
Higher Mathematics[SQA]27.[SQA]28.hsn.uk.netPage 6Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes
Higher Mathematics[SQA]29.[SQA]30.hsn.uk.netPage 7Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes
Higher Mathematics[SQA]31.[SQA]32.[END OF PAPER 1 SECTION B]hsn.uk.netPage 8Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes
Higher MathematicsPaper 2[SQA]1.(i) Write down the condition for the equation ax2 bx c 0 to have no realroots.1(ii) Hence or otherwise show that the equation x ( x 1) 3x 2 has no realroots.2[SQA]2. Show that the roots of the equation (k 2) x2 (3k 2) x 2k 0 are real.[SQA]3.[SQA]4. The roots of the equation ( x 1)( x k) 4 are equal.Find the values of k.hsn.uk.net45Page 9Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes
Higher Mathematics[SQA][SQA][SQA]5.6. Show that the equation (1 2k) x2 5kx 2k 0 has real roots for all integervalues of k.7. The diagram shows a sketch of aparabola passing through ( 1, 0) ,(0, p) and ( p, 0) .(a) Showthattheequationoftheparabolaisy p ( p 1) x x 2 .y(0, p)( 1, 0)O( p, 0)x(b) For what value of p will the liney x p be a tangent to thiscurve?[SQA](b) Hence solve the equation 2x3 x2 kx 2 0 when k takes this value.Page 10338. (a) Given that x 2 is a factor of 2x3 x2 kx 2, find the value of k.hsn.uk.net5Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes32
Higher Mathematics[SQA]y9. The diagram shows part of the graph of thecurve with equation y 2x3 7x2 4x 4.y f (x)(a) Find the x -coordinate of the maximumturning point.5(b) Factorise 2x3 7x2 4x 4.3A(c) State the coordinates of the point A andhence find the values of x for which2x3 7x2 4x 4 0.O(2, 0)x2[SQA]10. Find p if ( x 3) is a factor of x3 x2 px 15.[SQA]11. When f ( x ) 2x4 x3 px2 qx 12 is divided by ( x 2) , the remainder is 114.3One factor of f ( x ) is ( x 1) .Find the values of p and q.5[SQA]12.[SQA]13. The diagram shows a sketch of thegraph of y x3 3x2 2x .yy x3 3x2 2x(a) Find the equation of thetangent to this curve at thepoint where x 1.5Ox(b) The tangent at the point (2, 0)has equation y 2x 4. Findthe coordinates of the pointwhere this tangent meets thecurve again.5[END OF PAPER 2]hsn.uk.netPage 11Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes
Higher MathematicsGCC Quadratics and PolynomialsPaper 1 Section AEach correct answer in this section is worth two marks.1. Which of the following diagrams shows a parabola with equationy ax2 bx c, where a 0 b2 4ac eCFacility0Disc.0CalculatorCNPage 1ContentA7, A15, A17Source2010 P1 Q13Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes
Higher Mathematics2. The diagram shows the graph of a cubic.y(1, 2) –1O2xWhat is the equation of this cubic?A.y x ( x 1)( x 2)B.y x ( x 1)( x 2)C.y x ( x 1)( x 2)D.y x ( x 1)( x culatorCNPage 2ContentA7, A19Source2011 P1 Q17Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes
Higher Mathematics3. If f ( x ) ( x 3)( x 5) , for what values of x is the graph of y f ( x ) above thex -axis?A. 5 x 3B. 3 x 5C.x 5, x 3D.x 3, x ContentA16Source2011 P1 Q184. What is the solution of x2 4x 0, where x is a real number?A. 4 x 0B.x 4, x 0C.0 x 4D.x 0, x ulatorCNPage 3ContentA16Source2010 P1 Q18Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes
Higher Mathematics5. Solve 6 x x2 0.A. 3 x 2B.x 3, x 2C. 2 x 3D.x 2, x ntentA16Source2012 P1 Q196. The discriminant of a quadratic equation is 23.Here are two statements about this quadratic equation:I. the roots are real;II. the roots are rational.Which of the following is true?A. neither statement is correctB. only statement I is correctC. only statement II is correctD. both statements are sc.0CalculatorNCPage 4ContentA17Source2011 P1 Q9Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes
Higher Mathematics7. A function f is given by f ( x ) 2x2 x 9.Which of the following describes the nature of the roots of f ( x ) 0?A. No real rootsB. Equal rootsC. Real distinct rootsD. Rational distinct NCContentA17Source2009 P1 Q128. The roots of the equation kx2 3x 2 0 are equal.What is the value of k?A. 98B. isc.0CalculatorCNPage 5ContentA18Source2010 P1 Q6Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes
Higher Mathematics9. The diagram shows the graph with equation y k( x 1)2 ( x t) .y10O15xWhat are the values of k and t?ktA. 2 5B. 25C.2 CalculatorCNPage 6ContentA19Source2010 P1 Q16Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes
Higher Mathematics10. A parabola intersects the axes at x 2, x 1 and y 6, as shown in thediagram.y6–2OxCalculatorNCContentA19–1What is the equation of the parabola?A.y 6( x 1)( x 2)B.y 6( x 1)( x 2)C.y 3( x 1)( x 2)D.y 3( x 1)( x e 7Source2012 P1 Q13Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes
Higher Mathematics11. A function f is defined on the set of real numbers by f ( x ) x3 x2 x 3.What is the remainder when f ( x ) is divided by ( x 1) ?A. 0B. 2C. 3D. ntentA21Source2011 P1 Q7[END OF PAPER 1 SECTION A]Paper 1 Section B[SQA]12. (a) Express f ( x ) x2 4x 5 in the form f ( x ) ( x a)2 b.2(b) On the same diagram sketch:(i) the graph of y f ( x ) ;(ii) the graph of y 10 f ( x ) .4(c) Find the range of values of x for which 10 f ( x ) is positive.Part(a)(b)(c)Marks241LevelCCCCalc.NCNCNC 1 pd: process, e.g.square 2 pd: process, e.g.square 3 4 5 6ic:ic:ss:ss: 7 ic:ContentA5A3A16, A6completing thecompleting theinterpret minimuminterpret y-interceptreflect in x-axistranslate parallel to y-axisU1 OC22002 P1 Q7 1 a 2 2 b 1 3 any two from:parabola; min. t.p. (2, 1); (0, 5) 4 the remaining one from above list 5 reflecting in x-axis 6 translating 10 units, parallel toy-axis 7 ( 1, 5) i.e. 1 x 5interpret graphhsn.uk.netAnswera 2, b 1sketch 1 x 51Page 8Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes
Higher Mathematics[SQA]13. Find the values of x for which the function f ( x ) 2x3 3x2 36x is increasing.Part[SQA]LevelCA/BCalc.NCNCContentC7, A16C7, A16AnswerU2 OC11996 P1 Q1614. Given that k is a real number, show that the roots of the equation kx2 3x 3 kare always real ontentA17A17AnswerMarks3LevelCCalc.CNContentA18 1 ss: know to set disc. to zero 2 ic:substitute a, b and c intodiscriminant 3 pd: process equation in khsn.uk.netPage 95U2 OC11991 P1 Q1815. For what value of k does the equation x2 5x (k 6) 0 have equal roots?Part4Answerk 14U2 OC12001 P1 Q2 1 b2 4ac 0 stated or implied by 2 2 ( 5)2 4 (k 6) 3 k 14Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes3
Higher CNCContentA18A18AnswerU2 OC11992 P1 Q1717. Find the values of k for which the equation 2x2 4x k 0 has real age 10AnswerU2 OC11993 P1 Q3Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes2
Higher Mathematics18. (a) (i) Show that ( x 1) is a factor of f ( x ) 2x3 x2 8x 5.(ii) Hence factorise f ( x ) fully.5(b) Solve 2x3 x2 8x 5 0.1(c) The line with equation y 2x 3 is a tangent to the curve with equationy 2x3 x2 6x 2 at the point G.Find the coordinates of G.5(d) This tangent meets the curve again at the point H.Write down the coordinates of H.Part(a)(b)(c)(d) 1 2 3 4 5Marks5151ss:ic:ic:pd:pd: 6 ic: 7 8 9 10 22A23A23Answer( x 1)( x 1)(2x 5)x 1, 52(1, 1)( 52 , 8)know to use x 1complete evaluationstate conclusionfind quadratic factorfactorise completely 1 2 3 4 5state solutions 6 x 1 and x 52set ycurve ylineexpress in standard formcompare with (a) or factoriseidentify xGevaluate yG 7 8 9 10 11U2 OC12010 P1 Q22evaluating at x 1.2 1 8 5 0( x 1) is a factor( x 1)(2x2 3x 5)( x 1)( x 1)(2x 5)2x3 x2 6x 2 2x 32x3 x2 8x 5 0( x 1)( x 1)(2x 5) 0x 1y 1 12 ( 52 , 8) 12 pd: state solution[SQA]119. Factorise fully 2x3 5x2 4x 3.PartMarks4hsn.uk.netLevelCCalc.NC4ContentA21Page 11AnswerU2 OC11989 P1 Q2Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes
Higher lc.NCNCContentA21A21AnswerU2 OC11999 P1 Q121. One root of the equation 2x3 3x2 px 30 0 is 3.Find the value of p and the other 2LevelCCCalc.NCNCContentA21A214AnswerU2 OC11993 P1 Q722.Part(a)(b)hsn.uk.netPage 12AnswerU2 OC11995 P1 Q2Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes
Higher Mathematics[SQA]23. Express x4 x in its fully factorised CCalc.NCNCContentA21A21AnswerU2 OC11996 P1 Q724.Part(a)(b)[SQA]Marks44AnswerU2 OC11997 P1 Q525. Express x3 4x2 7x 10 in its fully factorised ge 13Answer4U2 OC11998 P1 Q2Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes
Higher LevelCCCCalc.NCNCNCContentA21A6C8Page 14AnswerU2 OC11992 P2 Q1Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes
Higher lCCCalc.NCNCContentA21A21Page 15AnswerU2 OC11994 P2 Q1Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes
Higher lCCCalc.NCNCContentC4, G3A23Page 16AnswerU2 OC11995 P2 Q2Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes
Higher .uk.netLevelCCCA/BCalc.NCNCNCNCContentA6C4, G3A23A23Page 17AnswerU2 OC11998 P2 Q5Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes
Higher LevelCCCCalc.NCNCNCContentA6G3A23Page 18AnswerU2 OC11999 P2 Q4Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes
Higher lCA/BCalc.NCNCContentA23, A21A24Page 19AnswerU2 OC11993 P2 Q7Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes
Higher CCA/BCalc.NCNCNCNCContentCGDCGDA26A26AnswerU2 OC11989 P2 Q3[END OF PAPER 1 SECTION B]hsn.uk.netPage 20Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes
Higher MathematicsPaper 2[SQA]1.(i) Write down the condition for the equation ax2 bx c 0 to have no realroots.1(ii) Hence or otherwise show that the equation x ( x 1) 3x 2 has no nswerU2 OC11999 P1 Q82. Show that the roots of the equation (k 2) x2 (3k 2) x 2k 0 are tA17A17Page 21AnswerU2 OC11990 P1 Q18Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes4
Higher netLevelCCA/BA/BCalc.CNCNCNCNContentA6C4, CGDC4, CGDA17Page 22AnswerU2 OC11994 P2 Q9Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes
Higher Mathematics[SQA]4. The roots of the equation ( x 1)( x k) 4 are equal.Find the values of 18A18Page 23AnswerU2 OC11995 P1 Q20k 5, 3Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes
Higher CCCalc.CNCNContentCGDA18, CGDPage 24AnswerU2 OC11993 P2 Q4Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes
Higher Mathematics[SQA]6. Show that the equation (1 2k) x2 5kx 2k 0 has real roots for all integervalues of k.Part 1 2 3 4 18, A16, CGDknow to use discriminantpick out discriminantsimplify to quadraticchoose to draw table or graphcomplete proof using disc. 0AnswerproofU2 OC12002 P2 Q9 1 2 3 4discriminant . . .disc ( 5k)2 4(1 2k)( 2k)9k2 8ke.g. draw a table, graph, completethe square5 complete proof and conclusionrelating to disc. 0y7. The diagram shows a sketch of aparabola passing through ( 1, 0) ,(0, p) and ( p, 0) .(a) Showthattheequationoftheparabolaisy p ( p 1) x x 2 .(0, p)( 1, 0)O( p, 0)x(b) For what value of p will the liney x p be a tangent to tentA19A24 1 ss: use a standard form of parabola 2 ss: use 3rd point to determine k 3 pd: complete proof 4 ss: equate and simplify to zero 5 ss: use discriminant for tangency 6 pd: processhsn.uk.netPage 25533Answerproof2U2 OC12001 P2 Q11 1 y k( x 1)( x p) 2 k 1 with justification (i.e.substitute (0, p)) 3 y 1( x 1)( x p) and complete 4 x2 2x px 0 5 b2 4ac (2 p)2 0or (2 p)2 4 0 06 p 2Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes
Higher Mathematics[SQA]8. (a) Given that x 2 is a factor of 2x3 x2 kx 2, find the value of k.3(b) Hence solve the equation 2x3 x2 kx 2 0 when k takes this value.Part(a)(b)Marks32LevelCCCalc.CNCN 1 ss:use synthf (evaluation) 2 pd: process 3 pd: processContentA21A22divisionAnswerk 5x 2, 12 , 1U2 OC12001 P2 Q1 1 f ( 2) 2( 2)3 · · · 2 2( 2)3 ( 2)2 2k 2 3 k 5or 4 2x2 3x 1 or 2x2 3x 2 orx2 x 25 (2x 1)( x 1) or (2x 1)( x 2) or( x 2)( x 1)and x 2, 12 , 1 4 ss: find a quadratic factor 5 pd: process[SQA]y9. The diagram shows part of the graph of thecurve with equation y 2x3 7x2 4x 4.y f (x)(a) Find the x -coordinate of the maximumturning point.5(b) Factorise 2x3 7x2 4x 4.3A(c) State the coordinates of the point A andhence find the values of x for which2x3 7x2 4x 4 0.Part(a)(b)(c) 1 2 3 4 C8A21A6know to differentiatedifferentiateknow to set derivative to zerostart solving process of equationcomplete solving process 6 ss: strategy for cubic, e.g. synth.division 7 ic: extract quadratic factor 8 pd: complete the cubic factorisation 9 ic: 10 ic:interpret the factorsinterpret the diagramhsn.uk.net2Ox(2, 0)2Answerx 13( x 2)(2x 1)( x 2)A( 12 , 0), x 12 1 2 3 4 5 6f ′ ( x) . . .6x2 14x 46x2 14x 4 0(3x 1)( x 2)x 13···2 7······4········· 3x 2 8 ( x 2)(2x 1)( x 2) 7U2 OC12002 P2 Q32x24···0 9 A( 12 , 0) 10 x 12Page 26Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes
Higher Mathematics[SQA]10. Find p if ( x 3) is a factor of x3 x2 px 15.Part[SQA]Marks3LevelCCalc.CNContentA21Answer3U2 OC11990 P1 Q111. When f ( x ) 2x4 x3 px2 qx 12 is divided by ( x 2) , the remainder is 114.One factor of f ( x ) is ( x 1) .Find the values of p and q.PartMarks5hsn.uk.netLevelCCalc.CN5ContentA21Page 27AnswerU2 OC11991 P1 Q6Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes
Higher CNContentA21, C8Page 28AnswerU2 OC11993 P2 Q1Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes
Higher Mathematics[SQA]y13. The diagram shows a sketch of thegraph of y x3 3x2 2x .y x3 3x2 2x(a) Find the equation of thetangent to this curve at thepoint where x 1.5Ox(b) The tangent at the point (2, 0)has equation y 2x 4. Findthe coordinates of the pointwhere this tangent meets thecurve 23, A22, A21 1 2 3 4 5ss:pd:ss:ss:ic:know to differentiatedifferentiate correctlyknow that gradient f ′ (1)know that y-coord f (1)state equ. of line 6 7 8 9 10ss:pd:ss:pd:ic:equate equationsarrange in standard formknow how to solve cubicprocessinterpretAnswerx y 1( 1, 6) 1 2 3 4 5U2 OC12000 P2 Q1y′ . . .3x2 6x 2y ′ ( 1) 1y ( 1) 0y 0 1( x 1) 6 2x 4 x3 3x2 2x 7 x3 3x2 4 01 304······ ··· ··· 8··· ··· ··· ···9 identify x 1 from working 10 ( 1, 6)[END OF PAPER 2]hsn.uk.netPage 29Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes
Higher MathematicsGCC Vectors[SQA]1. ABCD is a quadrilateral with vertices A (4, 1, 3) , B (8, 3, 1) , C (0, 4, 4) andD ( 4, 0, 8) .[SQA]2.[SQA]3.[SQA]4.[SQA]5.(a) Find the coordinates of M, the midpoint of AB.1(b) Find the coordinates of the point T, which divides CM in the ratio 2 : 1.3(c) Show that B, T and D are collinear and find the ratio in which T divides BD.4hsn.uk.netPage 1Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes
Higher Mathematics[SQA]6.[SQA]7.hsn.uk.netPage 2Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes
Higher Mathematics[SQA]8.[SQA]9.hsn.uk.netPage 3Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes
Higher Mathematics[SQA]10.[SQA]11. A box in the shape of a cuboidis designed with circles of differentsizes on each face.The diagram shows three of thecircles, where the origin representsone of the corners of the cuboid. Thecentres of the circles are A (6, 0, 7) ,B (0, 5, 6) and C (4, 5, 0) .zBFind the size of angle ABC.7AOyCx[SQA]12. The vectors p , q and r are defined as follows:p 3i 3 j 2k, q 4i j k , r 4i 2 j 3k.(a) Find 2 p q r in terms of i , j and k .1(b) Find the value of 2 p q r .2hsn.uk.netPage 4Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes
Higher Mathematics[SQA]13.[SQA]14. The vector ai b j k is perpendicular to both the vectors i j k and 2i j k .Find the values of a and b.3 [SQA]15. Calculate the length of the vector 2i 3 j [SQA]16. Show that the vectors a 2i 3 j k and b 3i j 3k are perpendicular.[SQA]17.[SQA]18.hsn.uk.netPage 53k .2Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes3
Higher Mathematics[SQA]19.[SQA] 3120. If u 3 and v 5 , write down the components of u v and u v .3 1 Hence show that u v and u v are perpendicular.[SQA]321. A cuboid measuring 11 cm by 5 cm by 7 cm is placed centrally on top of anothercuboid measuring 17 cm by 9 cm by 8 cm.Coordinates axes are taken as shown.zB7511CAx9817yO(a) The point A has coordinates (0, 9, 8) and C has coordinates (17, 0, 8) .Write down the coordinates of B.1(b) Calculate the size of angle ABC.hsn.uk.netPage 66Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes
Higher Mathematics[SQA]z22. The diagram shows a square-basedpyramid of height 8 units.D (3, 3, 8)ySquare OABC has a side length of 6 units.The coordinates of A and D are (6, 0, 0)and (3, 3, 8) .CBC lies on the y-axis.(a) Write down the coordinates of B.1OA (6, 0, 0) (b) Determine the components of DA and DB.x2(c) Calculate the size of angle ADB.423. D,OABC is a square based pyramid as shown in the diagram below.zD(2, 2, 6)yCBOxAMO is the origin, D is the point (2, 2, 6) and OA 4 units.M is the mid-point of OA.(a) State the coordinates of B.1 (b) Express DB and DM in component form.3(c) Find the size of angle BDM.5hsn.uk.netPage 7Questions marked ‘[SQA]’ c SQAAll others c Higher Still Notes
Higher Mathematics24. The diagram shows a cuboid OPQR,STUV relative to the coordinate axes.zP is the point (4, 0, 0) , Q is (4, 2, 0)and U is (4, 2, 3) .VM is the midpoint of OR.U (4, 2, 3)yN is the point on UQ such thatUN 13 UQ.SNTRQ (4, 2, 0)MOP (4, 0, 0)[SQA]x(a) State the coordinates of M and N.2 (b) Express the vectors VM and VN in component form.2(c) Calculate the size of angle MV
Higher Mathematics GCCStraightLine [SQA] 1. [SQA] 2. Find the size of the angle a that the line joining the points A(0, 1) and B(3 3,2) makes with the positive direction
Army training centers, and other training activities under the control of Headquarters (HQ), TRADOC and to all personnel, military and civilian, under the control of HQ TRADOC, to include Army elements stationed within Interservice Training Review Organizations (ITRO) for AIT, who interact with Trainees/Soldiers undergoing IET conducted on an installation, the commander of which is subordinate .
20172018 Brown Samuel Special Education Teacher Inclusion KILMER 89,815.00 20172018 Brown Deirdre MATHEMATICS TEACHER TCHS (Chambers) STEM Academy 89,015.00 20172018 Brown Elaine Special Education Teacher Resource WILSON 58,315.00 20172018 Brown Elizabeth PRE-KINDERGARTEN TEACHER WILSON 96,315.00
CREATING THE BLUEPRINT FOR YOUR “HOUSE” Cynthia Grant, PhD University of Colorado-Denver Azadeh Osanloo, PhD New Mexico State University The theoretical framework is one of the most important aspects in the research process, yet is often misunderstood by doctoral candidates as they prepare their dissertation research study. The importance of theory-driven thinking and acting is emphasized .
Jenis ikan hias yang banyak dibudidayakan antara lain Oscar, Tetra, Blackghost, Koki dan Cupang. Sedangkan untuk jenis ikan konsumsi terdiri dari Bawal Air Tawar, Gurami, Patin dan Tawes. Saat masih benih, ikan tersebut sangat memerlukan pakan alami/kutu air. Keberadaan pakan alami sangat diperlukan dalam budidaya ikan dan pembenihan,
Logistics Employment Work Labour markets Post-modernism Time Economic sociology Citation: Penn, R. (2016). Stratification and work in contemporary logistics. Turkish Journal of Sociology, 36, 125–142. 1 Correspondence to: Roger Penn (PhD), School of Sociology, Social Policy and Social Work, Queen’s University .
Bahagian Akaun Amanah / Cawangan 1 3. Penjelasan 1. Rekod Terimaan. 2. Keluarkan Resit Makluman kepada Fakulti / Bahagian / Cawangan 4. Pengurusan bayaran berdasarkan pengesahan Fakulti / Ketua / Bahagian 5. Pelarasan terimaan dan bayaran Lebihan pendapatan akan dimasukkan ke dalam akaun amanah berkenaan kekurangan akaun amanah fakulti /
Rates of Criminal Justice Action on Investigated Cases Study Sample N Rate Tjaden & Thoennes, 1992 CPS 833 4% prosecuted Finkelhor, 1983 State clearing - house data 6096 24% criminal justice action taken Stroud, Martens & Barker, 2000 &KLOGUHQ¶V Advocacy Center 1043 56% referred to p rosecutors
Introduction 4 UNHCR’s flexible funding 6 Sources of flexible funding 8 . Uganda’s progressive refugee policy is unique – it allows refugees to live a normal life just like Ugandan citizens, including freedom of movement, the right to work, access to . Crossing borders and sometimes continents, situations reflect the
