PSf Paper 1 Section A Polnomials and Quadratics Each correct answer in this section is worth two marks. 1. A parabola has equation = 2 2 + 4 + 5. Which of the following are true? I. The parabola has a minimum turning point. II. The parabola has no real roots. A. Neither I nor II is true B. nl I is true C. nl II is true D. Both I and II are true Ke utcome Grade Facilit Disc. Calculator Content Source A 2.1 C 0.37 0.4 CN A15, A17 HSN 070 PSf hsn.uk.net Page 1
PSf 2. Given p() = 2 + 6, which of the following are true? I. ( + 3) is a factor of p(). II. = 2 is a root of p() = 0. A. Neither I nor II is true B. nl I is true C. nl II is true D. Both I and II are true Ke utcome Grade Facilit Disc. Calculator Content Source D 2.1 C 0.78 0.67 NC A21 HSN 170 PSf hsn.uk.net Page 2
PSf 3. When 2a 3 + (a + 1) 6 is divided b + 2, the remainder is 2. What is the value of a? A. 5 3 B. 4 9 C. 5 9 D. 5 7 Ke utcome Grade Facilit Disc. Calculator Content Source C 2.1 C 0.41 0.77 NC A21 HSN 174 PSf [END F PAPER 1 SECTIN A] hsn.uk.net Page 3
PSf Paper 1 Section B 4. (a) Epress f () = 2 4 + 5 in the form f () = ( a) 2 + b. 2 (b) n the same diagram sketch: (i) the graph of = f (); (ii) the graph of = 10 f (). 4 (c) Find the range of values of for which 10 f () is positive. 1 Part Marks Level Calc. Content Answer U1 C2 (a) 2 C NC A5 a = 2, b = 1 2002 P1 Q7 (b) 4 C NC A3 sketch (c) 1 C NC A16, A6 1 < < 5 1 pd: process, e.g. completing the square 2 pd: process, e.g. completing the square 3 ic: interpret minimum 4 ic: interpret -intercept 5 ss: reflect in -ais 6 ss: translate parallel to -ais 7 ic: interpret graph 1 a = 2 2 b = 1 an two from: parabola; min. t.p. (2, 1); (0, 5) 4 the remaining one from above list 5 reflecting in -ais 6 translating +10 units, parallel to -ais 3 7 ( 1, 5) i.e. 1 < < 5 5. Find the values of for which the function f () = 2 3 3 2 36 is increasing. 4 hsn.uk.net Page 4
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PSf 8. For what value of k does the equation 2 5 + (k + 6) = 0 have equal roots? 3 Part Marks Level Calc. Content Answer U2 C1 3 C CN A18 k = 1 4 2001 P1 Q2 1 ss: know to set disc. to zero 2 ic: substitute a, b and c into discriminant 3 pd: process equation in k 1 b 2 4ac = 0 stated or implied b 2 2 ( 5) 2 4 (k + 6) 3 k = 1 4 9. 10. Find the values of k for which the equation 2 2 + 4 + k = 0 has real roots. 2 hsn.uk.net Page 6
PSf 11. For what value of a does the equation a 2 + 20 + 40 = 0 have equal roots? 2 12. Factorise full 2 3 + 5 2 4 3. 4 13. hsn.uk.net Page 7
PSf 14. ne root of the equation 2 3 3 2 + p + 30 = 0 is 3. Find the value of p and the other roots. 4 15. 16. Epress 4 in its full factorised form. 4 hsn.uk.net Page 8
17. PSf 18. Epress 3 4 2 7 + 10 in its full factorised form. 4 hsn.uk.net Page 9
PSf 19. (a) The function f is defined b f () = 3 2 2 5 + 6. The function g is defined b g() = 1. Show that f ( g() ) = 3 5 2 + 2 + 8. 4 (b) Factorise full f ( g() ). 3 (c) The function k is such that k() = 1 f ( g() ). For what values of is the function k not defined? 3 hsn.uk.net Page 10
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PSf 31. (a) Find the coordinates of the points of intersection of the curves with equations = 2 2 and = 4 2 2. 2 (b) Find the area completel enclosed between these two curves. 3 32. For what range of values of k does the equation 2 + 2 + 4k 2k k 2 = 0 represent a circle? 5 Part Marks Level Calc. Content Answer U2 C4 5 A NC G9, A17 for all k 2000 P1 Q6 1 ss: know to eamine radius 2 pd: process 3 pd: process 4 ic: interpret quadratic inequation 5 ic: interpret quadratic inequation 1 g = 2k, f = k, c = k 2 stated or implied b 2 2 r 2 = 5k 2 + k + 2 3 (real r ) 5k 2 + k + 2 > 0 (accept ) 4 use discr. or complete sq. or diff. 5 true for all k [END F PAPER 1 SECTIN B] hsn.uk.net Page 22
PSf Paper 2 1. hsn.uk.net Page 23
PSf 2. (i) Write down the condition for the equation a 2 + b + c = 0 to have no real roots. 1 (ii) Hence or otherwise show that the equation ( + 1) = 3 2 has no real roots. 2 3. Show that the roots of the equation (k 2) 2 (3k 2) + 2k = 0 are real. 4 hsn.uk.net Page 24
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PSf 7. Show that the equation (1 2k) 2 5k 2k = 0 has real roots for all integer values of k. 5 Part Marks Level Calc. Content Answer U2 C1 5 A/B CN A18, A16, 0.1 proof 2002 P2 Q9 1 ss: know to use discriminant 2 ic: pick out discriminant 3 pd: simplif to quadratic 4 ss: choose to draw table or graph 5 pd: complete proof using disc. 0 1 discriminant =... 2 disc = ( 5k) 2 4(1 2k)( 2k) 3 9k 2 + 8k 4 e.g. draw a table, graph, complete the square 5 complete proof and conclusion relating to disc. 0 8. The diagram shows part of the graph of the curve with equation = 2 3 7 2 + 4 + 4. = f () (a) Find the -coordinate of PSfrag the maimum turning point. 5 (b) Factorise 2 3 7 2 + 4 + 4. 3 (c) State the coordinates of the point A and A hence find the values of for which (2, 0) 2 3 7 2 + 4 + 4 < 0. 2 Part Marks Level Calc. Content Answer U2 C1 (a) 5 C NC C8 = 1 3 2002 P2 Q3 (b) 3 C NC A21 ( 2)(2 + 1)( 2) (c) 2 C NC A6 A( 1 2, 0), < 1 2 1 ss: know to differentiate 2 pd: differentiate 3 ss: know to set derivative to zero 4 pd: start solving process of equation 5 pd: complete solving process 6 ss: strateg for cubic, e.g. snth. division 7 ic: etract quadratic factor 8 pd: complete the cubic factorisation 9 ic: interpret the factors 10 ic: interpret the diagram hsn.uk.net Page 28 1 f () =... 2 6 2 14 + 4 3 6 2 14 + 4 = 0 4 (3 1)( 2) 5 = 1 3 6 2 7 4 4 0 7 2 2 3 2 8 ( 2)(2 + 1)( 2) 9 A( 1 2, 0) 10 < 1 2
PSf 9. Find p if ( + 3) is a factor of 3 2 + p + 15. 3 10. When f () = 2 4 3 + p 2 + q + 12 is divided b ( 2), the remainder is 114. ne factor of f () is ( + 1). Find the values of p and q. 5 11. Find k if 2 is a factor of 3 + k 2 4 12. 3 hsn.uk.net Page 29
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PSf 13. (a) Given that + 2 is a factor of 2 3 + 2 + k + 2, find the value of k. 3 (b) Hence solve the equation 2 3 + 2 + k + 2 = 0 when k takes this value. 2 Part Marks Level Calc. Content Answer U2 C1 (a) 3 C CN A21 k = 5 2001 P2 Q1 (b) 2 C CN A22 = 2, 1 2, 1 1 ss: use snth division or f (evaluation) 2 pd: process 3 pd: process 4 ss: find a quadratic factor 5 pd: process 1 f ( 2) = 2( 2) 3 + 2 2( 2) 3 + ( 2) 2 2k + 2 3 k = 5 4 5 2 2 3 + 1 or 2 2 + 3 2 or 2 + 2 (2 1)( 1) or (2 1)( + 2) or ( + 2)( 1) and = 2, 1 2, 1 hsn.uk.net Page 31
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PSf 15. The diagram shows a sketch of the graph of = 3 3 2 + 2. = 3 3 2 + 2 (a) Find the equation of the tangent to this curve at the point where = 1. PSf 5 (b) The tangent at the point (2, 0) has equation = 2 4. Find the coordinates of the point where this tangent meets the curve again. 5 Part Marks Level Calc. Content Answer U2 C1 (a) 5 C CN C5 + = 1 2000 P2 Q1 (b) 5 C CN A23, A22, A21 ( 1, 6) 1 ss: know to differentiate 2 pd: differentiate correctl 3 ss: know that gradient = f (1) 4 ss: know that -coord = f (1) 5 ic: state equ. of line 6 ss: equate equations 7 pd: arrange in standard form 8 ss: know how to solve cubic 9 pd: process 10 ic: interpret 1 =... 2 3 2 6 + 2 3 (1) = 1 4 (1) = 0 5 0 = 1( 1) 2 4 = 3 3 2 + 2 7 3 3 2 + 4 = 0 8 1 3 0 4 9 identif = 1 from working 10 ( 1, 6) 6 hsn.uk.net Page 33
PSf 16. The diagram shows a sketch PSfragof a parabola passing through ( 1, 0), (0, p) and (p, 0). (a) Show that the equation of the parabola is = p + (p 1) 2 ( 1, 0) (p, 0). 3 (b) For what value of p will the line = + p be a tangent to this curve? 3 (0, p) Part Marks Level Calc. Content Answer U2 C1 (a) 3 A/B CN A19 proof 2001 P2 Q11 (b) 3 A/B CN A24 2 1 ss: use a standard form of parabola 2 ss: use 3rd point to determine k 3 pd: complete proof 4 ss: equate and simplif to zero 5 ss: use discriminant for tangenc 6 pd: process 1 = k( + 1)( p) 2 k = 1 with justification (i.e. substitute (0, p)) 3 = 1( + 1)( p) and complete 4 2 + 2 p = 0 5 b 2 4ac = (2 p) 2 = 0 or (2 p) 2 4 0 = 0 6 p = 2 hsn.uk.net Page 34
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PSf 18. (a) n the same diagram, sketch the graphs of = log 10 and = 2 where 0 < < 5. Write down an approimation for the -coordinate of the point of intersection. 3 (b) Find the value of this -coordinate, correct to 2 decimal places. 3 hsn.uk.net Page 36
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PSf 20. The parabola shown crosses the -ais at (0, 0) and (4, 0), and has a maimum at (2, 4). PSf 4 The shaded area is bounded b the parabola, the -ais and the lines = 2 and = k. (a) Find the equation of the parabola. 2 k 4 2 (b) Hence show that the shaded area, A, is given b A = 1 3 k3 + 2k 2 16 3. 3 Part Marks Level Calc. Content Answer U2 C2 (a) 2 C CN A19 = 4 2 2000 P2 Q4 (b) 3 C CN C16 proof 1 ic: state standard form 2 pd: process for 2 coeff. 3 ss: know to integrate 4 pd: integrate correctl 5 pd: process limits and complete proof 1 a( 4) 2 a = 1 3 k 2 (function from (a)) 4 1 3 3 + 2 2 5 1 3 k3 + 2k 2 ( 8 3 + 8) 21. (a) Write the equation cos 2θ + 8 cos θ + 9 = 0 in terms of cos θ and show that, for cos θ, it has equal roots. 3 (b) Show that there are no real roots for θ. 1 hsn.uk.net Page 38
PSf 22. Find the possible values of k for which the line = k is a tangent to the circle 2 + 2 = 18. 5 [END F PAPER 2] hsn.uk.net Page 39