Questions Q1. f(x) = x4 − x3 + 3x2 + ax + b where a and b are constants. When f(x) is divided by (x − 1) the remainder is 4 When f(x) is divided by (x + 2) the remainder is 22 Find the value of a and the value of b. (5) (Total for question = 5 marks) Q2. f(x) = x3 + x2 − 12x − 18 (a) Use the factor theorem to show that (x + 3) is a factor of f(x). (2) (b) Factorise f(x). (2) (c) Hence find exact values for all the solutions of the equation f(x) = 0 (3) (Total for question = 7 marks) Q3. The equation x2 + (6k + 4)x + 3 = 0, where k is a constant, has no real roots. (a) Show that k satisfies the inequality 9k2 + 12k + 1 < 0 (3) (b) Find the range of possible values for k, giving your boundaries as fully simplified surds. (4) (Total for question = 7 marks) Q4. The equation k(3x2 + 8x + 9) = 2 − 6x, where k is a real constant, has no real roots. (a) Show that k satisfies the inequality 11k2 − 30k − 9 > 0 (4) (b) Find the range of possible values for k. (4) (Total for question = 8 marks) Q5. Given that express each of the following in the form kxn where k and n are constants. (a) y (1) (b) 3y−1 (1) (c) (1) (Total for question = 3 marks) Q6. Answer this question without a calculator, showing all your working and giving your answers in their simplest form. (i) Solve the equation 42x + 1 = 84x (3) (ii) (a) Express in the form k , where k is an integer. (2) (b) Hence, or otherwise, solve (2) (Total for question = 7 marks) Q7. (i) Given that = 7a, find the value of a. (2) (ii) Show that = 15 √2 + 20 You must show all stages of your working. (3) (Total for question = 5 marks) Q8. Answer this question without the use of a calculator and show all your working. (i) Show that (4) (ii) Show that (3) (Total for question = 7 marks) Q9. (i) Given that logax + loga 3 = loga 27 − 1, where a is a positive constant find, in its simplest form, an expression for x in terms of a. (4) (ii) Solve the equation (log5y)2 − 7(log5y) + 12 = 0 showing each step of your working. (4) (Total for question = 8 marks) Q10. Solve, giving each answer to 3 significant figures, the equations (a) 4a = 20 (2) (b) 3 + 2log2b = log2 (30b) (5) (Solutions based entirely on graphical or numerical methods are not acceptable.) (Total for question = 7 marks) Q11. Given that 2log4 (2x + 3) = 1 + log4x + log4 (2x – 1), x > ½ (a) show that 4x2 – 16x – 9 = 0 (5) (b) Hence solve the equation 2log4 (2x + 3) = 1 + log4x + log4 (2x – 1), x > ½ (2) (Total for question = 7 marks) Q12. (i) Use the laws of logarithms to solve the equation 3log8 2 + log8(7 − x) = 2 + log8x (5) (ii) Using algebra, find, in terms of logarithms, the exact value of y for which 32y + 3y + 1 = 10 (5) (Total for question = 10 marks) Q13. Figure 2 shows a sketch of part of the curve with equation y = f (x). The curve crosses the x-axis at the origin and at the point (6, 0). The curve has maximum points at (1, 6) and (5, 6) and has a minimum point at (3, 2). On separate diagrams sketch the curve with equation (a) y = −f (x) (3) (b) y = (3) (c) y = f (x + 4) (3) On each diagram show clearly the coordinates of the maximum and minimum points, and the coordinates of the points where the curve crosses the x-axis. (Total for question = 9 marks) Q14. Figure 3 Figure 3 shows a sketch of the curve with equation y = f(x) where The curve crosses the x-axis at (2, 0) and (8, 0) and has a minimum point at A. (a) Use calculus to find the coordinates of point A. (5) (b) State (i) the roots of the equation 2f(x) = 0 (ii) the coordinates of the turning point on the curve y = f(x) + 2 (iii) the roots of the equation f(4x) = 0 (3) (Total for question = 8 marks) Q15. Figure 2 shows a sketch of part of the curve with equation y = f(x). The curve crosses the coordinate axes at the points (2.5, 0) and (0, 9), has a stationary point at (1, 11), and has an asymptote y = 3 On separate diagrams, sketch the curve with equation (a) y = 3f(x) (3) (b) y = f(– x) (3) On each diagram show clearly the coordinates of the points of intersection of the curve with the two coordinate axes, the coordinates of the stationary point, and the equation of the asymptote. (Total for question = 6 marks) Q16. Figure 3 shows a sketch of the curve with equation . The curve crosses the y-axis at the point (0, 5) and crosses the x-axis at the point (6, 0). The curve has a minimum point at (1, 3) and a maximum point at (4, 7). On separate diagrams, sketch the curve with equation (a) y = f(−x) (3) (b) y = f(2x) (3) On each diagram, show clearly the coordinates of any points of intersection of the curve with the two coordinate axes and the coordinates of the stationary points. (Total for question = 6 marks) Q17. A circle, with centre C and radius r, has equation x2 + y2 − 8x + 4 y − 12 = 0 Find (a) the coordinates of C, (2) (b) the exact value of r. (2) The circle cuts the y-axis at the points A and B. (c) Find the coordinates of the points A and B. (3) (Total for question = 7 marks) Q18. The points A and B have coordinates (−8, −8) and (12, 2) respectively. AB is the diameter of a circle C. (a) Find an equation for the circle C. (6) The point (4, 8) also lies on C. (b) Find an equation of the tangent to C at the point (4, 8), giving your answer in the form ax + by + c = 0 (4) (Total for question = 10 marks) Q19. Figure 3 The points X and Y have coordinates (0, 3) and (6, 11) respectively. XY is a chord of a circle C with centre Z, as shown in Figure 3. (a) Find the gradient of XY. (2) The point M is the midpoint of XY. (b) Find an equation for the line which passes through Z and M. (5) Given that the y coordinate of Z is 10, (c) find the x coordinate of Z, (2) (d) find the equation of the circle C, giving your answer in the form x2 + y2 + ax + by + c = 0 where a, b and c are constants. (5) (Total for question = 14 marks) Q20. Figure 2 The straight line l1 has equation 2y = 3x + 5 The line l1 cuts the x-axis at the point A, as shown in Figure 2. (a) (i) State the gradient of l1 (ii) Write down the x coordinate of point A. (3) Another straight line l2 intersects l1 at the point B with x coordinate 1 and crosses the x-axis at the point C, as shown in Figure 2. Given that l2 is perpendicular to l1 (b) find an equation for l2 in the form ax + by + c = 0, where a, b and c are integers, (5) (c) find the exact area of triangle ABC. (3) (Total for question = 11 marks) Q21. Figure 3 Figure 3 shows a circle C C touches the y-axis and has centre at the point (a, 0) where a is a positive constant. (a) Write down an equation for C in terms of a (2) Given that the point P(4, –3) lies on C, (b) find the value of a (3) (Total for question = 5 marks) Q22. A sequence is defined by u1 = 3 un + 1 = un – 5, n≥1 Find the values of (a) u2, u3 and u4 (2) (b) u100 (3) (c) (3) (Total for question = 8 marks) Q23. A sequence is defined by where k is a constant. (a) Write down fully simplified expressions for u2, u3 and u4 in terms of k. (4) Given that u4 = 15 (b) find the value of k, (2) (c) find , giving an exact numerical answer. (3) (Total for question = 9 marks) Q24. A sequence is defined by (a) Find the exact values of u2, u3 and u4 (2) (b) Find the value of u20, giving your answer to 3 significant figures. (2) (c) Evaluate giving your answer to 3 significant figures. (3) (d) Explain why < 12 for all positive integer values of N. (1) (Total for question = 8 marks) Q25. A sequence of numbers u1, u2, u3, ... satisfies un+1 = 2un − 6, n≥1 Given that u1 = 2 (a) find the value of u3 (2) (b) evaluate (3) (Total for question = 5 marks) Q26. The 4th term of an arithmetic sequence is 3 and the sum of the first 6 terms is 27 Find the first term and the common difference of this sequence. (6) (Total for question = 6 marks) Q27. An arithmetic series has first term a and common difference d. Given that the sum of the first 9 terms is 54 (a) show that a + 4d = 6 (2) Given also that the 8th term is half the 7th term, (b) find the values of a and d. (4) (Total for question = 6 marks) Q28. An arithmetic sequence has first term 6 and common difference 10 Find (a) the 15th term of the sequence, (2) (b) the sum of the first 20 terms of the sequence. (2) (Total for question = 4 marks) Q29. (a) Prove that the sum of the first n terms of an arithmetic series is given by the formula where a is the first term of the series and d is the common difference between the terms. (4) (b) Find the sum of the integers which are divisible by 7 and lie between 1 and 500 (3) (Total for question = 7 marks) Q30. A 25-year programme for building new houses began in Core Town in the year 1986 and finished in the year 2010. The number of houses built each year form an arithmetic sequence. Given that 238 houses were built in the year 2000 and 108 were built in the year 2010, find (a) the number of houses built in 1986, the first year of the building programme, (5) (b) the total number of houses built in the 25 years of the programme. (2) (Total for question = 7 marks) Q31. The first three terms of an arithmetic series are 60, 4p and 2p – 6 respectively (a) Show that p = 9 (2) (b) Find the value of the 20th term of this series. (3) (c) Prove that the sum of the first n terms of this series is given by the expression 12n (6 − n) (3) (Total for question = 8 marks) Q32. The first term of a geometric series is 20 and the common ratio is 0.9 (a) Find the value of the fifth term. (2) (b) Find the sum of the first 8 terms, giving your answer to one decimal place. (2) Given that where SN is the sum of the first N terms of this series, (c) show that 0.9N < 0.0002 (4) (d) Hence find the smallest possible value of N. (2) (Total for question = 10 marks) Q33. (i) Find the value of (3 + 5r) (3) (ii) Given that = 16, find the value of the constant a. (4) (Total for question = 7 marks) Q34. A geometric series has a first term a and a common ratio r. (a) Prove that the sum of the first n terms of this series is given by (4) A liquid is to be stored in a barrel. Due to evaporation, the volume of the liquid in a barrel at the end of a year is 7% less than the volume at the start of the year. At the start of the first year, a barrel is filled with 180 litres of the liquid. (b) Show that the amount of the liquid in this barrel at the end of 5 years is approximately 125.2 litres. (2) At the start of each year a new identical barrel is filled with 180 litres of the liquid so that, at the end of 20 years, there are 20 barrels containing varying amounts of the liquid. (c) Calculate the total amount of the liquid, to the nearest litre, in the 20 barrels at the end of 20 years. (3) (Total for question = 9 marks) Q35. The resident population of a city is 130 000 at the end of Year 1 A model predicts that the resident population of the city will increase by 2% each year, with the populations at the end of each year forming a geometric sequence. (a) Show that the predicted resident population at the end of Year 2 is 132 600 (1) (b) Write down the value of the common ratio of the geometric sequence. (1) The model predicts that Year N will be the first year which will end with the resident population of the city exceeding 260 000 (c) Show that (4) (d) Find the value of N. (1) (Total for question = 7 marks) Q36. In the first month after opening, a mobile phone shop sold 300 phones. A model for future sales assumes that the number of phones sold will increase by 5% per month, so that 300 × 1.05 will be sold in the second month, 300 × 1.052 in the third month, and so on. Using this model, calculate (a) the number of phones sold in the 24th month, (2) (b) the total number of phones sold over the whole 24 months. (2) This model predicts that, in the Nth month, the number of phones sold in that month exceeds 3000 for the first time. (c) Find the value of N. (3) (Total for question = 7 marks) Q37. Find the first 3 terms in ascending powers of x of giving each term in its simplest form. (4) (Total for question = 4 marks) Q38. The binomial expansion, in ascending powers of x, of (1 + kx)n is 1 + 36x + 126kx2 + ... where k is a non-zero constant and n is a positive integer. (a) Show that nk (n − 1) = 252 (2) (b) Find the value of k and the value of n. (5) 3 (c) Using the values of k and n, find the coefficient of x in the binomial expansion of (1 + kx) n (3) (Total for question = 10 marks) Q39. (a) Find the first 4 terms in ascending powers of x of the binomial expansion of giving each term in its simplest form. (4) (b) Use your expansion to find an estimated value for 2.02510, stating the value of x which you have used and showing your working. (3) (Total for question = 7 marks) Q40. (a) Find the first 4 terms, in ascending powers of x, of the binomial expansion of giving each term in its simplest form. (4) (b) Use the answer to part (a) to find an estimated value for , stating the value of x that you have used and showing your working. Give your estimate to 4 decimal places. (3) (Total for question = 7 marks) Q41. (a) Find the first 3 terms, in ascending powers of x, of the binomial expansion of giving each term in its simplest form. (4) Given that the first two terms, in ascending powers of x in the series expansion of f(x), are 256 and 352x, (b) find the value of a, (2) (c) find the value of b. (2) (Total for question = 8 marks) Q42. Figure 1 shows a semicircle with centre O and radius 3cm. XY is the diameter of this semicircle. The point Z is on the circumference such that angle XOZ = 1.3 radians. The shaded region enclosed by the chord XZ, the arc ZY and the diameter XY is a template for a badge. Find, giving each answer to 3 significant figures, (a) the length of the chord XZ, (2) (b) the perimeter of the template XZYX, (4) (c) the area of the template. (4) (Total for question = 10 marks) Q43. Figure 2 Figure 2 shows a sketch of a design for a triangular garden ABC. The garden has sides BA with length 10 m, BC with length 6 m and CA with length 12 m. The point D lies on AC such that BD is an arc of the circle centre A, radius 10 m. A flowerbed BCD is shown shaded in Figure 2. (a) Find the size of angle BAC, in radians, to 4 decimal places. (2) (b) Find the perimeter of the flowerbed BCD, in m, to 2 decimal places. (3) (c) Find the area of the flowerbed BCD, in m2, to 2 decimal places. (4) (Total for question = 9 marks) Q44. In Figure 3, the points A and B are the centres of the circles C1 and C2 respectively. The circle C1 has radius 10 cm and the circle C2 has radius 5 cm. The circles intersect at the points X and Y, as shown in the figure. Given that the distance between the centres of the circles is 12 cm, (a) calculate the size of the acute angle XAB, giving your answer in radians to 3 significant figures, (2) (b) find the area of the major sector of circle C1, shown shaded in Figure 3, (3) (c) find the area of the kite AYBX. (3) (Total for question = 8 marks) Q45. Figure 1 shows a circle with centre O and radius 9 cm. The points A and B lie on the circumference of this circle. The minor sector OAB has perimeter 30cm and the angle between the radii OA and OB of this sector is θ radians. Find (a) the length of the arc AB, (1) (b) the value of θ, (2) (c) the area of the minor sector OAB, (2) (d) the area of triangle OAB, giving your answer to 3 significant figures. (2) (Total for question = 7 marks) Q46. Figure 1 The shape POQABCP, as shown in Figure 1, consists of a triangle POC, a sector OQA of a circle with radius 7 cm and centre O, joined to a rectangle OABC. The points P, O and Q lie on a straight line. PO = 4 cm, CO = 5 cm and angle AOQ = 0.8 radians. (a) Find the length of arc AQ. (2) (b) Find the size of angle POC in radians, giving your answer to 3 decimal places. (2) (c) Find the perimeter of the shape POQABCP, in cm, giving your answer to 2 decimal places. (4) (Total for question = 8 marks) Q47. The curve C has equation (a) In the space below, sketch the curve C. (2) (b) Write down the exact coordinates of the points at which C meets the coordinate axes. (3) (c) Solve, for x in the interval 0 ≤ x ≤ 2π, giving your answers in the form kπ, where k is a rational number. (4) (Total for question = 9 marks) Q48. Figure 4 shows a sketch of the curve C with equation y = sin(x – 60°), –360° x 360° (a) Write down the exact coordinates of the points at which C meets the two coordinate axes. (3) (b) Solve, for –360° x 360°, 4 sin(x – 60°) = √6 – √2 showing each stage of your working. (5) (Total for question = 8 marks) Q49. [In this question solutions based entirely on graphical or numerical methods are not acceptable.] (i) Solve for 0 ≤ x < 360°, 5sin(x + 65°) + 2 = 0 giving your answers in degrees to one decimal place. (4) (ii) Find, for 0 ≤ θ < 2π, all the solutions of 12sin2θ + cosθ = 6 giving your answers in radians to 3 significant figures. (6) (Total for question = 10 marks) Q50. In this question, solutions based entirely on graphical or numerical methods are not acceptable. (i) Solve, for 0 x < 360°, 3sin x + 7cos x = 0 Give each solution, in degrees, to one decimal place. (4) (ii) Solve, for 0 θ < 2π, 10 cos2θ + cos θ = 11sin2θ – 9 Give each solution, in radians, to 3 significant figures. (6) (Total for question = 10 marks) Q51. Solve, for 0 θ < 2π, 2cos2 θ = 5–13sin θ Give your answers in radians to 3 decimal places. (Solutions based entirely on graphical or numerical methods are not acceptable.) (5) (Total for question = 5 marks) Q52. In this question the angle θ is measured in degrees throughout. (a) Show that the equation may be rewritten as 6sin2θ + sin θ − 1 = 0 (3) (b) Hence solve, for –90° < θ < 90°, the equation Give your answers to one decimal place, where appropriate. (4) (Total for question = 7 marks) Q53. In this question solutions based entirely on graphical or numerical methods are not acceptable. (i) Solve, for 0 ≤ x < 2π, 3cos2x + 1 = 4sin2x giving your answers in radians to 2 decimal places. (5) (ii) Solve, for 0 ≤ θ < 360°, 5sin(θ + 10°) = cos(θ + 10°) giving your answers in degrees to one decimal place. (5) (Total for question = 10 marks) Q54. The curve C has equation (a) Find (2) (b) Hence find the coordinates of the stationary point on C. (5) (c) Use to determine the nature of this stationary point. (3) (Total for question = 10 marks) Q55. f(x) = , x>0 (a) Show that f(x) = Ax−1 + Bxk + C, where A, B, C and k are constants to be determined. (4) (b) Hence find f'(x). (2) (c) Find an equation of the tangent to the curve y = f(x) at the point where x = 4 (4) (Total for question = 10 marks) Q56. The curve C has equation (a) Find in a fully simplified form. (3) (b) Hence find the coordinates of the turning point on the curve C. (4) (c) Determine whether this turning point is a minimum or maximum, justifying your answer. (2) The point P, with x coordinate , lies on the curve C. (d) Find the equation of the normal at P, in the form ax + by + c = 0, where a, b and c are integers. (5) (Total for question = 14 marks) Q57. The finite region R, which is shown shaded in Figure 3, is bounded by the straight line l with equation y = 4x + 3 and the curve C with equation y = 2x − 2x + 3, x > 0 The line l meets the curve C at the point A on the y-axis and l meets C again at the point B, as shown in Figure 3. (a) Use algebra to find the coordinates of A and B. (4) (b) Use integration to find the area of the shaded region R. (6) (Total for question = 10 marks) Q58. Find, using calculus and showing each step of your working, (5) (Total for question = 5 marks) Q59. The straight line l with equation y = and Q, as shown in Figure 2 x + 1 cuts the curve C, with equation y = x2 − 4x + 3, at the points P (a) Use algebra to find the coordinates of the points P and Q. (5) The curve C crosses the x-axis at the points T and S. (b) Write down the coordinates of the points T and S. (2) The finite region R is shown shaded in Figure 2. This region R is bounded by the line segment PQ, the line segment TS, and the arcs PT and SQ of the curve. (c) Use integration to find the exact area of the shaded region R. (8) (Total for question = 15 marks) Q60. (a) Sketch the graph of y = ,x>0 (2) The table below shows corresponding values of x and y for y = decimal places where necessary. , with the values for y rounded to 3 (b) Use the trapezium rule with all the values of y from the table to find an approximate value, to 2 decimal places, for (4) (Total for question = 6 marks) Q61. Figure 1 Figure 1 shows a sketch of part of the curve with equation y = The finite region R, shown shaded in Figure 1, is bounded by the curve, the x-axis and the line x = 6 The table below shows corresponding values of x and y for y = (a) Complete the table above, giving the missing value of y to 4 decimal places. (1) (b) Use the trapezium rule, with all of the values of y in the completed table, to find an approximate value for the area of R, giving your answer to 3 decimal places. (3) Use your answer to part (b) to find approximate values of (c) (i) (ii) (4) (Total for question = 8 marks) Q62. The table below shows corresponding values of x and y for rounded to 3 decimal places where necessary. , with the values for y (a) Complete the table by giving the value of y corresponding to x = 15 (1) (b) Use the trapezium rule with all the values of y from the completed table to find an approximate value for giving your answer to 2 decimal places. (4) (Total for question = 5 marks)