Paper collated from year 2007 Content Pure Chapters 1-13 Marks 100 Time 2 hours

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1 1. Paper collated from year 2007 Content Pure Chapters 1-13 Marks 100 Time 2 hours

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8 Mark scheme Question 1 Question 2 Question 3 Question 4

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12 Paper collated from year 2008 Content Pure Chapters 1-13 Marks 100 Time 2 hours 1. Find the equation of the line passing through A( 1, 1) and B (3, 9). MEI C1 June 2008 Q-12(i) [3] 2. The curve with equation y = x 3 7x 6 is sketched below. [3] Find the gradient of the curve y = x 3 7x 6 at the point B(-1, 0). AQA C1 January 2008 Q-6 (iv) 3. The polynomial p(x) is given by p(x) = x 3 + x 2 8x 12. (a) Use the Factor Theorem to show that x + 2 is a factor of p(x). [2] (b) Express p(x) as the product of linear factors. [2] (c) Sketch the graph of y = x 3 + x 2 8x 12, indicating the values of x where the curve touches or crosses the x-axis. AQA C1 June 2008 Q-6 [3] 4. (a) [4] (b) [3] Edexcel C2 January 2008 Q-3 5. Given that point A has the position vector 4i + 7j and point B has the position vector 10i + qj, where q is a constant, given that AB = 2 13, find the two possible values of q showing detailed reasoning in your working. 6. The quadratic equation (k + 1)x 2 + 4kx + 9 = 0 has real roots. Unit Test 5: Vectors Q-5 (a) Show that 4k 2 9k 9 0. [3] (b) Hence find the possible values of k. Write your answer using set notation. AQA C1 June 2008 Q-8 [5] [4] 7. Differentiate 6x from first principles with respect to x. crashmaths practice paper1 SetB Q-5 [4]

13 8. The diagram shows a triangle ABC. The length of AC is 18.7 cm, and the sizes of angles BAC and ABC are 72 and 50 respectively. (a) Show that the length of BC = 23.2 cm, correct to the nearest 0.1 cm. [3] (b) Calculate the area of triangle ABC, giving your answer to the nearest cm 2. AQA C2 January 2008 Q-3 [3] 9. A curve, drawn from the origin O, crosses the x-axis at the point P(4, 0). The normal to the curve at P meets the y-axis at the point Q, as shown in the diagram. The curve, defined for x 0, has equation (a) (i) Find dy (b) y = 4x 1 2 x 3 2 dx. [3] (ii) Find an equation of the normal to the curve at P (4, 0) [3] (i) (ii) Find 4x 1 2 x 3 2 dx Find the total area of the region bounded by the curve and the lines PQ and QO. AQA C2 January 2008 Q (a) Sketch the graph of y = 3 x, stating the coordinates of the point where the graph crosses the y-axis (b) Describe a single geometrical transformation that maps the graph of y = 3 x : onto the graph of y = 3 x+1 (c) (i) Using the substitution Y = 3 x, show that the equation 9 x 3 x = 0 can be written as (Y 1)(Y 2) = 0 (ii) Hence show that the equation 9 x 3 x = 0 has a solution x = 0 and, by using logarithms, find the other solution, giving your answer to four decimal places. AQA C2 January 2008 Q-8 [3] [3] [2] [2] [2] [2]

14 11. Figure shows an open-topped water tank, in the shape of a cuboid, which is made of sheet metal. The base of the tank is a rectangle x metres by y metres. The height of the tank is x metres. The capacity of the tank is 100 m 3. (a) [4] (b) Use calculus to find the value of x for which A is stationary. [4] (c) Prove that this value of x gives a minimum value of A. [2] Edexcel C2 January 2008 Q (a) [2] (b) [3] Edexcel C2 January 2008 Q Use a counterexample to show that if n is an integer, n is not necessarily prime. crashmaths practice paper1 SetB Q-10 [2]

15 14. The circle S has centre C(8, 13), and touches the x-axis, as shown in the diagram. 15. (a) (b) Write down an equation for S, giving your answer in the form (x a) 2 + (y b) 2 = r 2 The point P with coordinates (3, 1) lies on the circle. (i) Find the gradient of the straight line passing through P and C. [1] (ii) (iii) Hence find an equation of the tangent to the circle S at the point P, giving your answer in the form ax + by = c, where a, b and c are integers. The point Q also lies on the circle S, and the length of PQ is 10. Calculate the shortest distance from C to the chord PQ. AQA C1 June 2008 Q-7 [2] [4] [3] MEI C2 June 2008 Q-13

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21 Paper collated from year 2010 Content Pure Chapters 1-13 Marks 103 Time 2 hours

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32 Paper collated from year 2011 Content Pure Chapters 1-13 Marks 100 Time 2 hours

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44 Paper collated from year 2012 Content Pure Chapters 1-13 Marks 100 Time 2 hours Q1. Q2. Q3.

45 Q4. Q5.

46 Q6. Q7.

47 Q8.

48 Q9.

49 Q10.

50 Q11. Q12.

51 Q13.

52 Mark scheme Q1. Problem Solving Q2. Surds Indices

53 Q3. Quadratic functions Q4. Equations and Inequalities

54 Q5. Coordinate Geometry

55 Q6. Coordinate Geometry

56 Q7. Polynomials

57 Q8. Graphs &Transformations Q9. The binomial expansion

58 Q10. Differentiation Q11. Integration

59 Q12. Vectors Q13. Logs and Exponentials

60 Paper collated from year 2014 Content Pure Chapters 1-13 Marks 100 Time 2 hours Q1. Factorise fully 25x 9x 3 Q2. (3) Solve the equation 10 + x 8 = Give your answer in the form a b where a and b are integers. (4) Q3. (a) Write 80 in the form c 5, where c is a positive constant. A rectangle R has a length of (1 + 5) cm and an area of 80 cm 2. (1) (b) Calculate the width of R in cm. Express your answer in the form p + q 5, where p and q are integers to be found. (4) Q4. Find the set of values of x for which (a) 3x 7 > 3 x (b) x 2 9x 36 (c) both 3x 7 > 3 x and x 2 9x 36 (2) (4) (1) Q5. f(x) = 2x 3 7x 2 + 4x + 4 (a) Use the factor theorem to show that (x 2) is a factor of f(x). (b) Factorise f(x) completely. (2) (4)

61 Q6. Figure 1 Figure 1 shows a sketch of the curve C with equation y = 1 x + 1, x 0 The curve C crosses the x-axis at the point A. (a) State the x coordinate of the point A. The curve D has equation y = x 2 (x 2), for all real values of x. (b) Add a sketch a graph of curve D to Figure 1. Show on the sketch the coordinates of each point where the curve D crosses the coordinate axes. (c) Using your sketch, state, giving a reason, the number of real solutions to the equation x 2 (x 2) = 1 x + 1. (1) (3) (1)

62 Q7. Figure 2 Figure 2 shows a right angled triangle LMN. The points L and M have coordinates ( 1, 2) and (7, 4) respectively. (a) Find an equation for the straight line passing through the points L and M. Give your answer in the form ax + by + c = 0, where a, b and c are integers. Given that the coordinates of point N are (16, p), where p is a constant, and angle LMN = 90, (b) find the value of p. Given that there is a point K such that the points L, M, N, and K form a rectangle, (c) find the y coordinate of K. (4) (3) (2)

63 Q8. The circle C, with centre A, passes through the point P with coordinates ( 9, 8) and the point Q with coordinates (15, 10). Given that PQ is a diameter of the circle C, (a) find the coordinates of A, (b) find an equation for C. A point R also lies on the circle C. Given that the length of the chord PR is 20 units, (c) find the length of the shortest distance from A to the chord PR. Give your answer as a surd in its simplest form. (d) Find the size of the angle ARQ, giving your answer to the nearest 0.1 of a degree. (2) (3) (2) (2) Q9. Differentiate with respect to x, giving each answer in its simplest form. (a) (1 2x) 2 (3) (b) (4)

64 Q10. Figure 4 Figure 4 shows the plan of a pool. The shape of the pool ABCDEFA consists of a rectangle BCEF joined to an equilateral triangle BFA and a semi-circle CDE, as shown in Figure 4. Given that AB = x metres, EF = y metres, and the area of the pool is 50 m 2, (a) show that (b) Hence show that the perimeter, P metres, of the pool is given by (3) (c) Use calculus to find the minimum value of P, giving your answer to 3 significant figures. (d) Justify, by further differentiation, that the value of P that you have found is a minimum. (3) (5) (2)

65 Q11. Use integration to find giving your answer in the form a + b 3, where a and b are constants to be determined. (5) Q12. Figure 3 Figure 3 shows a sketch of part of the curve C with equation The curve C has a maximum turning point at the point A and a minimum turning point at the origin O. The line l touches the curve C at the point A and cuts the curve C at the point B. The x coordinate of A is 4 and the x coordinate of B is 2. The finite region R, shown shaded in Figure 3, is bounded by the curve C and the line l. Use integration to find the area of the finite region R. (7)

66 Q13. (i) Solve, for 0 θ < 360, the equation giving your answers to 1 decimal place. You must show each step of your working. (ii) Solve, for π x < π, the equation 9sin(θ + 60 ) = 4 (4) 2tan x 3sin x = 0 giving your answers to 2 decimal places where appropriate. [Solutions based entirely on graphical or numerical methods are not acceptable.] Q14. A rare species of primrose is being studied. The population, P, of primroses at time t years after the study started is modelled by the equation (5) P = t 0, t (a) Calculate the number of primroses at the start of the study. (b) Find the exact value of t when P = 250, giving your answer in the form a ln(b) where a and b are integers. Q15. (2) (4) Find the exact solution, in its simplest form, to the equation 2 ln (2x + 1) 10 = 0 (2) Q16. Relative to a fixed origin O, the point A has position vector and the point B has position vector The line l 1 passes through the points A and B. (a) Find the vector. Q17 Calculate the derivative of g(x)=2x 3 from first principles. (2) (4)

67 Mark scheme Q1. Q2.

68 Q3. Q4.

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79 Q17. M1 M1 M1 A1dep

80 Paper collated from year 2015 Content Marks 100 Time Pure Chapters hours 4.

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83 Differentiate f(x) = 8x from first principles. (4)

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94 13 14 f f(x+h) f(x) (x) = lim h 0 h (1) States the formula for differentiation from first principles. f (x) = lim 8(x+h)3 +5 (8x3 +5) h 0 h f (x) = lim 8(x3 +3x2h+3xh2 +h3 )+5 8x3 5) h 0 h f (x) = lim (24x2h+24xh2 +8h3 ) h 0 h f (x) = lim h(24x2 +24xh+8h2 ) h 0 h f (x) = lim h 0 24x xh + 8h 2 (1) (1) Factorises the h out of the numerator and divides to simplify. Correctly applies the formula to the specific function and expands and simplifies. As h 0, f (x) 24x 2 (1)

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