1. Paper collated from year 2007 Content Pure Chapters 1-13 Marks 100 Time 2 hours
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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 2 + 1 from first principles with respect to x. crashmaths practice paper1 SetB Q-5 [4]
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-5 10. (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+1 + 2 = 0 can be written as (Y 1)(Y 2) = 0 (ii) Hence show that the equation 9 x 3 x+1 + 2 = 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]
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-9 12. (a) [2] (b) [3] Edexcel C2 January 2008 Q-4 13. Use a counterexample to show that if n is an integer, n 2 + 1 is not necessarily prime. crashmaths practice paper1 SetB Q-10 [2]
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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Paper collated from year 2010 Content Pure Chapters 1-13 Marks 103 Time 2 hours
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Paper collated from year 2011 Content Pure Chapters 1-13 Marks 100 Time 2 hours
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Paper collated from year 2012 Content Pure Chapters 1-13 Marks 100 Time 2 hours Q1. Q2. Q3.
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Mark scheme Q1. Problem Solving Q2. Surds Indices
Q3. Quadratic functions Q4. Equations and Inequalities
Q5. Coordinate Geometry
Q6. Coordinate Geometry
Q7. Polynomials
Q8. Graphs &Transformations Q9. The binomial expansion
Q10. Differentiation Q11. Integration
Q12. Vectors Q13. Logs and Exponentials
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)
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)
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)
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)
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)
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)
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)
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Paper collated from year 2015 Content Marks 100 Time Pure Chapters 1-13 2 hours 4.
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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 2 + 24xh + 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)