Legacy C2 — Core Mathematics 2: Edexcel A‑Level Maths questions

Edexcel A-Level Maths past-paper questions on L.C2 Legacy C2 — Core Mathematics 2, 2010–2025, with mark schemes and examiner’s reports where available. Add MadAsMaths questions to your practice, or browse them separately by topic and difficulty.

All 141 questions on Legacy C2 — Core Mathematics 2, as a list

June 2018, Legacy C2, Q1 (5 marks): y −2 O R 10 x Figure 1 Figure 1 shows a sketch of part of the curve with equation y x x = + − ( ) , 2 4 2 3 2  The finite region R, shown shaded in Figure 1, i
June 2018, Legacy C2, Q2 (7 marks): (a) Find the first 4 terms, in ascending powers of x, of the binomial expansion of (2 + k x)7 where k is a non-zero constant. Give each term in its simplest for
June 2018, Legacy C2, Q3 (8 marks): f ( x) = 24x 3 + Ax 2 − 3x + B where A and B are constants. When f ( x) is divided by (2x − 1) the remainder is 30 (a) Show that A + 4B = 114 (2) Given also tha
June 2018, Legacy C2, Q4 (9 marks): Not to scale X Z W Y 0.7 rad 7 cm 5 cm Figure 2 Figure 2 shows a flag XYWZX. The flag consists of a triangle XYZ joined to a sector ZYW of a circle with radius
June 2018, Legacy C2, Q5 (10 marks): The circle C has equation x 2 + y 2 − 2 x + 14 y = 0 Find (a) the coordinates of the centre of C, (2) (b) the exact value of the radius of C, (2) (c) the y coor
June 2018, Legacy C2, Q6 (7 marks): A geometric series with common ratio r = – 0.9 has sum to infinity 10 000 For this series, (a) find the first term, (2) (b) find the fifth term, (2) (c) find th
June 2018, Legacy C2, Q7 (8 marks): (i) Find the value of y for which 1.01y −1 = 500 Give your answer to 2 decimal places. (2) (ii) Given that 2 log4 (3 x + 5) = log4 (3 x + 8) + 1, x > −5 3 (a) s
June 2018, Legacy C2, Q8 (9 marks): 8 In this question solutions based entirely on graphical or numerical methods are not acceptable. (i) Solve for 0  x < 360°, 4 cos ( x + 70°) = 3 giving your a
June 2018, Legacy C2, Q9 (12 marks): y x O A R B Figure 3 Figure 3 shows a sketch of part of the curve with equation y = 7x 2 (5 − 2 x) , x  0 The curve has a turning point at the point A, where x
June 2017, Legacy C2, Q1 (4 marks): Find the first 4 terms, in ascending powers of x, of the binomial expansion of 3 1 3 5 − ⎛ ⎝⎜ ⎞ ⎠⎟ x giving each term in its simplest form. (4)
June 2017, Legacy C2, Q2 (4 marks): In the triangle ABC, AB = 16 cm, AC = 13 cm, angle ABC = 50q and angle BCA= xq Find the two possible values for x, giving your answers to one decimal place. (4)
June 2017, Legacy C2, Q3 (6 marks): (a) y = 5x + log2(x + 1), 0 - x - 2 Complete the table below, by giving the value of y when x = 1 x 0 0.5 1 1.5 2 y 1 2.821 12.502 26.585 (1) (b) Use the trapez
June 2017, Legacy C2, Q4 (8 marks): A E B D F C Diagram not drawn to scale Figure 1 Figure 1 is a sketch representing the cross-section of a large tent ABCDEF. AB and DE are line segments of equal
June 2017, Legacy C2, Q5 (7 marks): The circle C has equation x2 + y2 – 10x + 6y + 30 = 0 Find (a) the coordinates of the centre of C, (2) (b) the radius of C, (2) (c) the y coordinates of the poi
June 2017, Legacy C2, Q6 (9 marks): f(x) = –6x3 – 7x2 + 40x + 21 (a) Use the factor theorem to show that (x + 3) is a factor of f(x) (2) (b) Factorise f(x) completely. (4) (c) Hence solve the equa
June 2017, Legacy C2, Q7 (7 marks): (i) 2 log(x + a) = log(16a6), where a is a positive constant Find x in terms of a, giving your answer in its simplest form. (3) (ii) log3(9y + b) – log3(2y – b)
June 2017, Legacy C2, Q8 (8 marks): (a) Show that the equation cos2x = 8sin2x – 6sin x can be written in the form (3sin x – 1)2 = 2 (3) (b) Hence solve, for 0 - x 360q cos2x = 8sin2x – 6sin x giv
June 2017, Legacy C2, Q9 (12 marks): The first three terms of a geometric sequence are 7k – 5, 5k – 7, 2k + 10 where k is a constant. (a) Show that 11k2 – 130k + 99 = 0 (4) Given that k is not an i
June 2017, Legacy C2, Q10 (10 marks): A B C O R y x Figure 2 Figure 2 shows a sketch of part of the curve with equation y = 4x3 + 9x2 – 30x – 8, –0.5 - x - 2.2 The curve has a turning point at the p
June 2016, Legacy C2, Q1 (7 marks): A geometric series has first term a and common ratio r = 3 4 The sum of the first 4 terms of this series is 175 (a) Show that a = 64 (2) (b) Find the sum to inf
June 2016, Legacy C2, Q2 (6 marks): The curve C has equation y = 8 – 2x–1, 0 - x - 4 (a) Complete the table below with the value of y corresponding to x = 1 x 0 1 2 3 4 y 7.5 6 4 0 (1) (b) Use the
June 2016, Legacy C2, Q3 (8 marks): Figure 2 The circle C has centre P(7, 8) and passes through the point Q(10, 13), as shown in Figure 2. (a) Find the length PQ, giving your answer as an exact va
June 2016, Legacy C2, Q4 (8 marks): f(x) = 6x3 + 13x2 – 4 (a) Use the remainder theorem to find the remainder when f(x) is divided by (2x + 3). (2) (b) Use the factor theorem to show that (x + 2)
June 2016, Legacy C2, Q5 (9 marks): (a) Find the first 3 terms, in ascending powers of x, of the binomial expansion of (2 – 9x)4 giving each term in its simplest form. (4) f(x) = (1 + kx)(2 – 9x)4
June 2016, Legacy C2, Q6 (9 marks): (i) Solve, for –ʌ < ș - ʌ, 1 2cos 0 5 π θ ⎛ ⎞ − − = ⎜ ⎟ ⎝ ⎠ giving your answers in terms of ʌ. (3) (ii) Solve, for 0 - x < 360q, 4cos2 x + 7sin x – 2 = 0 giving
June 2016, Legacy C2, Q7 (6 marks): Figure 3 Figure 3 shows a sketch of part of the curve with equation y x x x = − 3 0 3 2, . The finite region S, bounded by the x-axis and the curve, is shown sh
June 2016, Legacy C2, Q8 (7 marks): (i) Given that log3(3b + 1) – log3(a – 2) = –1, a 2 express b in terms of a. (3) (ii) Solve the equation 22x+5 – 7(2x) = 0 giving your answer to 2 decimal place
June 2016, Legacy C2, Q9 (15 marks): Figure 4 Figure 4 shows a plan view of a sheep enclosure. The enclosure ABCDEA, as shown in Figure 4, consists of a rectangle BCDE joined to an equilateral tria
June 2015, Legacy C2, Q1 (4 marks): Find the first 3 terms, in ascending powers of x, of the binomial expansion of 2 4 10 − ⎛ ⎝⎜ ⎞ ⎠⎟ x giving each term in its simplest form. (4)
June 2015, Legacy C2, Q2 (7 marks): A circle C with centre at the point (2, –1) passes through the point A at (4, –5). (a) Find an equation for the circle C. (3) (b) Find an equation of the tangen
June 2015, Legacy C2, Q3 (9 marks): f(x) = 6x3 + 3x2 + Ax + B, where A and B are constants. Given that when f(x) is divided by (x + 1) the remainder is 45, (a) show that B – A = 48 (2) Given also
June 2015, Legacy C2, Q4 (8 marks): Figure 1 Figure 1 shows a sketch of a design for a scraper blade. The blade AOBCDA consists of an isosceles triangle COD joined along its equal sides to sectors
June 2015, Legacy C2, Q5 (10 marks): (i) All the terms of a geometric series are positive. The sum of the first two terms is 34 and the sum to infinity is 162 Find (a) the common ratio, (4) (b) the
June 2015, Legacy C2, Q6 (9 marks): (a) Find 10 2 1 2 x x ( ) ∫ − dx giving each term in its simplest form. (4) Figure 2 Figure 2 shows a sketch of part of the curve C with equation y x x x = − 10
June 2015, Legacy C2, Q7 (9 marks): (i) Use logarithms to solve the equation 82x +1 = 24, giving your answer to 3 decimal places. (3) (ii) Find the values of y such that log2(11y – 3) – log2 3 – 2
June 2015, Legacy C2, Q8 (9 marks): (i) Solve, for 0 -ș ʌ, the equation sin3 3 cos3 0 θ θ − = giving your answers in terms of ʌ. (3) (ii) Given that 4 4 0 3 2 sin cos , x x k k + = − - - (a) find
June 2015, Legacy C2, Q9 (10 marks): A solid glass cylinder, which is used in an expensive laser amplifier, has a volume of 75 ʌ cm3. The cost of polishing the surface area of this glass cylinder i
June 2014, Legacy C2, Q1 (5 marks): Figure 1 Figure 1 shows a sketch of part of the curve with equation y = ¥(x2 + 1), x . 0 The finite region R, shown shaded in Figure 1, is bounded by the curve,
June 2014, Legacy C2, Q2 (6 marks): f(x) = 2x3 – 7x2 + 4x + 4 (a) Use the factor theorem to show that (x – 2) is a factor of f(x). (2) (b) Factorise f(x) completely. (4)
June 2014, Legacy C2, Q3 (7 marks): (a) Find the first 3 terms, in ascending powers of x, of the binomial expansion of (2 – 3x)6 giving each term in its simplest form. (4) (b) Hence, or otherwise,
June 2014, Legacy C2, Q4 (5 marks): Use integration to find 1 3 3 2 6 1 3 x x x + ⎛ ⎝⎜ ⎞ ⎠⎟ d giving your answer in the form a b + 3, where a and b are constants to be determined. (5)
June 2014, Legacy C2, Q5 (9 marks): Figure 2 The shape ABCDEA, as shown in Figure 2, consists of a right-angled triangle EAB and a triangle DBC joined to a sector BDE of a circle with radius 5 cm
June 2014, Legacy C2, Q6 (8 marks): The first term of a geometric series is 20 and the common ratio is 7 8 The sum to infinity of the series is S (a) Find the value of S (2) The sum to N terms o
June 2014, Legacy C2, Q7 (9 marks): (i) Solve, for 0 - ș < 360°, the equation 9sin(ș + 60°) = 4 giving your answers to 1 decimal place. You must show each step of your working. (4) (ii) Solve, for
June 2014, Legacy C2, Q8 (7 marks): (a) Sketch the graph of y = 3x, x \ showing the coordinates of any points at which the graph crosses the axes. (2) (b) Use algebra to solve the equation 32x –
June 2014, Legacy C2, Q9 (5 marks): Figure 3 Figure 3 shows a circle C with centre Q and radius 4 and the point T which lies on C. The tangent to C at the point T passes through the origin O and O
June 2014, Legacy C2, Q10 (14 marks): Figure 4 Figure 5 Figure 4 shows a closed letter box ABFEHGCD, which is made to be attached to a wall of a house. The letter box is a right prism of length y cm
June 2014, Legacy C2 (01R), Q1 (4 marks): Find the first 4 terms, in ascending powers of x, of the binomial expansion of 1 3 2 8 + ⎛ ⎝⎜ ⎞ ⎠⎟ x giving each term in its simplest form. (4)
June 2014, Legacy C2 (01R), Q2 (8 marks): A geometric series has first term a, where a 0, and common ratio r. The sum to infinity of this series is 6 times the first term of the series. (a) Show that
June 2014, Legacy C2 (01R), Q3 (5 marks): Figure 1 Figure 1 shows a sketch of part of the curve with equation y = ¥(2x – 1), x .0.5 The finite region R, shown shaded in Figure 1, is bounded by the curv
June 2014, Legacy C2 (01R), Q4 (7 marks): f(x) = –4x3 + ax2 + 9x – 18, where a is a constant. Given that (x – 2) is a factor of f(x), (a) find the value of a, (2) (b) factorise f(x) completely, (3) (c)
June 2014, Legacy C2 (01R), Q5 (6 marks): Figure 2 Figure 2 shows the shape ABCDEA which consists of a right-angled triangle BCD joined to a sector ABDEA of a circle with radius 7 cm and centre B. A, B
June 2014, Legacy C2 (01R), Q6 (7 marks): Figure 3 Figure 3 shows a sketch of part of the curve C with equation y x x x = + ∈ 1 8 3 4 3 2, The curve C has a maximum turning point at the point A and a mi
June 2014, Legacy C2 (01R), Q7 (8 marks): (i) Solve, for 0 - ș 180°, the equation sin 2 1 (4sin 2 1) θ θ = − giving your answers to 1 decimal place. (3) (ii) Solve, for 0 - x 2ʌ, the equation 5sin2 x –
June 2014, Legacy C2 (01R), Q8 (8 marks): (i) Solve 5 y = 8 giving your answer to 3 significant figures. (2) (ii) Use algebra to find the values of x for which log ( ) log 2 2 15 4 1 2 x x + − = (6)
June 2014, Legacy C2 (01R), Q9 (13 marks): 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 C
June 2014, Legacy C2 (01R), Q10 (9 marks): 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
June 2013, Legacy C2, Q1 (4 marks): The first three terms of a geometric series are 18, 12 and p respectively, where p is a constant. Find (a) the value of the common ratio of the series, (1) (b)
June 2013, Legacy C2, Q2 (5 marks): (a) Use the binomial theorem to find all the terms of the expansion of (2 + 3x)4 Give each term in its simplest form. (4) (b) Write down the expansion of (2 – 3
June 2013, Legacy C2, Q3 (9 marks): f(x) = 2x3 – 5x2 + ax + 18 where a is a constant. Given that (x – 3) is a factor of f(x), (a) show that a = – 9 (2) (b) factorise f(x) completely. (4) Given tha
June 2013, Legacy C2, Q4 (7 marks): y x = + 5 1 2 ( ) (a) Complete the table below, giving the missing value of y to 3 decimal places. x 0 0.5 1 1.5 2 2.5 3 y 5 4 2.5 1 0.690 0.5 (1) Figure 1 show
June 2013, Legacy C2, Q5 (9 marks): Figure 2 shows a plan view of a garden. The plan of the garden ABCDEA consists of a triangle ABE joined to a sector BCDE of a circle with radius 12m and centre
June 2013, Legacy C2, Q6 (8 marks): Figure 3 shows a sketch of part of the curve C with equation y = x(x + 4)(x – 2) The curve C crosses the x-axis at the origin O and at the points A and B. (a) W
June 2013, Legacy C2, Q7 (7 marks): (i) Find the exact value of x for which log2(2x) = log2(5x + 4) – 3 (4) (ii) Given that logay + 3loga 2 = 5 express y in terms of a. Give your answer in its sim
June 2013, Legacy C2, Q8 (11 marks): (i) Solve, for –180° x < 180°, tan(x – 40°) = 1.5 giving your answers to 1 decimal place. (3) (ii) (a) Show that the equation sintan = 3cos+ 2 can be wri
June 2013, Legacy C2, Q9 (9 marks): The curve with equation y = x2 – 32x) x has a stationary point P. Use calculus (a) to find the coordinates of P, (6) (b) to determine the nat
June 2013, Legacy C2, Q10 (6 marks): The circle C has radius 5 and touches the y-axis at the point (0, 9), as shown in Figure 4. (a) Write down an equation for the circle C, that is shown in Figure
June 2013, Legacy C2 (01R), Q1 (6 marks): Using calculus, find the coordinates of the stationary point on the curve with equation y x x x = + + > 2 3 8 0 2 , (6)
June 2013, Legacy C2 (01R), Q2 (5 marks): y x x = + √( ) 1 (a) Complete the table below with the value of y corresponding to x = 1.3, giving your answer to 4 decimal places. (1) x 1 1.1 1.2 1.3 1.4 1.5
June 2013, Legacy C2 (01R), Q3 (4 marks): Find the first 4 terms, in ascending powers of x, of the binomial expansion of 2 1 2 8 − ⎛ ⎝⎜ ⎞ ⎠⎟ x giving each term in its simplest form. (4)
June 2013, Legacy C2 (01R), Q4 (9 marks): f(x) = ax3 – 11x2 + bx + 4, where a and b are constants. When f(x) is divided by (x – 3) the remainder is 55 When f(x) is divided by (x + 1) the remainder is –9
June 2013, Legacy C2 (01R), Q5 (11 marks): The first three terms of a geometric series are 4p, (3p + 15) and (5p + 20) respectively, where p is a positive constant. (a) Show that 11p2 – 10p – 225 = 0 (4)
June 2013, Legacy C2 (01R), Q6 (9 marks): Given that log3 x = a, find in terms of a, (a) log3 (9x) (2) (b) log3 5 81 x ⎛ ⎝⎜ ⎞ ⎠⎟ (3) giving each answer in its simplest form. (c) Solve, for x, log ( ) lo
June 2013, Legacy C2 (01R), Q7 (9 marks): Figure 1 The line with equation y = 10 cuts the curve with equation y = x2 + 2x + 2 at the points A and B as shown in Figure 1. The figure is not drawn to scale
June 2013, Legacy C2 (01R), Q8 (10 marks): Figure 2 Figure 2 shows the design for a triangular garden ABC where AB = 7 m, AC = 13 m and BC = 10 m. Given that angle BAC = radians, (a) show that, to 3 de
June 2013, Legacy C2 (01R), Q9 (12 marks): (i) Solve, for 0 180 θ ° sin (2– 30°) + 1 = 0.4 giving your answers to 1 decimal place. (5) (ii) Find all the values of x, in the interval 0 360 x ° ,
January 2013, Legacy C2, Q1 (4 marks): Find the first 3 terms, in ascending powers of x, in the binomial expansion of ( ) 2 5 6 −x Give each term in its simplest form. (4)
January 2013, Legacy C2, Q2 (6 marks): f ( ) x ax bx x = + − − 3 2 4 3, where a and b are constants. Given that (x – 1) is a factor of f(x), (a) show that a + b = 7 (2) Given also that, when f(x) is
January 2013, Legacy C2, Q3 (9 marks): A company predicts a yearly profit of £120 000 in the year 2013. The company predicts that the yearly profit will rise each year by 5%. The predicted yearly pro
January 2013, Legacy C2, Q4 (7 marks): Solve, for 0 180° x , cos( ) . 3 10 0 4 x − = − o giving your answers to 1 decimal place. You should show each step in your working. (7)
January 2013, Legacy C2, Q5 (9 marks): The circle C has equation x y x y 2 2 20 24 195 0 + − − + = The centre of C is at the point M. (a) Find (i) the coordinates of the point M, (ii) the radius of t
January 2013, Legacy C2, Q6 (7 marks): Given that 2 15 6 2 2 log ( ) log x x + − = (a) Show that x x 2 34 225 0 − + = (5) (b) Hence, or otherwise, solve the equation 2 15 6 2 2 log ( ) log x x + − =
January 2013, Legacy C2, Q7 (12 marks): 9 cm X W Z Y 6 cm 4 cm Figure 1 The triangle XYZ in Figure 1 has XY = 6 cm, YZ = 9 cm, ZX = 4 cm and angle ZXY = . The point W lies on the line XY. The circu
January 2013, Legacy C2, Q8 (9 marks): The curve C has equation y x x = − − 6 3 4 3 , x ≠0 (a) Use calculus to show that the curve has a turning point P when x = 2 (4) (b) Find the x-coordinate of
January 2013, Legacy C2, Q9 (12 marks): Figure 2 The finite region R, as shown in Figure 2, is bounded by the x-axis and the curve with equation y x x x x = − − − > 27 2 9 16 0 2 √ , The curve crosses
June 2012, Legacy C2, Q1 (4 marks): Find the first 3 terms, in ascending powers of x, of the binomial expansion of (2 – 3x)5 giving each term in its simplest form. (4)
June 2012, Legacy C2, Q2 (5 marks): Find the values of x such that 2 log3 x – log3(x – 2) = 2 (5)
June 2012, Legacy C2, Q3 (11 marks): r T L x y O C Q P Figure 1 The circle C with centre T and radius r has equation x2 + y2 – 20x – 16y + 139 = 0 (a) Find the coordinates of the centre of C. (3) (
June 2012, Legacy C2, Q4 (6 marks): f(x) = 2x3 – 7x2 – 10x + 24 (a) Use the factor theorem to show that (x + 2) is a factor of f(x). (2) (b) Factorise f(x) completely. (4)
June 2012, Legacy C2, Q5 (12 marks): O y A B x y = 10x – x2 – 8 y = 10 – x R Figure 2 Figure 2 shows the line with equation y = 10 – x and the curve with equation y = 10x – x2 – 8 The line and the
June 2012, Legacy C2, Q6 (7 marks): (a) Show that the equation tan 2x = 5 sin 2x can be written in the form (1 – 5 cos 2x) sin 2x = 0 (2) (b) Hence solve, for 0 x 180°, tan 2x = 5 sin 2x givin
June 2012, Legacy C2, Q7 (6 marks): y = (3x + x) (a) Complete the table below, giving the values of y to 3 decimal places. x 0 0.25 0.5 0.75 1 y 1 1.251 2 (2) (b) Use the trapezium rule with all
June 2012, Legacy C2, Q8 (13 marks): x mm h mm Figure 3 A manufacturer produces pain relieving tablets. Each tablet is in the shape of a solid circular cylinder with base radius x mm and height h m
June 2012, Legacy C2, Q9 (11 marks): A geometric series is a + ar + ar2 + ... (a) Prove that the sum of the first n terms of this series is given by S a r r n n = − − ( ) 1 1 (4) The third and fift
January 2012, Legacy C2, Q1 (6 marks): A geometric series has first term a = 360 and common ratio r = 7 8 Giving your answers to 3 significant figures where appropriate, find (a) the 20th term of the
January 2012, Legacy C2, Q2 (4 marks): A circle C has centre ( , ) −1 7 and passes through the point ( , ). 0 0 Find an equation for C. (4)
January 2012, Legacy C2, Q3 (7 marks): (a) Find the first 4 terms of the binomial expansion, in ascending powers of x, of 1 4 8 + ( ) x giving each term in its simplest form. (4) (b) Use your expansi
January 2012, Legacy C2, Q4 (6 marks): Given that y x = 3 2, (a) show that log log 3 3 1 2 y x = + (3) (b) Hence, or otherwise, solve the equation 1 2 28 9 3 3 + = − log log ( ) x x (3)
January 2012, Legacy C2, Q5 (6 marks): f ( ) x x ax bx = + + + 3 2 3, where a and b are constants. Given that when f ( ) x is divided by ( ) x + 2 the remainder is 7, (a) show that 2 6 a b − = (2) Gi
January 2012, Legacy C2, Q6 (11 marks): Figure 1 Figure 1 shows the graph of the curve with equation y x x = − + 16 2 1 2 , x 0 > The finite region R, bounded by the lines x = 1, the x-axis and the cu
January 2012, Legacy C2, Q7 (12 marks): Figure 2 Figure 2 shows ABC, a sector of a circle of radius 6 cm with centre A. Given that the size of angle BAC is 0.95 radians, find (a) the length of the arc
January 2012, Legacy C2, Q8 (13 marks): Figure 3 Figure 3 shows a flowerbed. Its shape is a quarter of a circle of radius x metres with two equal rectangles attached to it along its radii. Each rectan
January 2012, Legacy C2, Q9 (10 marks): (i) Find the solutions of the equation sin( ) 3 15 1 2 x − = o , for which 0 180 - - x o (6) (ii) Figure 4 Figure 4 shows part of the curve with equation y ax b
June 2011, Legacy C2, Q1 (8 marks): f ( ) x x x x = − − + 2 7 5 4 3 2 (a) Find the remainder when f (x) is divided by (x –1). (2) (b) Use the factor theorem to show that (x+1) is a factor of f (x)
June 2011, Legacy C2, Q2 (6 marks): (a) Find the first 3 terms, in ascending powers of x, of the binomial expansion of ( ) 3 5 + bx where b is a non-zero constant. Give each term in its simplest f
June 2011, Legacy C2, Q3 (4 marks): Find, giving your answer to 3 significant figures where appropriate, the value of x for which (a) 5 10 x = , (2) (b) log ( ) 3 2 1 x − = −. (2)
June 2011, Legacy C2, Q4 (8 marks): The circle C has equation x y x y 2 2 4 2 11 0 + + − − = Find (a) the coordinates of the centre of C, (2) (b) the radius of C, (2) (c) the coordinates of the po
June 2011, Legacy C2, Q5 (7 marks): Figure 1 The shape shown in Figure 1 is a pattern for a pendant. It consists of a sector OAB of a circle centre O, of radius 6 cm, and angle AOB = π 3. The circ
June 2011, Legacy C2, Q6 (10 marks): The second and third terms of a geometric series are 192 and 144 respectively. For this series, find (a) the common ratio, (2) (b) the first term, (2) (c) the s
June 2011, Legacy C2, Q7 (10 marks): (a) Solve for 0 - x 360°, giving your answers in degrees to 1 decimal place, 3sin(x+45°) = 2 (4) (b) Find, for 0 - x 2ʌ, all the solutions of 2 2 7 2 sin
June 2011, Legacy C2, Q8 (11 marks): Figure 2 A cuboid has a rectangular cross-section where the length of the rectangle is equal to twice its width, x cm, as shown in Figure 2. The volume of the c
June 2011, Legacy C2, Q9 (11 marks): y x A B R O 4 y x = + 2 2 24 y x x = − + + Figure 3 The straight line with equation y = x+ 4 cuts the curve with equation y x x = − + + 2 2 24 at the points A a
January 2011, Legacy C2, Q1 (7 marks): f(x) = x4 + x3 + 2x2 + ax + b where a and b are constants. When f(x) is divided by (x – 1), the remainder is 7. (a) Show that a + b = 3. (2) When f(x) is divide
January 2011, Legacy C2, Q2 (6 marks): In the triangle ABC, AB = 11 cm, BC = 7 cm and CA = 8 cm. (a) Find the size of angle C, giving your answer in radians to 3 significant figures. (3) (b) Find the
January 2011, Legacy C2, Q3 (7 marks): The second and fifth terms of a geometric series are 750 and –6 respectively. Find (a) the common ratio of the series, (3) (b) the first term of the series, (2)
January 2011, Legacy C2, Q4 (7 marks): y C A O R B x Figure 1 Figure 1 shows a sketch of part of the curve C with equation y = (x + 1)(x – 5) The curve crosses the x-axis at the points A and B. (a) W
January 2011, Legacy C2, Q5 (4 marks): Given that , (a) write down the value of b. (1) In the binomial expansion of (1 + x)40, the coefficients of x4 and x5 are p and q respectively. (b) Find the val
January 2011, Legacy C2, Q6 (9 marks): (a) Complete the table below, giving the values of y to 2 decimal places. x 2 2.25 2.5 2.75 3 y 0.5 0.38 0.2 (2) (b) Use the trapezium rule, with all the values
January 2011, Legacy C2, Q7 (7 marks): (a) Show that the equation 2 2 3sin 7sin cos 4 x x x + = − can be written in the form 2 4sin 7sin 3 0 x x + + = (2) (b) Hence solve, for 0 - x < 360°, 2 2 3sin
January 2011, Legacy C2, Q8 (8 marks): (a) Sketch the graph of 7 , x y x = ∈ , showing the coordinates of any points at which the graph crosses the axes. (2) (b) Solve the equation 2 7 4(7 ) 3 0 x x
January 2011, Legacy C2, Q9 (10 marks): The points A and B have coordinates (–2, 11) and (8, 1) respectively. Given that AB is a diameter of the circle C, (a) show that the centre of C has coordinates
January 2011, Legacy C2, Q10 (10 marks): The volume V cm3 of a box, of height x cm, is given by 2 4 (5 ) , V x x = − 0 5 x (a) Find d d V x . (4) (b) Hence find the maximum volume of the box. (4) (c) U
June 2010, Legacy C2, Q1 (6 marks): y x x = + 3 2 (a) Complete the table below, giving the values of y to 2 decimal places. x 0 0.2 0.4 0.6 0.8 1 y 1 1.65 5 (2) (b) Use the trapezium rule, with al
June 2010, Legacy C2, Q2 (7 marks): f ( ) x x x x = − − + 3 5 58 40 3 2 (a) Find the remainder when ) ( f x is divided by ) 3 ( − x . (2) Given that ) 5 ( − x is a factor of ) ( f x , (b) find all
June 2010, Legacy C2, Q3 (4 marks): 2 y x k x = − √ , where k is a constant. (a) Find x y d d . (2) (b) Given that y is decreasing at 4 x = , find the set of possible values of k. (2)
June 2010, Legacy C2, Q4 (6 marks): (a) Find the first 4 terms, in ascending powers of x, of the binomial expansion of ( ) 1 7 + ax , where a is a constant. Give each term in its simplest form. (4
June 2010, Legacy C2, Q5 (6 marks): (a) Given that 5 2 sin cos θ θ = , find the value of tanθ . (1) (b) Solve, for 0 360 x ° - < , 5 x x 2 cos 2 2 sin = , giving your answers to 1 decimal place. (
June 2010, Legacy C2, Q6 (9 marks): Figure 1 Figure 1 shows the sector OAB of a circle with centre O, radius 9 cm and angle 0.7 radians. (a) Find the length of the arc AB. (2) (b) Find the area of
June 2010, Legacy C2, Q7 (7 marks): (a) Given that 3 3 2log ( 5) log (2 13) 1 x x − − − = , show that x x 2 16 64 0 − + = . (5) (b) Hence, or otherwise, solve 2 5 2 13 1 3 3 log ( ) log ( ) x x −
June 2010, Legacy C2, Q8 (9 marks): Figure 2 Figure 2 shows a sketch of part of the curve C with equation y x x kx = − + 3 2 10 , where k is a constant. The point P on C is the maximum turning poi
June 2010, Legacy C2, Q9 (10 marks): The adult population of a town is 25 000 at the end of Year 1. A model predicts that the adult population of the town will increase by 3% each year, forming a g
June 2010, Legacy C2, Q10 (11 marks): The circle C has centre A(2,1) and passes through the point B(10, 7). (a) Find an equation for C. (4) The line 1l is the tangent to C at the point B. (b) Find a
January 2010, Legacy C2, Q1 (4 marks): Find the first 3 terms, in ascending powers of x, of the binomial expansion of (3 – x)6 and simplify each term. (4)
January 2010, Legacy C2, Q2 (6 marks): (a) Show that the equation 5 sin x = 1 + 2 cos2 x can be written in the form 2 sin2 x + 5 sin x – 3 = 0 (2) (b) Solve, for 0 - x < 360°, 2 sin2 x + 5 sin x – 3
January 2010, Legacy C2, Q3 (9 marks): f(x) x ax bx = + + − 2 6 3 2 where a and b are constants. When f(x) is divided by (2x – 1) the remainder is –5. When f(x) is divided by (x + 2) there is no rema
January 2010, Legacy C2, Q4 (7 marks): Figure 1 An emblem, as shown in Figure 1, consists of a triangle ABC joined to a sector CBD of a circle with radius 4 cm and centre B. The points A, B and D lie
January 2010, Legacy C2, Q5 (8 marks): (a) Find the positive value of x such that log x 64 = 2 (2) (b) Solve for x log 2 (11 – 6x) = 2 log 2 (x – 1) + 3 (6)
January 2010, Legacy C2, Q6 (9 marks): A car was purchased for £18 000 on 1st January. On 1st January each following year, the value of the car is 80% of its value on 1st January in the previous year
January 2010, Legacy C2, Q7 (10 marks): Figure 2 The curve C has equation y x x = − + 2 5 4 . It cuts the x-axis at the points L and M as shown in Figure 2. (a) Find the coordinates of the point L and
January 2010, Legacy C2, Q8 (12 marks): Figure 3 Figure 3 shows a sketch of the circle C with centre N and equation (a) Write down the coordinates of N. (2) (b) Find the radius of C. (1) The chord AB
January 2010, Legacy C2, Q9 (10 marks): The curve C has equation y x x = − − 12 10 3 2 G ( ) , √ x > 0 (a) Use calculus to find the coordinates of the turning point on C. (7) (b) Find d d 2 2 y x . (2

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