C4 "International A-level" (150 minute) papers: June 2014 and Specimen 1. C4 INTERNATIONAL A LEVEL PAPER JUNE 2014 1. f(x) = 2x 3 + x 10 (a) Show that the equation f(x) = 0 has a root in the interval [1.5, 2] (2) The only real root of f(x) = 0 is The iterative formula 1 3 xn+ 1 = xn x 5, 0 = 15. 2 can be used to find an approximate value for 1 (b) Calculate x 1, x 2 and x 3, giving your answers to 4 decimal places. (c) By choosing a suitable interval, show that = 1.6126 correct to 4 decimal places. (2) 2 *P44969A0248*
2. A curve C has the equation x 3 3xy x + y 3 11 = 0 Find an equation of the tangent to C at the point (2, 1), giving your answer in the form ax + by + c = 0, where a, b and c are integers. (6) 4 *P44969A0448*
3. Given that y cos 2θ =, 1 + sin2θ π 3π < θ < 4 4 show that dy a =, dθ 1 + sin2θ π 3π < θ < 4 4 where a is a constant to be determined. (4) 6 *P44969A0648*
4. Find (a) ( 2 x + 3) 12 dx (2) 5x (b) 2 4x + 1 dx (2) 8 *P44969A0848*
1 3 < 3 5. f( x) = ( 8 + 27x ), x Find the first three non-zero terms of the binomial expansion of f(x) in ascending powers of x. Give each coefficient as a simplified fraction. (5) 2 3 10 *P44969A01048*
6. (a) Express 5 4x ( 2x 1)( x + 1) in partial fractions. (b) (i) Find a general solution of the differential equation dy ( 2x 1)( x + 1) = ( 5 4xy ), dx x> 1 2 Given that y = 4 when x = 2, (ii) find the particular solution of this differential equation. Give your answer in the form y = f(x). (7) 12 *P44969A01248*
7. The function f is defined by x f : x 3 5, x, x x + 1 1 (a) Find an expression for f 1 (x) (b) Show that x + a ff( x) =, x, x 1, x 1 x 1 where a is an integer to be determined. (4) The function g is defined by g : x x 2 3x, x R, 0 x 5 (c) Find the value of fg(2) (2) (d) Find the range of g 16 *P44969A01648*
8. The volume V of a spherical balloon is increasing at a constant rate of 250 cm 3 s 1. Find the rate of increase of the radius of the balloon, in cm s 1, at the instant when the volume of the balloon is 12 000 cm 3. Give your answer to 2 significant figures. (5) [ ay a a V a a by 4 3 a V = πr.] 3 20 *P44969A02048*
9. y R O 4 9 x Figure 1 Figure 1 shows a sketch of part of the curve with equation y x = e, x > 0 The finite region R, shown shaded in Figure 1, is bounded by the curve, the x-axis and the lines x = 4 and x = 9 (a) Use the trapezium rule, with 5 strips of equal width, to obtain an estimate for the area of R, giving your answer to 2 decimal places. (4) (b) Use the substitution u = x to find, by integrating, the exact value for the area of R. (7) 24 *P44969A02448*
10. (a) Use the identity for sin(a + B) to prove that (b) Show that sin2a 2sinA cos A d dx [ln(tan( 1 2 x))] = cosecx (4) (2) A curve C has the equation y = ln(tan( 1 2 x)) 3sin x, 0 < x < (c) Find the x coordinates of the points on C where d y dx = 0 Give your answers to 3 decimal places. ( u ba y a ca u ca a acc ab ) (6) 28 *P44969A02848*
11. y O Q P C x Figure 2 Figure 2 shows a sketch of part of the curve C with equation where a is a constant and a > ln4 y = e a 3x 3e x, x The curve C has a turning point P and crosses the x -axis at the point Q as shown in Figure 2. (a) Find, in terms of a, the coordinates of the point P. (b) Find, in terms of a, the x coordinate of the point Q. (6) (c) Sketch the curve with equation y = e a 3x 3e x, x, a > ln 4 Show on your sketch the exact coordinates, in terms of a, of the points at which the curve meets or cuts the coordinate axes. 32 *P44969A03248*
12. y C S O 3 x Figure 3 Figure 3 shows a sketch of part of the curve C with parametric equations 2 x tan t, y 2sin t, 0 t = = < The finite region S, shown shaded in Figure 3, is bounded by the curve C, the line x = 3 and the x-axis. This shaded region is rotated through 2 radians about the x-axis to form a solid of revolution. (a) Show that the volume of the solid of revolution formed is given by π 4 3 2 (tan 2 sin )d 0 π t t t (b) Hence use integration to find the exact value for this volume. π 2 (6) (6) 36 *P44969A03648*
13. (a) Express 2sin + cos in the form Rsin ( ), where R and are constants, R > 0 and 0 < < 90. Give your value of to 2 decimal places. 2 m 1 4 m 4 m C D E Figure 4 2 Figure 4 shows the design for a logo that is to be displayed on the side of a large building. The logo consists of three rectangles, C, D and E, each of which is in contact with two horizontal parallel lines 1 and 2. Rectangle D touches rectangles C and E as shown in Figure 4. Rectangles C, D and E each have length 4 m and width 2 m. The acute angle between the line 2 and the longer edge of each rectangle is shown in Figure 4. Given that 1 and 2 are 4 m apart, (b) show that 2sin + cos = 2 (2) Given also that 0 < < 45, (c) solve the equation 2sin + cos = 2 giving the value of to 1 decimal place. Rectangles C and D and rectangles D and E touch for a distance m as shown in Figure 4. Using your answer to part (c), or otherwise, (d) find the value of, giving your answer to 2 significant figures. 40 *P44969A04048*
14. Relative to a fixed origin O, the line has vector equation where is a scalar parameter. r 1 2 = 4 + λ 1 6 1 Points A and B lie on the line, where A has coordinates (1, a, 5) and B has coordinates (b, 1, 3). (a) Find the value of the constant a and the value of the constant b. (b) Find the vector AB (2) The point C has coordinates (4, 3, 2) (c) Show that the size of the angle CAB is 30 (d) Find the exact area of the triangle CAB, giving your answer in the form k 3, where k is a constant to be determined. (2) The point D lies on the line so that the area of the triangle CAD is twice the area of the triangle CAB. (e) Find the coordinates of the two possible positions of D. (4) 44 *P44969A04448*
C4 INTERNATIONAL A LEVEL SPECIMEN PAPER 1. (a) Express 5cos2 12sin2 in the form Rcos(2 + ), where R > 0 and 0 < < 90 Give the value of to 2 decimal places. (b) Hence solve, for 0 < 180, the equation 5cos2 12sin2 = 10 giving your answers to 1 decimal place. (5) 58 *S45000A0244* Pearson Edexcel International Pearson Education Limited 2013 Sample Assessment Materials Advanced Level in Mathematics
2. y Q R O Figure 1 π x Figure 1 shows a sketch of the curve with equation y x = e sin x, 0 x π. The finite region R, shown shaded in Figure 1, is bounded by the curve and the x-axis. (a) Complete the table below with the values of y corresponding to x = 4 π and x = 2 π, giving your answers to 5 decimal places. x 0 π 4 π 3π 2 4 π y 0 8.87207 0 (2) (b) Use the trapezium rule, with all the values of y in the completed table, to obtain an estimate for the area of the region R. Give your answer to 4 decimal places. The curve y Figure 1. x = e sin x, 0 x π, has a maximum turning point at Q, shown in (c) Find the x coordinate of Q. (6) 60 *S45000A0444* Pearson Edexcel International Pearson Education Limited 2013 Sample Assessment Materials Advanced Level in Mathematics
3. Using the substitution u = cos x + 1, or otherwise, show that π 2 0 e (cos x + 1) sin x dx = e(e 1) (6) 64 *S45000A0844* Pearson Edexcel International Pearson Education Limited 2013 Sample Assessment Materials Advanced Level in Mathematics
4. (a) Use the binomial theorem to expand (2 3x) 2, ½ x ½ < 2 3 in ascending powers of x, up to and including the term in x 3. Give each coefficient as a simplified fraction. (5) f(x) = a + bx, ½ x ½ < 2, where a and b are constants. ( 2 3x) 2 3 In the binomial expansion of f(x), in ascending powers of x, the coefficient of x is 0 and the coefficient of x 2 is 9 16 Find (b) the value of a and the value of b, (5) (c) the coefficient of x 3, giving your answer as a simplified fraction. 66 *S45000A01044* Pearson Edexcel International Pearson Education Limited 2013 Sample Assessment Materials Advanced Level in Mathematics
5. The functions f and g are defined by f : x e x + 2, x g : x 2 ln x, x > 0 (a) Find fg(x), giving your answer in its simplest form. (b) Find the exact value of x for which f(2x + 3) = 6 (c) Find f 1, stating its domain. (4) (d) On the same axes, sketch the curves with equation y = f(x) and y = f 1 (x), giving the coordinates of all the points where the curves cross the axes. (4) 70 *S45000A01444* Pearson Edexcel International Pearson Education Limited 2013 Sample Assessment Materials Advanced Level in Mathematics
6. The curve C has equation 16y 3 + 9x 2 y x = 0 (a) Find d y dx in terms of x and y. (5) (b) Find the coordinates of the points on C where d y dx = 0 (7) 74 *S45000A01844* Pearson Edexcel International Pearson Education Limited 2013 Sample Assessment Materials Advanced Level in Mathematics
7. (a) Show that nπ cot x cot 2x cosec 2x, x, n 2 (5) (b) Hence, or otherwise, solve for 0 π π π 1 cosec 3θ + + cot 3θ + = 3 3 3 You must show your working. (Solutions based entirely on graphical or numerical methods are not acceptable.) (5) 78 *S45000A02244* Pearson Edexcel International Pearson Education Limited 2013 Sample Assessment Materials Advanced Level in Mathematics
8. h( x) = 2 4 18 +, x 0 2 2 x + 2 x + 5 ( x + 5)( x + 2) (a) Show that h( x) = 2x 2 x + 5 (4) x) in its simplest form. y y = h(x) O Figure 2 x Figure 2 shows a graph of the curve with equation y = h(x). (c) Calculate the range of h(x). (5) 82 *S45000A02644* Pearson Edexcel International Pearson Education Limited 2013 Sample Assessment Materials Advanced Level in Mathematics
9. The line l 1 has equation r = 2 1 3 + 2, where is a scalar parameter. 4 1 The line l 2 has equation r = 0 5 9 + 0, where is a scalar parameter. 3 2 Given that l 1 and l 2 meet at the point C, find (a) the coordinates of C. The point A is the point on l 1 where = 0 and the point B is the point on l 2 where = 1 (b) Find the size of the angle ACB. Give your answer in degrees to 2 decimal places. (4) (c) Hence, or otherwise, find the area of the triangle ABC. (5) 86 *S45000A03044* Pearson Edexcel International Pearson Education Limited 2013 Sample Assessment Materials Advanced Level in Mathematics
10. y l P C S O Figure 3 Q x Figure 3 shows part of the curve C with parametric equations x = tan, y = sin, 0 2 π The point P lies on C and has coordinates 3 1, 2 3 (a) Find the value of at the point P. (2) The line l is a normal to C at P. The normal cuts the x-axis at the point Q. (b) Show that Q has coordinates ( k 30, ), giving the value of the constant k. (6) The finite shaded region S shown in Figure 3 is bounded by the curve C, the line x = 3 and the x-axis. This shaded region is rotated through 2π radians about the x-axis to form a solid of revolution. (c) Find the volume of the solid of revolution, giving your answer in the form π 3 + qπ 2, where p and q are constants. (7) 90 *S45000A03444* Pearson Edexcel International Pearson Education Limited 2013 Sample Assessment Materials Advanced Level in Mathematics
11. A team of conservationists is studying the population of meerkats on a nature reserve. The population is modelled by the differential equation dp dt = 1 15 P(5 P), t 0 where P, in thousands, is the population of meerkats and t is the time measured in years since the study began. Given that when t = 0, P = 1, (a) solve the differential equation, giving your answer in the form P = a b + ce 1 t 3 where a, b and c are integers. (11) (b) Hence show that the population cannot exceed 5000 (1) 96 *S45000A04044* Pearson Edexcel International Pearson Education Limited 2013 Sample Assessment Materials Advanced Level in Mathematics