Questions Q1. (a) Find the values of the constants A, B and C. (4) b) Hence find (ii) Find, leaving your answer in the form a + ln b, where a and b are constants. (6) (Total 10 marks) Q2. (a) Find the values of the constants A, B and C. (4) (b) (i) Hence find (3) (ii) Find in the form 1nk, where k is a constant. (3) (Total 10 marks) Q3. (a) Express
in partial fractions. (3) A team of conservationists is studying the population of meerkats on a nature reserve. The population is modelled by the differential equation 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, (b) solve the differential equation, giving your answer in the form, where a, b and c are integers. (8) (c) Hence show that the population cannot exceed 5000 (1) (Total 12 marks) Q4. (a) Express in partial fractions. (3) (b) Hence find dx, where x > 1. (3) (c) Find the particular solution of the differential equation (x 1)(3x + 2) = 5y, x > 1, for which y = 8 at x = 2. Give your answer in the form y = f (x). (6)
(Total 12 marks) Q5. Figure 2 shows a sketch of part of the curve C with parametric equations x=1 t, y = 2t 1 The curve crosses the y-axis at the point A and crosses the x-axis at the point B. (a) Show that A has coordinates (0, 3). (2) (b) Find the x coordinate of the point B. (2) (c) Find an equation of the normal to C at the point A. (5) The region R, as shown shaded in Figure 2, is bounded by the curve C, the line x = 1 and the x-axis. (d) Use integration to find the exact area of R. (6) (Total 15 marks) Q6.
Figure 1 Figure 1 shows the curve with equation The finite region S, shown shaded in Figure 1, is bounded by the curve, the x-axis and the line x =2 The region S is rotated 360 about the x-axis. Use integration to find the exact value of the volume of the solid generated, giving your answer in the form k ln a, where k and a are constants. (5) (Total 5 marks) Q7. (a) Find tan2x dx. (2) (b) Use integration by parts to find ln x dx. (4) (c) Use the substitution u = 1 + ex to show that dx = e2x ex + ln(1 + ex) + k, where k is a constant.
(7) (Total 13 marks) Q8. (a) Given that y =, complete the table below with values of y corresponding to x = 3 and x = 5. Give your values to 4 decimal places. (2) (b) Use the trapezium rule, with all of the values of y in the completed table, to obtain an estimate of I, giving your answer to 3 decimal places. (4) (c) Using the substitution x = (u 4)2 + 1, or otherwise, and integrating, find the exact value of I. (8) (Total 14 marks) Q9. Using the substitution u = cos x +1, or otherwise, show that (6) (Total 6 marks) Q10.
(a) Using the substitution x = 2 cos u, or otherwise, find the exact value of (7) Figure 3 Figure 3 shows a sketch of part of the curve with equation y =, 0 < x < 2. The shaded region S, shown in Figure 3, is bounded by the curve, the x-axis and the lines with equations x = 1 and x = 2. The shaded region S is rotated through 2π radians about the x-axis to form a solid of revolution. (b) Using your answer to part (a), find the exact volume of the solid of revolution formed. (3) (Total 10 marks) Q11. Using the substitution u = 2 + (2x + 1), or other suitable substitutions, find the exact value of giving your answer in the form A + 2ln B, where A is an integer and B is a positive constant. (8) (Total 8 marks) Q12.
Figure 1 Figure 1 shows part of the curve with equation x = 4te t + 3. The finite region R shown shaded in Figure 1 is bounded by the curve, the x-axis, the t-axis and the line t = 8. (a) Complete the table with the value of x corresponding to t = 6, giving your answer to 3 decimal places. (1) (b) Use the trapezium rule with all the values of x in the completed table to obtain an estimate for the area of the region R, giving your answer to 2 decimal places. (3) (c) Use calculus to find the exact value for the area of R. (6) (d) Find the difference between the values obtained in part (b) and part (c), giving your answer to 2 decimal places. (1) (Total 11 marks) Q13.
Figure 1 Figure 1 shows a sketch of part of the curve with equation y =, x>0 The finite region R, shown shaded in Figure 1, is bounded by the curve, the x-axis, and the lines with equations x = 1 and x = 4 The table below shows corresponding values of x and y for y = (a) Complete the table above by giving the missing value of y to 5 decimal places. (1) (b) Use the trapezium rule, with all the values of y in the completed table, to find an estimate for the area of R, giving your answer to 4 decimal places. (3) (c) By reference to the curve in Figure 1, state, giving a reason, whether your estimate in part (b) is an overestimate or an underestimate for the area of R. (1) (d) Use the substitution u = x, or otherwise, to find the exact value of (6) (Total 11 marks)
Q14. (i) Find x e4x dx (3) (ii) Find dx, x > 1 2 (2) Given that y = π 6 at x = 0, solve the differential equation dy dx = ex cosec2y cosecy (7) (Total 12 marks) Q15. Figure 1 Figure 1 shows a sketch of part of the curve with equation y = (2 x)e2x, x The finite region R, shown shaded in Figure 1, is bounded by the curve, the x-axis and the y-axis. The table below shows corresponding values of x and y for y = (2 x)e2x
(a) Use the trapezium rule with all the values of y in the table, to obtain an approximation for the area of R, giving your answer to 2 decimal places. (3) (b) Explain how the trapezium rule can be used to give a more accurate approximation for the area of R. (1) (c) Use calculus, showing each step in your working, to obtain an exact value for the area of R. Give your answer in its simplest form. (5) (Total 9 marks) Q16. Integrate the following with respect to x. (a) x2ln x (4) (b) sec 2x tan 2x + sec2x (3) Using the substitution u = 2 + cosθ, or otherwise, (c) find the exact value of giving your answer in the form a ln b + c, where a, b and c are constants to be found. (8) (Total 15 marks) Q17. (a) Use the substitution x = u2, u > 0, to show that
(3) (b) Hence show that where a and b are integers to be determined. (7) (Total 10 marks) Q18. (a) Find (2) (b) Given that y =1.5 at x = 2, solve the differential equation giving your answer in the form y = f(x). (6) (Total 8 marks) Q19.
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, the line with equation x = 1 and the line with equation x = 4. (a) Complete the table with the value of y corresponding to x = 3, giving your answer to 4 decimal places. (1) (b) Use the trapezium rule, with all the values of y in the completed table, to obtain an estimate of the area of the region R, giving your answer to 3 decimal places. (3) (c) Use the substitution u = 1 + x, to find, by integrating, the exact area of R. (8) (Total 12 marks) Q20.
Figure 3 Figure 3 shows a sketch of the curve with equation y =. The finite region R, shown shaded in Figure 3, is bounded by the curve and the x-axis. The table below shows corresponding values of x and y for y = (a) Complete the table above giving the missing value of y to 5 decimal places. (1) (b) Use the trapezium rule, with all the values of y in the completed table, to obtain an estimate for the area of R, giving your answer to 4 decimal places. (3) (c) Using the substitution u = 1 + cos x, or otherwise, show that where k is a constant. (5) (d) Hence calculate the error of the estimate in part (b), giving your answer to 2 significant figures. (3)
(Total 12 marks) Q21. Figure 2 shows a sketch of the curve with equation The finite region R, shown shaded in Figure 2, is bounded by the curve, the x-axis and the line x = 2. The table below shows corresponding values of x and y for y = x3 ln (x2 + 2). (a) Complete the table above giving the missing values of y to 4 decimal places. (2) (b) Use the trapezium rule, with all the values of y in the completed table, to obtain an estimate for the area of R, giving your answer to 2 decimal places. (3) (c) Use the substitution u = x2 + 2 to show that the area of R is (4) (d) Hence, or otherwise, find the exact area of R. (6)
(Total 15 marks) Mark Scheme Q1. Q2.
Q3.
Q4. Q5.
Q6. Q7.
Q8. Q9. Q10.
Q11.
Q12.
Q13.
Q14.
Q15.
Q16.
Q17.
Q18.
Q19.
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