Past common Test 5AAHL Term 3 GDC required 1. [Maximum mark: 6] Do not use calculator for this question Find the exact value of ∫ 3 2 1 2 + sin ( 3πx ) dx (2 x − 3) + x +1 2. [Maximum mark: 7] (a) Show that 5𝑥𝑥−1 𝑥𝑥 2 −4𝑥𝑥−5 ≡ 𝐴𝐴 𝑥𝑥+1 + 𝐵𝐵 𝑥𝑥−5 ,where 𝐴𝐴 and 𝐵𝐵 have to be found. 2 (b) Hence find the value of k such that ∫1 3. [Maximum mark: 7] (a) Evaluate ∫ 𝑥𝑥tan2 𝑥𝑥 d𝑥𝑥. 5𝑥𝑥−1 𝑥𝑥 2 −4𝑥𝑥−5 [3] d𝑥𝑥 = ln𝑘𝑘 . [4] [5] m (b) Determine the value of m if ∫0 𝑥𝑥tan2 𝑥𝑥 d𝑥𝑥 = 0.5, where m > 0 . 4. [Maximum mark: 6] Find ∫ ln(ln 𝑥𝑥) 𝑥𝑥 [2] 𝑑𝑑𝑑𝑑 , where 𝑥𝑥 > 1. 5. [Maximum mark: 7] By using the substitution 𝑥𝑥 3 = 4 cosec θ , show that ∫ 6. [Maximum mark: 5] ( x) Find from first principles the derivative of f = 7. [Maximum mark: 7] d𝑥𝑥 𝑥𝑥√𝑥𝑥 6 −16 =− 1 12 4 arc sin � 3 � + 𝑐𝑐 𝑥𝑥 2x − 3 π 3 A curve has equation arcsin( x 2 ) + arcsin( y 2 ) = . dy in terms of x and y . dx 1 dy (b) Find the value of when x = , y<0 . dx 2 (a) Find [4] [3] 8. [Maximum mark: 4] x . 2x +1 1 Using first principles, show that f ′( x) = . (2 x + 1) 2 The function f is defined by f ( x) = Page 1 of 8 9. [Maximum mark: 7] 2 The curve C is defined by equation x ln y = 2 x − 1, y > 0. (a) Find dy in terms of x and y. dx (b) Find the value of [4] dy at the point on C where y = e and x > 0. dx [3] 10. [Maximum mark: 7] Differentiate the following with respect to x ( ) (a) y = x 5 x (b) y = x2 arctan kx 11. [Maximum mark: 5] By using differentiation from first principles, obtain the derivative of f ( x) = x . 2x −1 12. [Maximum mark: 8] The curve below shows the graph of y = f ( x) . (a) Find the value of the following: (i) [3] f ′(−2) (ii) f ′(1.5) Page 2 of 8 13. [Maximum mark: 11] x . y A curve C is defined implicitly by the equation xy = y 2 + ln dy . dx (a) Find an expression for (b) Find the equation of the normal to the curve C at the point ( 2, 2 ) . [4] [3] Set P contains the x-coordinates of all the intersection points between the graph of C and y = x . (c) By showing clear reasoning, determine the elements of set P. [2] (d) Find the asymptotes of the graph of C. [2] 14. [Maximum mark: 6] Find the coordinates of the points on the curve 3𝑥𝑥 2 + 𝑥𝑥𝑥𝑥 + 𝑦𝑦 2 = 33 at which the tangent is parallel to the 𝑥𝑥-axis. 15. [Maximum mark: 9] 3 x dx ∫ tan x sec = sec3 x +c. 3 (a) Show that (b) Using the substitution 𝑥𝑥 = 2 tan 𝑢𝑢 , find ∫ 𝑥𝑥√4 + 𝑥𝑥 2 𝑑𝑑𝑑𝑑 . 16. [Maximum mark: 7] [3] [6] 𝜋𝜋 Using integration by parts, find ∫02 e−2𝑥𝑥 sin 𝑥𝑥 d𝑥𝑥 in exact form. 17. [Maximum mark: 10] Find an expression for (a) dy for the following: dx y = x sin y [4] 1 x [6] 2 (b) arctan = y + ln( xy ) 18. [Maximum mark: 5] Let y = cos x . Show that (a) 2 y dy + sin x = 0, dx (b) 2 y d2 y dy + 2 + y2 = 0. 2 dx dx [2] 2 [3] Page 3 of 8 19. [Maximum mark: 7] Diagram not to scale In the diagram, POQ is a rail where OQ is horizontal and ∠𝑃𝑃𝑃𝑃𝑃𝑃 = 2𝜋𝜋 3 . AB is a straight rod of length 3 metres which is free to slide on the rail with one end A on OP and the other end B on OQ, till end B is at point O. End A is initially at point O and end B is pushed towards O at a constant speed of √3 3 𝑚𝑚𝑠𝑠 −1 . After 𝑡𝑡 seconds, B is 𝑥𝑥 metres away from point O and the rod makes angle 𝜃𝜃 radians with the horizontal. (a) (b) Express 𝑥𝑥 in terms of 𝜃𝜃 . [2] Let 𝑆𝑆 𝑚𝑚2 be the area of ∆𝐴𝐴𝐴𝐴𝐴𝐴 . Show that 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝜋𝜋 = 3√3 sin � − 2𝜃𝜃� . 20. [Maximum mark: 4] 3 [5] The equation of the gradient at any point of a curve is given by: dy = e2 x + x 2 . dx If the y-intercept of the curve is 6, find its equation. 21. [Maximum mark: 7] A B 3 − x2 = 2 + 2 (a) Given that , find the values of A and B 2 2 ( x + 1)( x + 9 ) x + 1 x + 9 where 𝐴𝐴, 𝐵𝐵 ∈ ℚ . 3 − x2 (b) Hence obtain the indefinite integral of ∫ 2 dx ( x + 1)( x 2 + 9 ) [2] [5] Page 4 of 8 22. [Maximum mark: 11] Consider the function f ( x) = xe − x for x ≥ 0. (a) (i) Find an expression for f ′( x) . (ii) Determine the coordinates of the point A where f ′( x) = 0 . [3] (b) (i) Find an expression for f ′′( x) . (ii) Hence show that A is a maximum point. [4] (c) Find the coordinates of B, the point of inflexion on f ( x) . [4] 23. [Maximum mark: 7] Using integration by substitution with x = sin u , find 3 k Hence or otherwise, find k such that 1 ∫ 1− 9x 0 2 1 ∫ 1− 9x 2 dx . dx = 0.4 . 24. [Maximum mark: 7] Using integration by parts, show that π ∫x 2 ( −1) cos nx dx = n 0 n 2π 2 for n ∈ + . 25. [Maximum mark: 6] Find dy y if ln ( x 2 + y 2 ) = arctan dx x 26. [Maximum mark: 5] (Do not use a calculator for this question) 2 −x The graph of the function defined by f ( x) = xe has a local minimum at the point A and a local maximum at the point B. Find the coordinates of the points A and B. 27. [Maximum mark: 6] Find the equation of the normal to the curve whose equation is given by x2 + 2 y − 2 = e xy at the point where y = 0 and x > 0. 28. [Maximum mark: 6] Consider the curve y = points of inflection. d2 y 1+ x . Find . Hence show that the curve has three dx 2 1 + x2 Page 5 of 8 29. [Maximum mark: 8] x cm B A 10 cm C In the diagram above, triangle ABC has AC = 10 cm, AB = x cm, and angle ACB = θ π radian. Line AC is horizontal and the point B is moving away 6 π radians with the from point A at a constant rate of 2 cms − 1 , making an angle of 6 radians, and angle BAC = horizontal. (a) Find the value of x when θ = π radians. 2 5π 10sin θ . (b) Show that x sin − θ = 6 (c) Using parts (a) and (b), find the rate of change of θ when θ = [2] [2] π radians. 2 [4] 30. [Maximum mark: 11] Find (a) ∫ ln(e 2 x ) dx x2 [2] (b) ∫ cot α dα [2] (c) ∫3 dx [3] (d) ∫ ln x dx x2 [4] 2 2 −3 x 31. [Maximum mark: 5] Find the remainder when x 2 + 6 is divided by x 2 + 2 x + 10 . Hence, find x2 + 6 ∫ x 2 + 2 x + 10 dx Page 6 of 8 32. [Maximum mark: 2] Given that ∫ 4 1 f ( x) dx = 6, find the value of ∫ 4 1 f (5 − x) dx . 33. [Maximum mark: 6] Using the substitution x = sin y , find the exact value of 1 2 0 ∫ x dx . 1− x 34. [Maximum mark: 6] Find ∫e sin x sin 2 x dx . 35. [Maximum mark: 14] The graph of the first derivative f ′ of a function f is shown below. (a) Find all possible x -coordinate(s) of the maximum point(s). Justify your answer. [2] (b) State the range of values of x for which the function is decreasing. [3] (c) State the x -coordinates of the inflection points of f . Give reasons for your answers. [3] (d) State the range of values of x for which the function f is (i) Concave upwards (ii) Concave downwards (e) Sketch the graph of f ′′( x) for 0 ≤ x ≤ 5 . [3] [3] Page 7 of 8 36. [Maximum mark: 7] A cone is formed by joining the two straight edges of a sector from a circle of radius r. Let θ be the angle between the two straight edges. r θ r (a) Show that the radius of the cone is rθ . 2π (b) Find an expression for the height of the cone in terms of r and θ . [2] [2] (c) Given that θ varies, find the value of θ which makes the volume of the cone a maximum. [3] 37. [Maximum mark: 8] (a) Find the root of the equation e 2− 2 x = 2e − x . (b) The curve = y e 2− 2 x − 2e − x has a minimum point. Find the coordinates of this minimum. (c) The curve = y e 2− 2 x − 2e − x is shown below. Write down the coordinates of the points A, B and C. k (d) Hence state the set of values of k for which the equation e 2− 2 x − 2e − x = has two distinct positive roots. [2] [3] [1] [2] Page 8 of 8