6 4 (a) Show that (b) Find © UCLES 2022 y 5 2 2x + 3 1 4 can be written as . + 2x - 1 (2x - 1) 2 (2x - 1) 2 [1] 2x + 3 dx , giving your answer in the form a + 1n b , where a and b are constants. [5] (2x - 1) 2 0606/12/F/M/22 7 5 Variables x and y are such that y = (a) Find dy . dx 1n (2x 2 - 3) . 3x [3] (b) Hence find the approximate change in y when x increases from 2 to 2 + h , where h is small. [2] (c) At the instant when x = 2 , y is increasing at the rate of 4 units per second. Find the corresponding rate of increase in x. [2] © UCLES 2022 0606/12/F/M/22 [Turn over 8 6 r The normal to the curve y = 1 + tan 3x at the point P with x-coordinate , meets the x-axis at the 12 point Q. The line x = © UCLES 2022 r 12 meets the x-axis at the point R. Find the area of the triangle PQR. 0606/12/F/M/22 [8] 9 7 d2y - 31 . The curve passes through the point (- 2, 10.2) . The 2 = (2 - 3x) dx gradient of the tangent to the curve at (- 2, 10.2) is –6. Find f (x) . [8] A curve y = f (x) is such that © UCLES 2022 0606/12/F/M/22 [Turn over 13 11 y A B y = 6 + e 4x − 5 0 2 x The diagram shows part of the graphs of y = 6 + e 4x - 5 and x = 2 . The line x = 2 meets the curve at the point B(2, b) and the line AB is parallel to the x-axis. Find the area of the shaded region. [7] © UCLES 2022 0606/22/F/M/22 [Turn over 14 12 In this question all lengths are in centimetres. O R P A C 12 Q h B 8 The diagram shows a right triangular prism of height h inside a right pyramid. The pyramid has a height of 12 and a base that is an equilateral triangle, ABC, of side 8. The base of the prism sits on the base of the pyramid. Points P, Q and R lie on the edges OA, OB and OC, respectively, of the pyramid OABC. Pyramids OABC and OPQR are similar. (a) Show that the volume, V, of the triangular prism is given by b and c are integers to be found. © UCLES 2022 0606/22/F/M/22 V= 3 (ah 3 + bh 2 + ch) where a, 9 [4] 15 (b) It is given that, as h varies, V has a maximum value. Find the value of h that gives this maximum value of V. [3] © UCLES 2022 0606/22/F/M/22 14 9 A circle, centre O and radius r cm, has a sector OAB of fixed area 10cm 2 . Angle AOB is i radians and the perimeter of the sector is P cm. (a) Find an expression for P in terms of r. [3] (b) Find the value of r for which P has a stationary value. [3] (c) Determine the nature of this stationary value. [2] (d) Find the value of i at this stationary value. [1] © UCLES 2022 0606/11/M/J/22 15 10 The normal to the curve y = tan b3x + l at the point P with coordinates (p, -1), where 0 1 p G , 6 2 meets the x-axis at the point A and the y-axis at the point B. Find the exact coordinates of the mid-point of AB. [10] r r © UCLES 2022 0606/11/M/J/22 9 7 (a) Show that 8 - 3x 2 1 1 can be written as . + 2x + 3 x - 1 (x - 1) 2 (x - 1) 2 (2x + 3) [2] 8 - 3x dx where a 2 2 . Give your answers in the form c + ln d , where c and d 2 2 (x - 1) (2x + 3) are functions of a. [6] (b) Find © UCLES 2022 y a 0606/12/M/J/22 [Turn over 11 9 ln (3x 2 + 2) , at the point A on the curve where x = 0 , meets the x-axis at x+1 point B. Point C has coordinates (0, 3 ln 2) . Find the gradient of the line BC in terms of ln 2. [9] The normal to the curve y = © UCLES 2022 0606/12/M/J/22 [Turn over 14 11 In this question all lengths are in kilometres and time is in hours. A particle P moves in a straight line such that its displacement, s, from a fixed point at time t is given by s = (t + 2) (t - 5) 2 , for t H 0 . (a) Find the values of t for which the velocity of P is zero. [4] (b) On the axes, draw the displacement–time graph for P for 0 G t G 6 , stating the coordinates of the points where the graph meets the coordinate axes. [2] s 0 © UCLES 2022 1 2 3 0606/12/M/J/22 4 5 6 t 15 (c) On the axes below, draw the velocity–time graph for P for 0 G t G 6 , stating the coordinates of the points where the graph meets the coordinate axes. [2] velocity 0 1 2 3 4 5 t 6 (d) (i) Write down an expression for the acceleration of P at time t. [1] (ii) Hence, on the axes below, draw the acceleration–time graph for P for 0 G t G 6 , stating the coordinates of the points where the graph meets the coordinate axes. [2] acceleration 0 © UCLES 2022 1 2 3 0606/12/M/J/22 4 5 6 t
