8 (a) It is given that e tan y x x 3 2 = + . Use calculus to find the approximate change in y as x increases from 0.1 to . h 0 1+ , where h is small. [5] Ĭ Ċ Ġ í õ Ī Ċ ü Ā (b) A curve is such that ( ) sin x y x 3 d d r = + . The curve passes through the point , 9 3 4 r e o. Find the exact y-coordinate of the point on the curve where x 12 5r = . [4] , ,

8(a) 3 2 3 2 2 d 3e tan e sec d + + = + x x y x x x 3 B1 for 3 2 3 2 d(e ) 3e d + + = x x x M1 for correct structure of product rule A1 FT their 3 2 3 2 d(e ) 3e d + + = x x x for all other components of product rule correct Correct small changes relationship e.g. 0.1 δ d d = = x y y their h x M1 13.1h or 13.07[68…]h A1 dep on first three marks awarded 8(b) 1 : cos(3 π) 3 − + y x + c B2 B1 for 1cos(3 π) 3 − + x or for cos(3 π) + k x +c where k is1 3 or k < 0 4 1 3π cos π 3 3 9   = − + +     c M1 FT their k providing B1 has been awarded 1 5π 7 7 2 cos π 3 4 6 6   −   = − + + =         y or exact equivalent A1

2 Find ( ) d x x x 1 1 1 1 2 3 5 - - - e o y , giving your answer in the form ln a b + , where a and b are rational numbers. [5]

2 > x B1 2 ( ) 5 3 1 ln 1 1   − +   −   x x 2 B1 for ( ) ln 1 − x B1 for 1 1 + − x 1 1 ln 4 ln 2 4 2     + − +         M1 Dep on at least one B mark, for correct use of limits 1 ln2 4 − 2 A1 for ln2 A1 for 1 4 − oe To earn marks, you must correctly integrate the given expression, showing the correct form for the natural logarithm term and the power term. Then, substitute the limits of integration correctly into your integrated expression and simplify to get the final answer in the specified form, with separate marks for the rational number and the natural logarithm term.

2 (a) Find d d e x x x 2 3 ` j. [3] (b) (i) Find d d x x 3 4 2 3 1 + ` j . [2] (ii) Hence find dx x x 3 4 2 0 2 3 2 + - ` j y . [3]

2(a) ( ) 2 3 3 3e 2 e + x x x x 3 M1 for differentiation of a product A1 for ( ) 2 3 3e x x A1 for 3 2 e + x x 2(b)(i) ( ) 2 2 3 2 3 4 − + x x 2 M1 for ( ) 2 2 3 3 4 − + kx x 2(b)(ii) ( ) 2 1 2 3 0 1 3 4 2   +       x M1 For ( ) 1 2 3 3 4 + k x 1 1 3 3 1 1 16 4 2 2       −                   M1 Dep for correct substitution of limits into their integral 0.466 A1 (a) To earn marks, you must correctly differentiate the product of the two functions. One mark is for correctly differentiating the first part of the product, and another for the second part. (b)(i) You need to correctly apply the power rule for differentiation, specifically for a term like (ax+b)^n, to get the correct form. (b)(ii) To score marks, you must first correctly integrate the expression from part (i), then substitute the given limits into your integrated expression and calculate the final numerical value.

2 A particle moves in a straight line such that its velocity, v ms 1 - , at time t seconds after passing through a fixed point O, is given by v 25 e t3 = - . Find the speed of the particle when t 1 = . [2]

2 4.91 =− v soi B1 Speed = 4.91 B1

4 (a) Show that x x 2 5 3 2 + - can be written in the form a x b c 2 + + ` j , where a, b and c are constants. [3] (b) Hence write down the coordinates of the stationary point on the curve with equation y x x 2 5 3 2 = + - . [2] (c) On the axes below, sketch the graph of y x x 2 5 3 2 = + - , stating the coordinates of the intercepts with the axes. [3] y x O (d) Write down the value of k for which the equation x x k 2 5 3 2 + - = has exactly 3 distinct solutions. [1]

4(a) 2 5 49 2 4 8   + −     x 3 B1 for 2 5 4   = +     b x or ( ) 2 1.25 + x B1 for 49 8 = − c or 6.125 − 4(b) 5 49 , 4 8   − −     oe 2 B1 for 5 4 − as part of a set of coordinates or 5 4 = − x , FT on − their b B1 for 49 8 − as part of a set of coordinates or 49 8 = − y FT on their c Need to be using their answer to (a) and not using differentiation as ‘Hence’. B1 for 5 49 , 4 8 − − 4(c) 3 B1 for correct shape, with maximum in the second quadrant and cusps on the x-axes and reasonable curvature for 3 x B1 for ( ) 3, 0 − and ( ) 0.5, 0 either seen on the graph or stated, must have attempted a correct shape B1 for ( ) 0, 3 either seen on the graph or stated, must have attempted a correct shape 4(d) 49 8 oe B1 FT on their c from (a) Allow 49 8 from other methods To earn marks, you need to correctly identify the 'b' term (the coefficient of x inside the squared term) and the 'c' term (the constant outside the squared term) when rewriting the expression in the specified form. To earn marks, you must use your answer from part (a) to directly state the x and y coordinates of the stationary point. The x-coordinate comes from the term inside the squared bracket, and the y-coordinate comes from the constant term outside the bracket. To earn marks, you must draw a graph with the correct 'W' shape, showing a maximum in the second quadrant and cusps on the x-axis. You also need to correctly label the x-intercepts and the y-intercept, either on the graph or stated separately. To earn the mark, you need to state the value of k that corresponds to the y-coordinate of the maximum point of the graph, which is the 'c' value from part (a).