O-LevelMathematicsAlgebra and graphs4024/22 · May/June 2016 · Q812 Marks

The table below shows some values of x and the corresponding values of y for y = ¼ x 2^x. x | 0 | 1 | 2 | 3 | 4 | 5 --|-----|---|---|---|---|--- y | ¼ | 1 | | 2 | 4 | 8

(a)Complete the table.[1]
(b)On the grid below, draw the graph of y = ¼ x 2^x. [Figure 8.1][2]
(c)By drawing a suitable line, find the gradient of your graph where x = 4.[2]
(d)(i)Show that the line 2x + y = 6, together with the graph of y = ¼ x 2^x, can be used to solve the equation 2^x + 8x - 24 = 0.[1]
(d)(ii)Hence solve 2^x + 8x - 24 = 0.[2]
(e)(i)The points P and Q are (2, 3) and (5, 4) respectively. Find the gradient of PQ.[1]
(e)(ii)On the grid, draw the line l, parallel to PQ, that touches the curve y = ¼ x 2^x. [Figure 8.1][1]
(e)(iii)Write down the equation of l.[2]

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Mark scheme

Marking points from the official mark scheme (May/June 2016). The mark or mark type is shown in brackets.

[1] (a) 0.5
[2] (b) Correct graph with smooth curve
[2] (b) B1 for at least 4 correct points
[2] (c) Tangent drawn and gradient = 2.3 to 3.0
[2] (c) B1 for tangent drawn at x = 4
[2] (c) or B1 for gradient 2.3 to 3.0
[1] (d)(i) Correct method to eliminate y and reaching the given equation without error including at least one intermediate line
[1] (d)(i) isw
[2] (d)(ii) 2.3 to 2.4 dep on line drawn
[2] (d)(ii) B1 for 2x + y = 6 drawn
[1] (e)(i) 1/3 or 0.33..
[1] (e)(ii) Tangent gradient roughly 1/3
[2] (e)(iii) y = 1/3 x + k oe where 0 < k < 0.25
[2] (e)(iii) Ft from their e(i)
[2] (e)(iii) B1 for 1/3 x + k oe where 0 < k < 0.25
[2] (e)(iii) or y = 1/3 x + k oe (any k outside range)

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