O-LevelMathematicsAlgebra and graphs4024/22 · May/June 2020 · Q714 Marks

The table shows some values for y = 4^x . [Table] x: 0, 0.5, 1, 1.5, 2, 2.5, 3. y: [blank], [blank], 4, 8, 16, 32, 64.

(a)The table shows some values for y = 4^x . [Table] x: 0, 0.5, 1, 1.5, 2, 2.5, 3. y: [blank], [blank], 4, 8, 16, 32, 64.
(a)(i)Complete the table.[1]
(a)(ii)Draw the graph of y = 4^x for 0 ≤ x ≤ 3. [Figure 7.1, Graph grid][3]
(a)(iii)By drawing a tangent, estimate the gradient of the curve when x = 2.[2]
(a)(iv)The solutions of the equation 3(4^x)+ax+b = 0 can be found from the points of intersection of y = 4^x and y = 20x−12.
(a)(iv)(a)Find the value of a and the value of b.[2]
(a)(iv)(b)By drawing the line y = 20x-12 on the grid opposite, find all the solutions of 3(4^x) + ax+b = 0 .[3]
(b)[Figure 7.2] Here is a sketch of the graph of a quadratic function. The curve has a maximum point (p, q). Find the value of p and the value of q.[3]

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Mark scheme

Marking points from the official mark scheme (May/June 2020). The mark or mark type is shown in brackets.

[1] (a)(i) 1,2
[3] (a)(ii) Correct curve
[3] B2FT for 6 or 7 points correctly plotted or B1FT for 4 or 5 points correctly plotted
[2] (a)(iii) Tangent drawn at (2, 16)
[2] B1 18 to 27
[2] B1 Dependent on correct tangent or close attempt
[2] (a)(iv)(a) a=-60, b=36
[2] B1 for either correct or 3(4^x)-60x+36 [= 0]
[3] (a)(iv)(b) y = 20x - 12 ruled line
[3] M2 M1 for one correct coordinate soi
[3] 0.7 to 0.8, 2.65 to 2.75
[3] B1
[3] (b) p=1
[3] B1 q=9
[3] B2 M1 for [y =] (4−x)(x + 2) oe or [y=] q−(x – 1)² oe or two correct equations in x and y using (-2, 0), (4, 0) or (0, 8) or SC1 for q = _9

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