O-LevelAdditional MathematicsLogarithmic and exponential functions4037/22 · May/June 2016 · Q109 Marks
The graph of the curve y = p(4^(2x)) - q(4^x) passes through the points (0, 2) and (0.5, 14). Find the value of p and of q.
- (a)The graph of the curve y = p(4^(2x)) - q(4^x) passes through the points (0, 2) and (0.5, 14). Find the value of p and of q.[3]
- (b)The variables x and y are connected by the equation y = 10^(2x) - 2(10^x). Using the substitution u = 10^x, or otherwise, find the exact value of x when y = 24.[3]
- (c)Solve log₂(x + 1) – log₂x = 3.[3]
Mark scheme
Marking points from the official mark scheme (May/June 2016). The mark or mark type is shown in brackets.
- [M1] 2= p-q and 14 = 4p-2q oe
- [A1] p = 5
- [A1] q = 3
- [M1] Factorise 10^(2x) – 2(10^x) – 24 [= 0] or factorise u² – 2u – 24 [= 0] or applies the formula or completes the square
- [A1] 10^x = 6 ignore 10^x = -4 for this mark
- [A1] x = lg6 cao as final answer or exact equivalent
- [M2] combines logs and anti-logs or B1 for one correct log move e.g. log₂( (x+1)/x ) = 3 or log₂(x+1)-log₂(x) = log₂ 8 or log₂(x+1)-log₂(x) = 3log₂ 2
- [A1] x = 1/7 or 0.143 or 0.1428 to 0.1429
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