A-LevelMathematicsDifferentiation9709/21 · May/June 2025 · Q610 Marks

The parametric equations of a curve are x = (2t+1)/(3t+4), y = 2ln(3t+4), where t > -4/3.

(a)Show that dy/dx can be expressed in the form c(3t+4) and state the value of the constant c.[5]
(b)It is given that the gradient of the curve at the point (a, ln100) is m. Find the values of a and m.[4]
(c)State whether the curve represents a decreasing function or an increasing function or neither. Give a reason for your answer.[1]

Mark scheme

Marking points from the official mark scheme (May/June 2025). The mark or mark type is shown in brackets.

[M1] Attempt use of quotient rule (or equivalent) for derivative of x.
[A1] Obtain 5/(3t + 4)² or unsimplified equivalent.
[B1] Obtain 6/(3t + 4) for derivative of y (Allow unsimplified).
[M1] Attempt to express dy/dx in terms of t.
[A1] Obtain 6/5 (3t + 4) stating or implying that c = 6/5 (Form of answer given. Necessary detail needed).
[M1] Attempt solution of 2ln(3t + 4) = ln100 (Must be using logarithms correctly).
[A1] Obtain 3t+4=10 (or equivalent) and hence t = 2 (only) (May be implied by only one (correct) value for a).
[A1] Substitute to obtain a = 1/2.
[A1] Substitute to obtain m=12.
[B1] State increasing, with reference to positive gradient.

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