A-LevelMathematicsDifferentiation9709/22 · Oct/Nov 2025 · Q78 Marks

A curve has equation 5x²y +4e² - 7x+10 = 0.

(a)Find an expression in terms of x and y for dy dx and hence find the gradient of the curve at the point for which y = 0. [6]
(b)Show that there is no point on the curve at which the tangent is parallel to the y-axis. [2]

Mark scheme

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

[6] (a) Attempt use of product rule for differentiating 5x²y
[6] (a) Obtain 10xy + 5x² ⁿ ⁿ
[6] (a) Differentiate 4e² and obtain 8e²ⁿ ⁿ ⁿ
[6] (a) Obtain 10xy + 5x² ⁿ ⁿ + 8e²ⁿ ⁿ ⁿ − 7 = 0 and hence ⁿ ⁿ = ⁿ−ⁿⁿⁿ ⁿⁿⁿ² +ⁿⁿⁿ
[6] (a) Attempt to find relevant value of x and substitute that and y = 0 to find gradient
[6] (a) Obtain x=2 and gradient ¼
[2] (b) Attempt to show that denominator of derivative cannot be zero
[2] (b) State that both terms are non-negative or equivalent and conclude appropriately

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