CAIE A-Level Mathematics AS 1.8.2 Constant of Integration Questions

Practise recovering a curve from its derivative, retaining + c and using a given point or stationary-point condition to determine the constant before finding further coordinates.

Exam points

use vector notation and operations
solve geometric problems with vectors
interpret vector relationships

[Maximum number: 4]

A curve passes through the point $\left(\frac{4}{5},-3\right)$ and is such that $\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{-20}{(5 x-3)^{2}}$ ．

Find the equation of the curve．

Integrate to obtain form $k(5 x-3)^{-1}$

*M1

OE

$4(5 x-3)^{-1}$
Or unsimplified equivalent.
Condone absence of $\ldots+c$ so far.

Substitute $x=\frac{4}{5}$ and y=-3 to attempt value of c
DM0 for substituting $\left(-3, \frac{4}{5}\right)$ .

$y=4(5 x-3)^{-1}-7$ allow f(x) or $f=4(5 x-3)^{-1}-7$
OE
Condone c=-7 as the final answer providing
y= or $f(x)=\frac{4}{(5 x-3)}+c$ OE is seen earlier.
Attempts to write equation in y=m x+c form
scores A0. Do not ISW. Gains max 3/4.

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CAIE A-Level Mathematics AS 1.8.2 Constant of Integration Questions question 1