Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 2 (P2-9709/02) | Year 2019 | Feb-Mar | (P2-9709/22) | Q#3

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Question

The variables x and y satisfy the equation y = Aepx+p, where A and p are constants.  The graph of ln y against x is a straight line passing through the points (1, 2.835) and  (6, 6.585), as shown in the diagram.

Find the values of A and p.

Solution

We are given;

Taking natural logarithm of both sides;

Multiplication Rule;

Since  for any ;

Slope-Intercept form of the equation of the line;

Where is the slope of the line.

Comparing the standard equation with above obtained equation, it is evident that it is  equation of a straight line and;

We are given that a straight line passing through the points (1, 2.835) and (6, 6.585)  is line of graph against .

We can find slope of this line to equate with slope of line whose equation is obtained  above.

Expression for slope of a line joining points  and ;

Therefore;

Similarly we can equate the y-intercepts of given line through the points (1, 2.835)  and (6, 6.585).

The point at which curve (or line) intercepts y-axis, the value of .

It is evident that coordinates of y-intercept will be (0,y).

It is also evident that slope of the line obtained from point (1, 2.835) and y-intercept  (0,y) will be same.

Expression for slope of a line joining points  and ;

Therefore;

Hence coordinates of y-intercept point are (0, 2.085).

Therefore

Therefore;

Taking anti-logarithm of both sides;

 for any

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