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Date May 2010 Marks available 8 Reference code 10M.2.hl.TZ1.10
Level HL only Paper 2 Time zone TZ1
Command term Determine and Find Question number 10 Adapted from N/A

Question

The diagram below shows the graphs of y=|32x3|, y=3 and a quadratic function, that all intersect in the same two points.


 

Given that the minimum value of the quadratic function is −3, find an expression for the area of the shaded region in the form t0(ax2+bx+c)dx, where the constants a, b, c and t are to be determined. (Note: The integral does not need to be evaluated.)

Markscheme

|32x3|=0 when x = 2     (A1)

the equation of the parabola is y=p(x2)23     (M1)

through (0, 3)3=4p3p=32     (M1)

the equation of the parabola is y=32(x2)23 (=32x26x+3)     A1

area =220(332x)(32x26x+3)dx     M1M1A1

Note: Award M1 for recognizing symmetry to obtain 220,

M1 for the difference,

A1 for getting all parts correct.

 

=20(3x2+9x)dx     A1

[8 marks]

Examiners report

This was a difficult question and, although many students obtained partial marks, there were few completely correct solutions.

Syllabus sections

Topic 6 - Core: Calculus » 6.5 » Area of the region enclosed by a curve and the x-axis or y-axis in a given interval; areas of regions enclosed by curves.
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