SPM.1.SL.TZ0.10

pestleMathematicsAISLPaper 1SPM· sl-5-5-integration-introduction-areas-between-curve-and-x-axissource ↗

The following diagram shows part of the graph of $f(x) = \left( {6 - 3x} \right)\left( {4 + x} \right)$ , $x \in \mathbb{R}$ . The shaded region R is bounded by the $x$ -axis, $y$ -axis and the graph of $f$ .

Write down an integral for the area of region R.

[2]

Find the area of region R.

[1]

The three points A(0, 0) , B(3, 10) and C( $a$ , 0) define the vertices of a triangle.

Find the value of $a$ , the $x$ -coordinate of C, such that the area of the triangle is equal to the area of region R.

[2]

Markscheme / solution

A = $\int_0^2 {\left( {6 - 3x} \right)\left( {4 + x} \right){\text{d}}x}$ A1A1

Note: Award A1 for the limits $x$ = 0, $x$ = 2. Award A1 for an integral of $f(x)$ .

[2 marks]

28 A1

[1 mark]

$28 = 0.5 \times a \times 10$ M1

$5.6\left( {\frac{{28}}{5}} \right)$ A1

[2 marks]

Examiners’ report

It was pleasing to see that, for those candidates who made a reasonable attempt at the paper, many were able to identify the correct values on the tree diagram.

[N/A]