22M.1.SL.TZ1.2

pestleMathematicsAASLPaper 122M· sl-1-7-laws-of-exponents-and-logs, sl-5-5-integration-introduction-areas-between-curve-and-x-axissource ↗

The expression $\frac{3 \sqrt{x} - 5}{\sqrt{x}}$ can be written as $3 - 5 x^{p}$ . Write down the value of $p$ .

[1]

Hence, find the value of $\int_{1}^{9} (\frac{3 \sqrt{x} - 5}{\sqrt{x}}) d x$ .

[4]

Markscheme / solution

$\frac{3 \sqrt{x} - 5}{\sqrt{x}} = 3 - 5 x^{- \frac{1}{2}}$ A1

$p = - \frac{1}{2}$

[1 mark]

$\int \frac{3 \sqrt{x} - 5}{\sqrt{x}} d x = 3 x - 10 x^{\frac{1}{2}} (+ c)$ A1A1

substituting limits into their integrated function and subtracting (M1)

$3 (9) - 10 {(9)}^{\frac{1}{2}} - (3 (1) - 10 {(1)}^{\frac{1}{2}})$ OR $27 - 10 \times 3 - (3 - 10)$

$= 4$ A1

[4 marks]

Examiners’ report

Many candidates could give the value of p correctly. However, many did struggle with the integration, including substituting limits into the integrand, without integrating at all. An incorrect value of p often resulted in arithmetic of greater complexity.

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