Pre-U Pre-U 9794/2 2017 June — Question 9 12 marks

Exam BoardPre-U
ModulePre-U 9794/2 (Pre-U Mathematics Paper 2)
Year2017
SessionJune
Marks12
TopicStationary points and optimisation
TypeFind stationary points coordinates
DifficultyStandard +0.8 This is a multi-step problem requiring integration by parts verification, then differentiation of a complex expression involving fractional powers, followed by solving the resulting equation. While the techniques are standard A-level content, the algebraic complexity and the need to recognize that dy/dx equals the integrand from part (i) makes this moderately challenging, above average difficulty.
Spec1.07n Stationary points: find maxima, minima using derivatives1.08i Integration by parts
9
  1. Show that \(\int x ( x - 2 ) ^ { \frac { 3 } { 2 } } \mathrm {~d} x = \frac { 2 } { 35 } ( 5 x + 4 ) ( x - 2 ) ^ { \frac { 5 } { 2 } } + c\).
  2. Hence find the coordinates of the stationary points of the curve $$y = \frac { 2 } { 35 } ( 5 x + 4 ) ( x - 2 ) ^ { \frac { 5 } { 2 } } + x ^ { 2 } - \frac { 1 } { 3 } x ^ { 3 }$$
Question 9: \(\int x(x-2)^{\frac{3}{2}}\,\mathrm{d}x\)
(i)
Method 1 (Substitution \(u = x-2\)):
- \(\int(u+2)u^{\frac{3}{2}}\,\mathrm{d}u\) M1\* Substitute \(u = x-2\); A1 Correct integrand in terms of \(u\)
- \(= \int u^{\frac{5}{2}} + 2u^{\frac{3}{2}}\,\mathrm{d}u\) M1d\* Expand brackets and attempt integration
- \(= \frac{2}{7}u^{\frac{7}{2}} + \frac{4}{5}u^{\frac{5}{2}} + c\) A1 Obtain correct integral (in terms of \(u\) or \(x\)) as long as consistent
- M1 Attempt to use algebraic highest common factor on expression of form \(au^{\frac{7}{2}} + bu^{\frac{5}{2}}\)
- \(= \frac{2}{35}(5x+4)(x-2)^{\frac{5}{2}} + c\) A1 Obtain correct integral AG
Method 2 (Integration by parts):
- \(\int x(x-2)^{\frac{3}{2}}\,\mathrm{d}x = \frac{2}{5}x(x-2)^{\frac{5}{2}} - \int\frac{2}{5}(x-2)^{\frac{5}{2}}\,\mathrm{d}x\) M1\* Attempt integration by parts; A1 Obtain correct expression
- M1d\* Attempt integration
- \(= \frac{2}{5}x(x-2)^{\frac{5}{2}} - \frac{4}{35}(x-2)^{\frac{7}{2}} + c\) A1 Obtain correct integral
- M1 Attempt to use algebraic highest common factor on expression of form \(ax(x-2)^{\frac{5}{2}} - b(x-2)^{\frac{7}{2}}\)
- \(= \frac{2}{35}(x-2)^{\frac{5}{2}}(5x+4) + c\) A1 Obtain correct integral AG
Method 3 (Differentiation verification):
- \(\frac{\mathrm{d}}{\mathrm{d}x}\left(\frac{2}{35}(x-2)^{\frac{5}{2}}(5x+4)+c\right) = \frac{5}{2}\times\frac{2}{35}(x-2)^{\frac{3}{2}}(5x+4) + \frac{2}{35}(x-2)^{\frac{5}{2}}\times 5\) M1\* Attempt use of product rule; A1 Obtain one correct term; A1 Obtain fully correct derivative
- M1d\* Attempt to use algebraic highest common factor
- \(= \frac{1}{7}(x-2)^{\frac{3}{2}}(5x+4+2(x-2))\) A1 Obtain correct unsimplified expression
- \(= x(x-2)^{\frac{3}{2}}\) A1 Obtain correct expression AG
(ii)
- \(\frac{\mathrm{d}y}{\mathrm{d}x} = x(x-2)^{\frac{3}{2}} - x^2 + 2x\) M1\* Attempt differentiation, using part (i)
- \(x(x-2)\left(\sqrt{x-2}-1\right) = 0\) A1 Obtain correct derivative
- M1d\* Equate to zero and attempt to solve, as far as non-zero value for \(x\) – allow inspection
- \(x = 0\) not valid B1 Obtain \(x = 0\) and deduce no solution oe
- \(x = 2,\ y = \frac{4}{3}\) A1 Obtain \(x=2\), \(y=\frac{4}{3}\)
- \(x = 3,\ y = \frac{38}{35}\) A1 Obtain \(x=3\), \(y=\frac{38}{35}\) (allow 1.09 or better)
Total: 12 marks
**Question 9: $\int x(x-2)^{\frac{3}{2}}\,\mathrm{d}x$**

**(i)**

**Method 1 (Substitution $u = x-2$):**
- $\int(u+2)u^{\frac{3}{2}}\,\mathrm{d}u$ **M1\*** Substitute $u = x-2$; **A1** Correct integrand in terms of $u$
- $= \int u^{\frac{5}{2}} + 2u^{\frac{3}{2}}\,\mathrm{d}u$ **M1d\*** Expand brackets and attempt integration
- $= \frac{2}{7}u^{\frac{7}{2}} + \frac{4}{5}u^{\frac{5}{2}} + c$ **A1** Obtain correct integral (in terms of $u$ or $x$) as long as consistent
- **M1** Attempt to use algebraic highest common factor on expression of form $au^{\frac{7}{2}} + bu^{\frac{5}{2}}$
- $= \frac{2}{35}(5x+4)(x-2)^{\frac{5}{2}} + c$ **A1** Obtain correct integral **AG**

**Method 2 (Integration by parts):**
- $\int x(x-2)^{\frac{3}{2}}\,\mathrm{d}x = \frac{2}{5}x(x-2)^{\frac{5}{2}} - \int\frac{2}{5}(x-2)^{\frac{5}{2}}\,\mathrm{d}x$ **M1\*** Attempt integration by parts; **A1** Obtain correct expression
- **M1d\*** Attempt integration
- $= \frac{2}{5}x(x-2)^{\frac{5}{2}} - \frac{4}{35}(x-2)^{\frac{7}{2}} + c$ **A1** Obtain correct integral
- **M1** Attempt to use algebraic highest common factor on expression of form $ax(x-2)^{\frac{5}{2}} - b(x-2)^{\frac{7}{2}}$
- $= \frac{2}{35}(x-2)^{\frac{5}{2}}(5x+4) + c$ **A1** Obtain correct integral **AG**

**Method 3 (Differentiation verification):**
- $\frac{\mathrm{d}}{\mathrm{d}x}\left(\frac{2}{35}(x-2)^{\frac{5}{2}}(5x+4)+c\right) = \frac{5}{2}\times\frac{2}{35}(x-2)^{\frac{3}{2}}(5x+4) + \frac{2}{35}(x-2)^{\frac{5}{2}}\times 5$ **M1\*** Attempt use of product rule; **A1** Obtain one correct term; **A1** Obtain fully correct derivative
- **M1d\*** Attempt to use algebraic highest common factor
- $= \frac{1}{7}(x-2)^{\frac{3}{2}}(5x+4+2(x-2))$ **A1** Obtain correct unsimplified expression
- $= x(x-2)^{\frac{3}{2}}$ **A1** Obtain correct expression **AG**

**(ii)**
- $\frac{\mathrm{d}y}{\mathrm{d}x} = x(x-2)^{\frac{3}{2}} - x^2 + 2x$ **M1\*** Attempt differentiation, using part (i)
- $x(x-2)\left(\sqrt{x-2}-1\right) = 0$ **A1** Obtain correct derivative
- **M1d\*** Equate to zero and attempt to solve, as far as non-zero value for $x$ – allow inspection
- $x = 0$ not valid **B1** Obtain $x = 0$ and deduce no solution oe
- $x = 2,\ y = \frac{4}{3}$ **A1** Obtain $x=2$, $y=\frac{4}{3}$
- $x = 3,\ y = \frac{38}{35}$ **A1** Obtain $x=3$, $y=\frac{38}{35}$ (allow 1.09 or better)

**Total: 12 marks**
9 (i) Show that $\int x ( x - 2 ) ^ { \frac { 3 } { 2 } } \mathrm {~d} x = \frac { 2 } { 35 } ( 5 x + 4 ) ( x - 2 ) ^ { \frac { 5 } { 2 } } + c$.\\
(ii) Hence find the coordinates of the stationary points of the curve

$$y = \frac { 2 } { 35 } ( 5 x + 4 ) ( x - 2 ) ^ { \frac { 5 } { 2 } } + x ^ { 2 } - \frac { 1 } { 3 } x ^ { 3 }$$

\hfill \mbox{\textit{Pre-U Pre-U 9794/2 2017 Q9 [12]}}
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