WJEC Unit 3 Specimen — Question 1 4 marks

Exam BoardWJEC
ModuleUnit 3 (Unit 3)
SessionSpecimen
Marks4
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Mark schemeDownload PDF ↗
TopicSmall angle approximation
TypeEstimate root of equation
DifficultyStandard +0.3 This is a straightforward numerical methods question requiring iteration or graphical approximation to find a root. It involves standard A-level techniques (substitution into an equation, possibly rearrangement) with minimal conceptual challenge—slightly easier than average since it's a direct application of a single method with no proof or multi-step reasoning required.
Spec1.05a Sine, cosine, tangent: definitions for all arguments1.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.09a Sign change methods: locate roots
Find a small positive value of \(x\) which is an approximate solution of the equation. $$\cos x - 4\sin x = x^2.$$ [4]
AnswerMarks Guidance
\(1 - \frac{x^2}{2} - 4x = x^2\)M1 (Attempt to substitute for \(\cos x, \sin x\))
\(\frac{3x^2}{2} + 4x - 1 = 0\)A1 (Correct)
\(3x^2 + 8x - 2 = 0\)B1
\(x = \frac{-8 \pm \sqrt{64 + 24}}{6} = \frac{-8 \pm \sqrt{88}}{6}\)B1
\(x = 0.230(1385...), (-2.896805...)\)B1
Total: [4]
$1 - \frac{x^2}{2} - 4x = x^2$ | M1 | (Attempt to substitute for $\cos x, \sin x$)

$\frac{3x^2}{2} + 4x - 1 = 0$ | A1 | (Correct)

$3x^2 + 8x - 2 = 0$ | B1 |

$x = \frac{-8 \pm \sqrt{64 + 24}}{6} = \frac{-8 \pm \sqrt{88}}{6}$ | B1 |

$x = 0.230(1385...), (-2.896805...)$ | B1 |

**Total: [4]**
Find a small positive value of $x$ which is an approximate solution of the equation.
$$\cos x - 4\sin x = x^2.$$ [4]

\hfill \mbox{\textit{WJEC Unit 3  Q1 [4]}}
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