| Exam Board | WJEC |
|---|---|
| Module | Unit 3 (Unit 3) |
| Year | 2019 |
| Session | June |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Arithmetic Sequences and Series |
| Type | Recurrence relation: find specific terms |
| Difficulty | Moderate -0.8 This question asks students to work backwards from x₃ using a linear recurrence relation (requiring simple algebraic manipulation), then identify the sequence type. The recurrence relation is linear and straightforward to reverse: x₂ = (113+3)/4 = 29, x₁ = (29+3)/4 = 8. Checking for arithmetic/geometric is routine. This is below average difficulty as it involves only basic algebraic manipulation and standard sequence classification with no problem-solving insight required. |
| Spec | 1.02l Modulus function: notation, relations, equations and inequalities1.02s Modulus graphs: sketch graph of |ax+b|1.02w Graph transformations: simple transformations of f(x)1.04e Sequences: nth term and recurrence relations1.04f Sequence types: increasing, decreasing, periodic1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc |
| 0 | 5 |
The $n$th term of a number sequence is denoted by $x _ { n }$. The $( n + 1 )$ th term is defined by $x _ { n + 1 } = 4 x _ { n } - 3$ and $x _ { 3 } = 113$.
a) Find the values of $x _ { 2 }$ and $x _ { 1 }$.\\
b) Determine whether the sequence is an arithmetic sequence, a geometric sequence or neither. Give reasons for your answer.\\
a) Express $5 \sin x - 12 \cos x$ in the form $R \sin ( x - \alpha )$, where $R > 0$ and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$.\\
b) Find the minimum value of $\frac { 4 } { 5 \sin x - 12 \cos x + 15 }$.\\
c) Solve the equation
$$5 \sin x - 12 \cos x + 3 = 0$$
for values of $x$ between $0 ^ { \circ }$ and $360 ^ { \circ }$.
\begin{center}
\begin{tabular}{ | l | l | }
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0 & 5 \\
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\end{tabular}
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a) Find the range of values of $x$ for which $| 1 - 3 x | > 7$.\\
b) Sketch the graph of $y = | 1 - 3 x | - 7$. Clearly label the minimum point and the points where the graph crosses the $x$-axis.
\hfill \mbox{\textit{WJEC Unit 3 2019 Q3}}