Standard +0.3 This question requires sketching a parabola and a V-shaped modulus function, then solving by considering cases where the modulus expression is positive/negative. While it involves multiple steps and careful case analysis, these are standard techniques for modulus equations at A-level. The algebraic manipulation is straightforward once cases are identified, making it slightly easier than average.
7. (a) Sketch on the same diagram in the space provided the graphs of \(y = 4 a ^ { 2 } - x ^ { 2 }\) and \(y = | 2 x - a |\), where \(a\) is a positive constant. Show, in terms of \(a\), the coordinates of any points where each graph meets the coordinate axes.
(b) Find the exact solutions of the equation
$$4 - x ^ { 2 } = | 2 x - 1 |$$
7. (a) Sketch on the same diagram in the space provided the graphs of $y = 4 a ^ { 2 } - x ^ { 2 }$ and $y = | 2 x - a |$, where $a$ is a positive constant. Show, in terms of $a$, the coordinates of any points where each graph meets the coordinate axes.\\
(b) Find the exact solutions of the equation
$$4 - x ^ { 2 } = | 2 x - 1 |$$
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\hfill \mbox{\textit{SPS SPS FM 2022 Q7 [12]}}