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Simultaneous linear equations: NYJC 2024 Promo Q1

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Nanyang Junior College2024 PromoPromo Q1●○○ Foundational4 marks

What this question tests

Systems of linear equations.

Question

A curve $C$ has equation $y = \frac{a}{x^2} + b\mathrm{e}^{2x} + c,$ where $a$ , $b$ and $c$ are constants.

It is given that $C$ passes through the point with coordinates $\left(1, 2\mathrm{e}^2 - 1\right)$ . When $C$ undergoes a scaling of factor $2$ parallel to the $x$ -axis, the transformed curve passes through the point with coordinates $\left(-1, 8 + \dfrac{2}{\mathrm{e}}\right)$ . When $C$ undergoes a translation by negative $1$ unit in the direction of the $y$ -axis, the transformed curve passes through the point with coordinates $\left(\sqrt{3}, 2\mathrm{e}^{2\sqrt{3}} - 4\right)$ . Find the values of $a$ , $b$ and $c$ . [4]

Show full worked solution▾

(a)

$y=\frac{a}{x^2}+b\mathrm{e}^{2x}+c$ Sub $x=1$ , $y=2\mathrm{e}^2-1$ $a+b\mathrm{e}^2+c=2\mathrm{e}^2-1 \text{ --- (1)}$ After scaling, $y=\frac{a}{\left(\frac{x}{2}\right)^2}+b\mathrm{e}^{2\left(\frac{x}{2}\right)}+c =\frac{4a}{x^2}+b\mathrm{e}^x+c$ Sub $x=-1$ , $y=8+\dfrac{2}{\mathrm{e}}$ $4a+b\mathrm{e}^{-1}+c=8+\frac{2}{\mathrm{e}} \text{ --- (2)}$ After translation $y=\frac{a}{x^2}+b\mathrm{e}^{2x}+c-1$ Sub $x=\sqrt{3},\quad y=2\mathrm{e}^{2\sqrt{3}}-4$ $\frac{a}{3}+b\mathrm{e}^{2\sqrt{3}}+c=2\mathrm{e}^{2\sqrt{3}}-3 \text{ --- (3)}$ Using GC, $a=3,\quad b=2,\quad c=-4$

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