2015 H2 Mathematics Paper 1 Question 1

Equations and Inequalities

Answers

(i)

a=−3.593,  {a=-3.593,\;}b=−5.187,  {b=-5.187,\;}c=7.303{c=7.303}

(ii)

x=−0.589{x= -0.589}

(iii)

y=−5.187x+7.303{y=-5.187x + 7.303}

Full solutions

(i)

C{C} passes through (1.6,−2.4){(1.6, -2.4)} and (−0.7,3.6):{(-0.7, 3.6):}
11.62a+1.6b+c=−2.41(−0.7)2a−0.7b+c=3.6\begin{align} && \quad \frac{1}{1.6^2} a + 1.6 b + c &= -2.4 \\ && \quad \frac{1}{(-0.7)^2} a -0.7 b + c &= 3.6 \\ \end{align}
dydx=−2ax3+b\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{-2a}{x^3} + b
Gradient of C{C} is 2{2} where x=1:{x=1:}
−2a+b=2\begin{align} && \quad -2a + b &= 2 \\ \end{align}
Solving (1),(2){(1), (2)} and (3){(3)} with a GC,
a=−3.593 (3 dp)  ■b=−5.187 (3 dp)  ■c=7.303 (3 dp)  ■\begin{align*} a&=-3.593 \textrm{ (3 dp)} \; \blacksquare \\ b&=-5.187 \textrm{ (3 dp)} \; \blacksquare \\ c&=7.303 \textrm{ (3 dp)} \; \blacksquare \end{align*}

(ii)

Using a GC, when C{C} crosses the x-{x\textrm{-}}axis,
x=−0.589 (3 dp)  ■x=-0.589 \textrm{ (3 dp)} \; \blacksquare

(iii)

Other (oblique) asymptote of C:{C:}
y=bx+cy=−5.187x+7.303  ■\begin{gather*}y=bx+c\\y=-5.187x + 7.303 \; \blacksquare\end{gather*}