Standard +0.8 Part (a) requires algebraic manipulation to prove an inequality involving surds (squaring both sides). Part (b) is more demanding: students must complete the square for both circles to find centres and radii, then use the distance formula and apply a geometric criterion (distance between centres plus smaller radius less than larger radius) to prove one circle lies inside another. This requires synthesis of multiple techniques and geometric insight beyond routine circle problems.
11.
a) Without using a calculator, show that \(5 > 3 \sqrt { 2 }\)
b) Two circles \(C _ { 1 }\) and \(C _ { 2 }\) have equations
$$x ^ { 2 } + y ^ { 2 } + 6 x - 5 y = \frac { 39 } { 4 } \text { and } x ^ { 2 } + y ^ { 2 } + 2 x - y = \frac { 3 } { 4 }$$
respectively. Show that \(C _ { 2 }\) lies completely inside \(C _ { 1 }\)
[0pt]
End of Examination
11.\\
a) Without using a calculator, show that $5 > 3 \sqrt { 2 }$\\
b) Two circles $C _ { 1 }$ and $C _ { 2 }$ have equations
$$x ^ { 2 } + y ^ { 2 } + 6 x - 5 y = \frac { 39 } { 4 } \text { and } x ^ { 2 } + y ^ { 2 } + 2 x - y = \frac { 3 } { 4 }$$
respectively. Show that $C _ { 2 }$ lies completely inside $C _ { 1 }$\\[0pt]
End of Examination
\hfill \mbox{\textit{SPS SPS FM 2022 Q11 [6]}}