| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2002 |
| Session | November |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Inequalities |
| Type | Complete the square technique |
| Difficulty | Moderate -0.3 This is a structured multi-part question testing standard completing the square technique and its applications. Parts (i)-(iii) are routine algebraic manipulation, (iv) requires understanding of one-one functions (vertex location), and (v) involves finding an inverse function. All parts follow predictable patterns with no novel problem-solving required, making it slightly easier than average but still requiring multiple techniques. |
| Spec | 1.02e Complete the square: quadratic polynomials and turning points1.02v Inverse and composite functions: graphs and conditions for existence |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(2x^2 + 8x - 10 = 2(x+2)^2 + c\). \(c = -18\) | B1 B1, B1 | \(a = 2\) gets B1, \(b = 2\) gets B1; correct only |
| (ii) Least value = \(-18\) when \(x = -2\) | B1∨/B1∨ | follow through for c and for \(-b\). Calculus ok |
| (iii) \(2x^2 + 8x - 10 \geq 14\) or \(2(x+2)^2 - 18 \geq 14\). \(x^2 + 4x - 12 = 0\) or \((x+2)^2 = 16\). Limit points 2 and \(-6\). \(x \geq 2\) and \(x \leq -6\) | M1, A1, A1 | setting the inequality to 0; correct only – irrespective of what they do; correct only (condone \(>\) or \(<\)) |
| (iv) Smallest k is \(-2\) | B1 | Follow through |
| (v) Makes x the subject and replaces x by y. \(f^{-1}(x) = \sqrt{\frac{x+18}{2}} - 2\) | M1, M1, A1∨ | x the subject – reasonable attempt from completion of square; x,y interchanged; Correct form his answer to (i) |
**(i)** $2x^2 + 8x - 10 = 2(x+2)^2 + c$. $c = -18$ | B1 B1, B1 | $a = 2$ gets B1, $b = 2$ gets B1; correct only
**(ii)** Least value = $-18$ when $x = -2$ | B1∨/B1∨ | follow through for c and for $-b$. Calculus ok
**(iii)** $2x^2 + 8x - 10 \geq 14$ or $2(x+2)^2 - 18 \geq 14$. $x^2 + 4x - 12 = 0$ or $(x+2)^2 = 16$. Limit points 2 and $-6$. $x \geq 2$ and $x \leq -6$ | M1, A1, A1 | setting the inequality to 0; correct only – irrespective of what they do; correct only (condone $>$ or $<$)
**(iv)** Smallest k is $-2$ | B1 | Follow through
**(v)** Makes x the subject and replaces x by y. $f^{-1}(x) = \sqrt{\frac{x+18}{2}} - 2$ | M1, M1, A1∨ | x the subject – reasonable attempt from completion of square; x,y interchanged; Correct form his answer to (i)11 (i) Express $2 x ^ { 2 } + 8 x - 10$ in the form $a ( x + b ) ^ { 2 } + c$.\\
(ii) For the curve $y = 2 x ^ { 2 } + 8 x - 10$, state the least value of $y$ and the corresponding value of $x$.\\
(iii) Find the set of values of $x$ for which $y \geqslant 14$.
Given that $\mathrm { f } : x \mapsto 2 x ^ { 2 } + 8 x - 10$ for the domain $x \geqslant k$,\\
(iv) find the least value of $k$ for which f is one-one,\\
(v) express $\mathrm { f } ^ { - 1 } ( x )$ in terms of $x$ in this case.
\hfill \mbox{\textit{CAIE P1 2002 Q11 [12]}}