| Exam Board | OCR MEI |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2009 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discriminant and conditions for roots |
| Type | Prove/show always positive |
| Difficulty | Moderate -0.8 This is a straightforward C1 question testing standard techniques: equating expressions to find intersections (solving a quadratic), completing the square (routine algebraic manipulation), and using completed square form to show the parabola is always positive. All parts are textbook exercises requiring only direct application of learned methods with no problem-solving insight needed. |
| Spec | 1.02c Simultaneous equations: two variables by elimination and substitution1.02d Quadratic functions: graphs and discriminant conditions1.02e Complete the square: quadratic polynomials and turning points |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(3x^2 + 6x + 10 = 2 - 4x\) | M1 | for subst for \(x\) or \(y\) or subtraction attempted |
| \(3x^2 + 10x + 8 [= 0]\) | M1 | or \(3y^2 - 52y + 220 [=0]\); for rearranging to zero (condone one error) |
| \((3x + 4)(x + 2) [=0]\) | M1 | or \((3y-22)(y-10)\); for sensible attempt at factorising or completing the square |
| \(x = -2\) or \(-4/3\) o.e. | A1 | or A1 for each of \((-2, 10)\) and \((-4/3, 22/3)\) o.e. |
| \(y = 10\) or \(22/3\) o.e. | A1 | |
| Total: 5 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(3(x+1)^2 + 7\) | 4 | 1 for \(a = 3\), 1 for \(b = 1\), 2 for \(c = 7\) or M1 for \(10 - 3 \times\) their \(b^2\) soi or for \(7/3\) or for \(10/3 -\) their \(b^2\) soi |
| Total: 4 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| min at \(y = 7\) or ft from (ii) for positive \(c\) (ft for (ii) only if in correct form) | B2 | may be obtained from (ii) or from good symmetrical graph or identified from table of values showing symmetry; condone error in \(x\) value in stated min; ft from (iii) [getting confused with 3 factor]; B1 if say turning pt at \(y = 7\) or ft without identifying min; or M1 for min at \(x = -1\) [e.g. may start again and use calculus to obtain \(x = -1\)] or min when \((x+1)^{[2]} = 0\); and A1 for showing \(y\) positive at min; or M1 for showing discriminant neg. so no real roots and A1 for showing above axis not below e.g. positive \(x^2\) term or goes though \((0, 10)\); or M1 for stating bracket squared must be positive [or zero] and A1 for saying other term is positive |
| Total: 2 marks |
# Question 12:
## Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $3x^2 + 6x + 10 = 2 - 4x$ | M1 | for subst for $x$ or $y$ or subtraction attempted |
| $3x^2 + 10x + 8 [= 0]$ | M1 | or $3y^2 - 52y + 220 [=0]$; for rearranging to zero (condone one error) |
| $(3x + 4)(x + 2) [=0]$ | M1 | or $(3y-22)(y-10)$; for sensible attempt at factorising or completing the square |
| $x = -2$ or $-4/3$ o.e. | A1 | or A1 for each of $(-2, 10)$ and $(-4/3, 22/3)$ o.e. |
| $y = 10$ or $22/3$ o.e. | A1 | |
| **Total: 5 marks** | | |
## Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $3(x+1)^2 + 7$ | 4 | 1 for $a = 3$, 1 for $b = 1$, 2 for $c = 7$ or M1 for $10 - 3 \times$ their $b^2$ soi or for $7/3$ or for $10/3 -$ their $b^2$ soi |
| **Total: 4 marks** | | |
## Part (iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| min at $y = 7$ or ft from (ii) for positive $c$ (ft for (ii) only if in correct form) | B2 | may be obtained from (ii) or from good symmetrical graph or identified from table of values showing symmetry; condone error in $x$ value in stated min; ft from (iii) [getting confused with 3 factor]; B1 if say turning pt at $y = 7$ or ft without identifying min; or M1 for min at $x = -1$ [e.g. may start again and use calculus to obtain $x = -1$] or min when $(x+1)^{[2]} = 0$; and A1 for showing $y$ positive at min; or M1 for showing discriminant neg. so no real roots and A1 for showing above axis not below e.g. positive $x^2$ term or goes though $(0, 10)$; or M1 for stating bracket squared must be positive [or zero] and A1 for saying other term is positive |
| **Total: 2 marks** | | |
---12 (i) Find algebraically the coordinates of the points of intersection of the curve $y = 3 x ^ { 2 } + 6 x + 10$ and the line $y = 2 - 4 x$.\\
(ii) Write $3 x ^ { 2 } + 6 x + 10$ in the form $a ( x + b ) ^ { 2 } + c$.\\
(iii) Hence or otherwise, show that the graph of $y = 3 x ^ { 2 } + 6 x + 10$ is always above the $x$-axis.
\hfill \mbox{\textit{OCR MEI C1 2009 Q12 [11]}}