| Exam Board | OCR MEI |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2015 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discriminant and conditions for roots |
| Type | Find range for two distinct roots |
| Difficulty | Moderate -0.8 This is a straightforward multi-part question testing standard discriminant and completing-the-square techniques. Part (i) requires setting up a quadratic equation and applying b²-4ac > 0, part (ii) is routine completing the square with a guided proof, and part (iii) involves finding the minimum point and substituting. All parts follow textbook procedures with no problem-solving insight required, making it easier than average. |
| 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 points1.02f Solve quadratic equations: including in a function of unknown |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(3x^2 + 12x + 13 = 2x + k\) | M1 | oe e.g. M1 for \(3x^2 + 10x + 13 = k\); condone \(3x^2+10x+13-k=y\) |
| \(3x^2 + 10x + 13 - k\ [=0]\) | M1 | for rearranging to 0; condone one error in adding/subtracting; but M0 for \(3x^2+10x+13=k\) or \(3x^2+10x+13-k=y\) |
| \(b^2 - 4ac > 0\) oe soi | M1 | may be earned near end with correct inequality sign used; allow '\(b^2-4ac\) is positive' oe; 0 for just 'discriminant \(> 0\)' unless implied by later work |
| \(100 - 4 \times 3 \times (13-k) > 0\) oe | M1 | for correct substitution ft into \(b^2-4ac\), dep on second M1 earned; brackets/signs must be correct; can be earned with equality or wrong inequality; M0 for trials of values of \(k\) in \(b^2-4ac\) |
| \(k > \frac{14}{3}\) oe | A1 | accept \(k > \frac{56}{12}\) or better; isw incorrect conversion of fraction but not wrong use of inequalities; if A0, allow B1 for \(\frac{56}{12}\) oe obtained with equality or wrong inequality |
| [5] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(3(x+2)^2 + 1\) www as final answer | B4 | B1 for \(a=3\) and B1 for \(b=2\); and B2 for \(c=1\) or M1 for \(13 - 3 \times \text{their } b^2\) or for \(\frac{13}{3} - \text{their } b^2\); or B3 for \(3\left[(x+2)^2 + \frac{1}{3}\right]\); condone omission of square symbol |
| \(y\)-minimum \(= 1\) [hence curve is above \(x\)-axis] | B1 | Stating min pt is \((-2, 1)\) is sufficient; allow ft if their \(c > 0\); B0 for only showing discriminant is negative; B0 for stating min point \(= 1\) or ft |
| [5] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(5\) cao | B2 | M1 for substitution of their \((-2, 1)\) in \(y = 2x + k\); allow M1 ft their \(3(x+2)^2+1\); or use of \((-2,1)\) found using calculus; M0 if they use an incorrect minimum point inconsistent with their completed square form |
| [2] |
## Question 12:
### Part (i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $3x^2 + 12x + 13 = 2x + k$ | M1 | oe e.g. M1 for $3x^2 + 10x + 13 = k$; condone $3x^2+10x+13-k=y$ |
| $3x^2 + 10x + 13 - k\ [=0]$ | M1 | for rearranging to 0; condone one error in adding/subtracting; but M0 for $3x^2+10x+13=k$ or $3x^2+10x+13-k=y$ |
| $b^2 - 4ac > 0$ oe soi | M1 | may be earned near end with correct inequality sign used; allow '$b^2-4ac$ is positive' oe; 0 for just 'discriminant $> 0$' unless implied by later work |
| $100 - 4 \times 3 \times (13-k) > 0$ oe | M1 | for correct substitution ft into $b^2-4ac$, dep on second M1 earned; brackets/signs must be correct; can be earned with equality or wrong inequality; M0 for trials of values of $k$ in $b^2-4ac$ |
| $k > \frac{14}{3}$ oe | A1 | accept $k > \frac{56}{12}$ or better; isw incorrect conversion of fraction but not wrong use of inequalities; if A0, allow **B1** for $\frac{56}{12}$ oe obtained with equality or wrong inequality |
| | **[5]** | |
### Part (ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $3(x+2)^2 + 1$ www as final answer | B4 | **B1** for $a=3$ and **B1** for $b=2$; and **B2** for $c=1$ or **M1** for $13 - 3 \times \text{their } b^2$ or for $\frac{13}{3} - \text{their } b^2$; or **B3** for $3\left[(x+2)^2 + \frac{1}{3}\right]$; condone omission of square symbol |
| $y$-minimum $= 1$ [hence curve is above $x$-axis] | B1 | Stating min pt is $(-2, 1)$ is sufficient; allow ft if their $c > 0$; B0 for only showing discriminant is negative; B0 for stating min point $= 1$ or ft |
| | **[5]** | |
### Part (iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $5$ cao | B2 | **M1** for substitution of their $(-2, 1)$ in $y = 2x + k$; allow M1 ft their $3(x+2)^2+1$; or use of $(-2,1)$ found using calculus; M0 if they use an incorrect minimum point inconsistent with their completed square form |
| | **[2]** | |12 (i) Find the set of values of $k$ for which the line $y = 2 x + k$ intersects the curve $y = 3 x ^ { 2 } + 12 x + 13$ at two distinct points.\\
(ii) Express $3 x ^ { 2 } + 12 x + 13$ in the form $a ( x + b ) ^ { 2 } + c$. Hence show that the curve $y = 3 x ^ { 2 } + 12 x + 13$ lies completely above the $x$-axis.\\
(iii) Find the value of $k$ for which the line $y = 2 x + k$ passes through the minimum point of the curve $y = 3 x ^ { 2 } + 12 x + 13$.
\hfill \mbox{\textit{OCR MEI C1 2015 Q12 [12]}}