OCR C1 2013 June — Question 4 7 marks

Exam BoardOCR
ModuleC1 (Core Mathematics 1)
Year2013
SessionJune
Marks7
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicCompleting the square and sketching
TypeComplete square then find vertex/turning point
DifficultyModerate -0.8 This is a straightforward C1 question testing routine completing the square technique, reading off the minimum point, and calculating a discriminant using a standard formula. All three parts are direct applications of basic procedures with no problem-solving or insight required, making it easier than the average A-level question.
Spec1.02d Quadratic functions: graphs and discriminant conditions1.02e Complete the square: quadratic polynomials and turning points
  1. Express \(3x^2 + 9x + 10\) in the form \(3(x + p)^2 + q\). [3]
  2. State the coordinates of the minimum point of the curve \(y = 3x^2 + 9x + 10\). [2]
  3. Calculate the discriminant of \(3x^2 + 9x + 10\). [2]
(i)
Answer: \(3(x^2 + 3x) + 10 = 3\left(x + \frac{3}{2}\right)^2 - \frac{27}{4} + 10 = 3\left(x + \frac{3}{2}\right)^2 + \frac{13}{4}\)
AnswerMarks Guidance
Marks: B1M1 A1 [3]
Guidance: \((x + \frac{3}{2})^2\)\(10 - 3p^2\) or \(\frac{10}{3} - p^2\) Allow \(p = \frac{3}{2}, q = \frac{13}{4}\) A1 www
Additional notes: If \(p, q\) found correctly, then ISW slips in format. 3(x + 1.5)² - 3.25 B1 M0 A0; 3(x + 1.5)² + 3.25 B1 M1 A1 (BOD); 3(x² + 1.5)² + 3.25 B0 M1 A0; 3(x - 1.5)² + 3.25 B0 M1 A1 (BOD); 3 x (x + 1.5)² + 3.25 B0M1A0
(ii)
Answer: \(\left(-\frac{3}{2}, \frac{13}{4}\right)\)
AnswerMarks
Marks: B1B1 [2]
Guidance: FT i.e. – their \(p\)FT i.e. their \(q\)
Additional notes: If restarted e.g. by differentiation: Correct method to find \(x\) value of minimum point M1; Fully correct answer www A1
(iii)
Answer: \(9^2 - (4 \times 3 \times 10) = -39\)
AnswerMarks
Marks: M1A1 [2]
Guidance: Uses \(b^2 - 4ac\)Ignore >0, <0 etc. ISW comments about number of roots
Additional notes: Use of \(\sqrt{b^2 - 4ac}\) is M0 unless recovered
## (i)
**Answer:** $3(x^2 + 3x) + 10 = 3\left(x + \frac{3}{2}\right)^2 - \frac{27}{4} + 10 = 3\left(x + \frac{3}{2}\right)^2 + \frac{13}{4}$

**Marks:** B1 | M1 | A1 [3]

**Guidance:** $(x + \frac{3}{2})^2$ | $10 - 3p^2$ or $\frac{10}{3} - p^2$ | Allow $p = \frac{3}{2}, q = \frac{13}{4}$ A1 www

**Additional notes:** If $p, q$ found correctly, then ISW slips in format. 3(x + 1.5)² - 3.25 B1 M0 A0; 3(x + 1.5)² + 3.25 B1 M1 A1 (BOD); 3(x² + 1.5)² + 3.25 B0 M1 A0; 3(x - 1.5)² + 3.25 B0 M1 A1 (BOD); 3 x (x + 1.5)² + 3.25 B0M1A0

## (ii)
**Answer:** $\left(-\frac{3}{2}, \frac{13}{4}\right)$

**Marks:** B1 | B1 [2]

**Guidance:** FT i.e. – their $p$ | FT i.e. their $q$

**Additional notes:** If restarted e.g. by differentiation: Correct method to find $x$ value of minimum point M1; Fully correct answer www A1

## (iii)
**Answer:** $9^2 - (4 \times 3 \times 10) = -39$

**Marks:** M1 | A1 [2]

**Guidance:** Uses $b^2 - 4ac$ | Ignore >0, <0 etc. ISW comments about number of roots

**Additional notes:** Use of $\sqrt{b^2 - 4ac}$ is M0 unless recovered

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\begin{enumerate}[label=(\roman*)]
\item Express $3x^2 + 9x + 10$ in the form $3(x + p)^2 + q$. [3]
\item State the coordinates of the minimum point of the curve $y = 3x^2 + 9x + 10$. [2]
\item Calculate the discriminant of $3x^2 + 9x + 10$. [2]
\end{enumerate}

\hfill \mbox{\textit{OCR C1 2013 Q4 [7]}}
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