Question 8 - A-Level Maths

OCR C3 2014 June — Question 8 11 marks

Exam Board	OCR
Module	C3 (Core Mathematics 3)
Year	2014
Session	June
Marks	11
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Modulus function
Type	Find range of k for number of roots
Difficulty	Standard +0.8 This question requires differentiation using the quotient rule, solving a cubic equation for stationary points, understanding modulus function transformations, and analyzing the number of intersections between a horizontal line and a reflected curve. The final part demands careful geometric reasoning about when exactly two roots occur, which is non-trivial and goes beyond routine application of techniques.
Spec	1.02k Simplify rational expressions: factorising, cancelling, algebraic division 1.02l Modulus function: notation, relations, equations and inequalities 1.02m Graphs of functions: difference between plotting and sketching 1.07i Differentiate x^n: for rational n and sums 1.07n Stationary points: find maxima, minima using derivatives 1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates

8 \includegraphics[max width=\textwidth, alt={}, center]{33a2b09d-0df9-48d6-9ee9-e0a1ec345f41-4_616_1024_296_516} The diagram shows the curve $y = \frac { 2 x + 4 } { x ^ { 2 } + 5 }$.

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ and hence find the coordinates of the two stationary points.
The function g is defined for all real values of $x$ by $$\mathrm { g } ( x ) = \left| \frac { 2 x + 4 } { x ^ { 2 } + 5 } \right| .$$
1. Sketch the curve $y = \mathrm { g } ( x )$ and state the range of g .
2. It is given that the equation $\mathrm { g } ( x ) = k$, where $k$ is a constant, has exactly two distinct real roots. Write down the set of possible values of $k$.

Show mark scheme Show mark scheme source

Question 8:

Part (i):

Answer	Marks	Guidance
Answer	Mark	Guidance
Attempt use of quotient rule or equiv	M1	Condone one slip only but must be subtraction in numerator; condone absence of necessary brackets; or equiv
Obtain $\dfrac{2(x^2+5)-2x(2x+4)}{(x^2+5)^2}$	A1	Or correct equiv; now with brackets as necessary. Correct numerator but error in denominator: max M1A0A1M1A1A1; numerator wrong way round: max M0A0A0M1A1A1
Obtain $-2x^2-8x+10=0$	A1	Or equiv involving three terms
Attempt solution of three-term quadratic equation based on numerator of derivative (even if their equation has no real roots)	M1	Implied by no working but 2 correct values obtained. M1 for factorisation awarded if attempt is such that $x^2$ term and one other term correct upon expansion; if formula used, M1 awarded as per Qn 2
Obtain $-5$ and $1$	A1
Obtain $(-5, -\frac{1}{5})$ and $(1, 1)$	A1	Allow $-\frac{6}{30}$
[6]

Part (ii)(a):

Answer	Marks	Guidance
Answer	Mark	Guidance
Sketch (more or less) correct curve	B1	Showing negative part reflected in $x$-axis and positive part unchanged; ignore intercept values on axes, right or wrong
State values between $0$ and their $y$-value of maximum point lying in first quadrant	M1	Accept $\leq$ or $<$ signs here
State correct $0 \leq y \leq 1$	A1ft	Following their $y$-value of maximum point in first quadrant; now with $\leq$ signs; or equiv perhaps involving $g$ or $g(x)$. For "$y\geq 0$ and $y\leq 1$", award M1A1; for separate statements $y\geq 0$, $y\leq 1$, award M1A0
[3]

Part (ii)(b):

Answer	Marks	Guidance
Answer	Mark	Guidance
Indicate, in some way, values between $y$-coordinates of maximum point and reflected minimum point (provided their $y$-coordinate of minimum point is negative)	M1	Allow $\leq$ sign(s) here; could be clear indication on graph. For "$k>\frac{1}{5}$ and $k<1$", award M1A1; for separate statements, award M1A0
State $\frac{1}{5} < k < 1$	A1	Or correct equiv; not $\leq$ now; correct answer only earns M1A1
[2]

# Question 8:

## Part (i):

| Answer | Mark | Guidance |
|--------|------|----------|
| Attempt use of quotient rule or equiv | M1 | Condone one slip only but must be subtraction in numerator; condone absence of necessary brackets; or equiv |
| Obtain $\dfrac{2(x^2+5)-2x(2x+4)}{(x^2+5)^2}$ | A1 | Or correct equiv; now with brackets as necessary. Correct numerator but error in denominator: max M1A0A1M1A1A1; numerator wrong way round: max M0A0A0M1A1A1 |
| Obtain $-2x^2-8x+10=0$ | A1 | Or equiv involving three terms |
| Attempt solution of three-term quadratic equation based on numerator of derivative (even if their equation has no real roots) | M1 | Implied by no working but 2 correct values obtained. M1 for factorisation awarded if attempt is such that $x^2$ term and one other term correct upon expansion; if formula used, M1 awarded as per Qn 2 |
| Obtain $-5$ and $1$ | A1 | |
| Obtain $(-5, -\frac{1}{5})$ and $(1, 1)$ | A1 | Allow $-\frac{6}{30}$ |
| | **[6]** | |

## Part (ii)(a):

| Answer | Mark | Guidance |
|--------|------|----------|
| Sketch (more or less) correct curve | B1 | Showing negative part reflected in $x$-axis and positive part unchanged; ignore intercept values on axes, right or wrong |
| State values between $0$ and their $y$-value of maximum point lying in first quadrant | M1 | Accept $\leq$ or $<$ signs here |
| State correct $0 \leq y \leq 1$ | A1ft | Following their $y$-value of maximum point in first quadrant; now with $\leq$ signs; or equiv perhaps involving $g$ or $g(x)$. For "$y\geq 0$ and $y\leq 1$", award M1A1; for separate statements $y\geq 0$, $y\leq 1$, award M1A0 |
| | **[3]** | |

## Part (ii)(b):

| Answer | Mark | Guidance |
|--------|------|----------|
| Indicate, in some way, values between $y$-coordinates of maximum point and reflected minimum point (provided their $y$-coordinate of minimum point is negative) | M1 | Allow $\leq$ sign(s) here; could be clear indication on graph. For "$k>\frac{1}{5}$ and $k<1$", award M1A1; for separate statements, award M1A0 |
| State $\frac{1}{5} < k < 1$ | A1 | Or correct equiv; not $\leq$ now; correct answer only earns M1A1 |
| | **[2]** | |

---

Show LaTeX source

8\\
\includegraphics[max width=\textwidth, alt={}, center]{33a2b09d-0df9-48d6-9ee9-e0a1ec345f41-4_616_1024_296_516}

The diagram shows the curve $y = \frac { 2 x + 4 } { x ^ { 2 } + 5 }$.\\
(i) Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ and hence find the coordinates of the two stationary points.\\
(ii) The function g is defined for all real values of $x$ by

$$\mathrm { g } ( x ) = \left| \frac { 2 x + 4 } { x ^ { 2 } + 5 } \right| .$$
\begin{enumerate}[label=(\alph*)]
\item Sketch the curve $y = \mathrm { g } ( x )$ and state the range of g .
\item It is given that the equation $\mathrm { g } ( x ) = k$, where $k$ is a constant, has exactly two distinct real roots. Write down the set of possible values of $k$.
\end{enumerate}

\hfill \mbox{\textit{OCR C3 2014 Q8 [11]}}

This paper (9 questions)

View full paper

Q1 5 Q2 6 Q3 6 Q4 7 Q5 7 Q6 8 Q7 10 Q8 11 Q9 12