Standard +0.3 This is a straightforward curve sketching question involving transformations of a basic reciprocal function. Part (a) is routine recall, part (b) requires recognizing a vertical translation, and part (c) involves setting up and solving an inequality using the discriminant—all standard P1 techniques with no novel insight required. Slightly easier than average due to the scaffolded structure and familiar content.
6. (a) Sketch the curve with equation
$$y = - \frac { k } { x } \quad k > 0 \quad x \neq 0$$
(b) On a separate diagram, sketch the curve with equation
$$y = - \frac { k } { x } + k \quad k > 0 \quad x \neq 0$$
stating the coordinates of the point of intersection with the \(x\)-axis and, in terms of \(k\), the equation of the horizontal asymptote.
(c) Find the range of possible values of \(k\) for which the curve with equation
$$y = - \frac { k } { x } + k \quad k > 0 \quad x \neq 0$$
does not touch or intersect the line with equation \(y = 3 x + 4\)
\includegraphics[max width=\textwidth, alt={}, center]{6a5d0ffc-a725-404b-842a-f3b6000e6fed-21_72_47_2615_1886}
Uses \(b^2-4ac<0\) and selects inside region for critical values
dM1
\(10-2\sqrt{21} < k < 10+2\sqrt{21}\)
A1
(5) (10 marks)
Question 6(c):
Answer
Marks
Guidance
Answer/Working
Mark
Guidance
Sets \(3x+4 = -\frac{k}{x}+k\) and proceeds to a 3TQ in \(x\) on one side
M1
Terms do not need to be collected (=0 may be omitted)
\(3x^2+(4-k)x+k=0\)
A1
With \(x\) terms collected together or implied by values for \(a\), \(b\) and \(c\)
Uses \(b^2-4ac...0\) giving \(k^2-20k+16\)
M1
Dependent on \(a=3\) with both \(b\) and \(c\) expressions in \(k\)
\(10-2\sqrt{21} < k < 10+2\sqrt{21}\)
dM1
Uses \(b^2-4ac<0\) OR \(b^2-4ac\leq 0\), selects inside region; dependent on both previous M marks
Accept \(10-\sqrt{84} < k < 10+\sqrt{84}\) or equivalent
A1
Must be in terms of \(k\); allow AND form but not OR form
# Question 6(a):
Negative reciprocal shape (top left/bottom right) in any position, no clear vertical or horizontal overlaps | M1 |
Correct sketch in quadrants 2 and 4 only | A1 | **(2)** Condone pen slips at ends; ignore scaling
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# Question 6(b):
Graph is part (a) translated $\uparrow$ | B1ft | For translating their graph from (a) upward
Correct asymptote or intercept: $\left(1,\frac{k}{k}\right)$ or $(1,0)$ on correct axis, OR $y=k$ stated | B1 |
Correct asymptote AND intercept both present | B1 | **(3)** Do not allow just $k$ marked on asymptote line
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# Question 6(c):
Sets $3x+4=-\frac{k}{x}+k \Rightarrow 3x^2+(4-k)x+k=0$ | M1, A1 |
Attempts $b^2-4ac=0$ to find critical values | M1 |
Uses $b^2-4ac<0$ and selects inside region for critical values | dM1 |
$10-2\sqrt{21} < k < 10+2\sqrt{21}$ | A1 | **(5) (10 marks)**
## Question 6(c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Sets $3x+4 = -\frac{k}{x}+k$ and proceeds to a 3TQ in $x$ on one side | M1 | Terms do not need to be collected (=0 may be omitted) |
| $3x^2+(4-k)x+k=0$ | A1 | With $x$ terms collected together or implied by values for $a$, $b$ and $c$ |
| Uses $b^2-4ac...0$ giving $k^2-20k+16$ | M1 | Dependent on $a=3$ with both $b$ and $c$ expressions in $k$ |
| $10-2\sqrt{21} < k < 10+2\sqrt{21}$ | dM1 | Uses $b^2-4ac<0$ OR $b^2-4ac\leq 0$, selects inside region; dependent on both previous M marks |
| Accept $10-\sqrt{84} < k < 10+\sqrt{84}$ or equivalent | A1 | Must be in terms of $k$; allow AND form but not OR form |
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6. (a) Sketch the curve with equation
$$y = - \frac { k } { x } \quad k > 0 \quad x \neq 0$$
(b) On a separate diagram, sketch the curve with equation
$$y = - \frac { k } { x } + k \quad k > 0 \quad x \neq 0$$
stating the coordinates of the point of intersection with the $x$-axis and, in terms of $k$, the equation of the horizontal asymptote.\\
(c) Find the range of possible values of $k$ for which the curve with equation
$$y = - \frac { k } { x } + k \quad k > 0 \quad x \neq 0$$
does not touch or intersect the line with equation $y = 3 x + 4$
\includegraphics[max width=\textwidth, alt={}, center]{6a5d0ffc-a725-404b-842a-f3b6000e6fed-21_72_47_2615_1886}\\
\hfill \mbox{\textit{Edexcel P1 2021 Q6 [10]}}