The diagram shows the graph of $y = kx - x^2 + 2$, where $k$ is an integer - OCR - GCSE Maths - Question 7 - 2023 - Paper 6
Question 7
The diagram shows the graph of $y = kx - x^2 + 2$, where $k$ is an integer.
(a) Show that $k = 3$.
(b) Use the graph to solve $3 - x^2 + 2 = 3$.
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Worked Solution & Example Answer:The diagram shows the graph of $y = kx - x^2 + 2$, where $k$ is an integer - OCR - GCSE Maths - Question 7 - 2023 - Paper 6
Step 1
Show that $k = 3$
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Answer
To show that k=3, we can select a point on the graph. Let's use the point (2,4) which lies on the curve.
Substituting x=2 and y=4 into the equation:
4 &= k(2) - (2)^2 + 2 \
4 &= 2k - 4 + 2 \
4 &= 2k - 2 \
6 &= 2k \
k &= 3.
\end{align*}$$
Thus, we have shown that $k = 3$.
Step 2
Use the graph to solve $3 - x^2 + 2 = 3$
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Answer
First, simplify the equation:
This simplifies to:
x^2 = 2.$$
Taking the square root gives:
$$x = ext{±} rac{ ext{sqrt}{2}}{1} ≈ ±1.4.$$
Thus, the answers are:
$$x = 1.4 ext{ or } x = -1.4.$$
Both answers are given to 1 decimal place.
