A-Level Edexcel Maths Pure Solving Equations: The curve C has equation $y = \f

The curve C has equation $y = \frac{1}{3}x^2 + 8$ - Edexcel - A-Level Maths Pure - Question 10 - 2014 - Paper 2

Question 10

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The curve C has equation $y = \frac{1}{3}x^2 + 8$. The line L has equation $y = 3x + k$, where $k$ is a positive constant. (a) Sketch C and L on separate diagrams,... show full transcript

Worked Solution & Example Answer:The curve C has equation $y = \frac{1}{3}x^2 + 8$ - Edexcel - A-Level Maths Pure - Question 10 - 2014 - Paper 2

Step 1

(a) Sketch C and L on separate diagrams, showing the coordinates of the points at which C and L cut the axes.

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Answer

To sketch the curve C defined by the equation $y = \frac{1}{3}x^2 + 8$ , we first note its shape. It is a parabola that opens upwards and is symmetric about the y-axis. The vertex of the parabola is at the point (0, 8).

The curve intersects the y-axis at (0, 8). To find the x-intercepts, we set $y = 0$ : $0 = \frac{1}{3}x^2 + 8$ Solving this gives $x^2 = -24$ , meaning there are no real x-intercepts. Therefore, the graph does not cut the x-axis.

Next, we plot the line L given by $y = 3x + k$ . This line is also drawn on a separate diagram. Since $k$ is positive, the y-intercept $(0, k)$ is above the x-axis.

The line L will cut the y-axis at (0, k) and the x-axis can be found by setting $y=0$ : $0 = 3x + k \Rightarrow x = -\frac{k}{3}$ Thus, the intercepts for line L are (0, k) and (-k/3, 0).

Step 2

(b) find the value of k.

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Answer

To find the value of $k$ when line L is tangent to curve C, we will equate the two equations: $\frac{1}{3}x^2 + 8 = 3x + k$ Rearranging gives: $\frac{1}{3}x^2 - 3x + (8 - k) = 0$ For L to be a tangent to C, this quadratic equation must have a double root, meaning the discriminant must be zero: $D = b^2 - 4ac = 0\Rightarrow (-3)^2 - 4 \times \frac{1}{3}(8 - k) = 0$ This simplifies to: $9 - \frac{4}{3}(8 - k) = 0$ Multiplying through by 3 to clear the fraction yields: $27 - 4(8 - k) = 0\Rightarrow 27 - 32 + 4k = 0\Rightarrow 4k = 5\Rightarrow k = \frac{5}{4}$ Thus, the value of $k$ is $rac{5}{4}$ .

The curve C has equation $y = \frac{1}{3}x^2 + 8$ - Edexcel - A-Level Maths Pure - Question 10 - 2014 - Paper 2

Worked Solution & Example Answer:The curve C has equation $y = \frac{1}{3}x^2 + 8$ - Edexcel - A-Level Maths Pure - Question 10 - 2014 - Paper 2

(a) Sketch C and L on separate diagrams, showing the coordinates of the points at which C and L cut the axes.

(b) find the value of k.

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