The curve with equation $y = 3x^2$ meets the curve with equation $y = 15 - 2^{-x}$ at the point P - Edexcel - A-Level Maths Pure - Question 7 - 2020 - Paper 2
Question 7
The curve with equation $y = 3x^2$ meets the curve with equation $y = 15 - 2^{-x}$ at the point P.
Find, using algebra, the exact x coordinate of P.
Worked Solution & Example Answer:The curve with equation $y = 3x^2$ meets the curve with equation $y = 15 - 2^{-x}$ at the point P - Edexcel - A-Level Maths Pure - Question 7 - 2020 - Paper 2
Step 1
Combine the Equations
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Answer
To find the x-coordinate of point P, we begin by setting the equations equal to each other:
3x2=15−2−x
This can be rearranged to:
2−x=15−3x2
Step 2
Rearranging to Form a Power Equation
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Answer
Next, we can rewrite the equation in terms of powers of 2:
15−3x2=2−x
To eliminate the negative exponent, we can rearrange the equation to:
15 - 3x^2 = rac{1}{2^x}
Step 3
Multiplying Through by 2^x
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Answer
Multiply both sides by 2x:
2x(15−3x2)=1
Which leads to:
15imes2x−3x2imes2x=1
Step 4
Logarithmic Transformation
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Answer
From the equation 3x2=15−2−x, we can isolate 2−x and switch to a logarithmic form:
2^x = rac{15}{3}
Thus, we find:
x=−extlog2(3)
The exact x-coordinate of P is:
x=extlog32
