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Question 3 (a) Factorise fully: 3xy − 9x + 4y − 12 - Leaving Cert Mathematics - Question 3 - 2019
Question 3
Question 3 (a) Factorise fully: 3xy − 9x + 4y − 12. (b) g(x) = 3xy ln x − 9x + 4ln x − 12. Using your answer to part (a) or otherwise, solve g(x) = 0. (c) Evaluat... show full transcript
Worked Solution & Example Answer:Question 3 (a) Factorise fully: 3xy − 9x + 4y − 12 - Leaving Cert Mathematics - Question 3 - 2019
Step 1
Factorise fully: 3xy − 9x + 4y − 12.
Answer
To factorise the expression (3xy - 9x + 4y - 12), we can group the terms:
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Group terms: ((3xy + 4y) + (-9x - 12))
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Factor out the common factors in each group:
- From the first group (3xy + 4y), we can factor out (y): (y(3x + 4))
- From the second group (-9x - 12), we can factor out (-3): (-3(3x + 4))
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Putting it together gives us: (y(3x + 4) - 3(3x + 4))
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Now factor out (3x + 4): ((3x + 4)(y - 3))
So, the fully factored form is ((3x + 4)(y - 3)).
Step 2
Using your answer to part (a) or otherwise, solve g(x) = 0.
Answer
To solve (g(x) = 0) given: [ g(x) = 3xy \ln x - 9x + 4 \ln x - 12 ]
Using the factorised form from part (a):
(g(x)) can be simplified or factored further. From part (a), we have ((3x + 4)(y - 3) = 0).
Setting each factor to zero gives:
- (3x + 4 = 0)
- (y - 3 = 0)
From (3x + 4 = 0), we find:
[ 3x = -4 \quad \Rightarrow \quad x = -\frac{4}{3} ]
(This solution is not valid in the context of logarithms.)
From (y - 3 = 0):
[ y = 3 ]
This indicates that ( \ln x = 3 ), and therefore:
[ x = e^3 ]
Step 3
Evaluate g′(e) correct to 2 decimal places.
Answer
To find (g'(e)), we need to differentiate (g(x)):
Given: [ g(x) = 3xy \ln x - 9x + 4 \ln x - 12 ]
Using the product rule on the first term: [ g'(x) = 3y \left( 1 + \ln x \right) - 9 + 4 \cdot \frac{1}{x} ]
Substituting (x = e): [ g'(e) = 3y\left(1 + 1\right) - 9 + 4 \cdot 1] [ g'(e) = 3y(2) - 9 + 4] [ g'(e) = 6y - 5]
Finally, substituting (y = 3) from the earlier part: [ g'(e) = 6(3) - 5 = 18 - 5 = 13]
To summarize, (g'(e) = 13) is already in the correct format, but if needed to decimal form, it is 13.00.