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Evaluate \( \int_{3}^{4} (x+2)\sqrt{3-x} \, dx \) using the substitution \( u = x - 3 \) - HSC - SSCE Mathematics Extension 1 - Question 12 - 2023 - Paper 1
Question 12
Evaluate \( \int_{3}^{4} (x+2)\sqrt{3-x} \, dx \) using the substitution \( u = x - 3 \). (b) Use mathematical induction to prove that \( (1 \times 2^2) + (2 \time... show full transcript
Worked Solution & Example Answer:Evaluate \( \int_{3}^{4} (x+2)\sqrt{3-x} \, dx \) using the substitution \( u = x - 3 \) - HSC - SSCE Mathematics Extension 1 - Question 12 - 2023 - Paper 1
Step 1
Evaluate \( \int_{3}^{4} (x+2)\sqrt{3-x} \, dx \) using the substitution \( u = x - 3 \)
Answer
To solve the integral, we will use the substitution ( u = x - 3 ). Thus, the bounds change as follows: when ( x = 3 ), ( u = 0 ) and when ( x = 4 ), ( u = 1 ).
The integral becomes:
Next, we'll simplify and compute this integral. We have:
Breaking that down:
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Calculate ( \int_{0}^{1} -u\sqrt{u} , du ):
- This can be computed as ( \int_{0}^{1} -u^{3/2} , du = -\frac{2}{5}u^{5/2} \Big|_{0}^{1} = -\frac{2}{5} )
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Calculate ( \int_{0}^{1} 5\sqrt{u} , du ):
- This becomes ( 5\cdot \frac{2}{3}u^{3/2} \Big|_{0}^{1} = \frac{10}{3} )
Combining both results, we get:
Step 2
Use mathematical induction to prove that \( (1 \times 2^2) + (2 \times 2^2) + (3 \times 2^n) + \cdots + (n \times 2^n) = 2 + (n-1)2^{n+1} \)
Answer
Base Case (n=1): Evaluating the left-hand side: ( LHS = 1 \times 2^2 = 2 ) For the right-hand side: ( RHS = 2 + (1-1)2^{1+1} = 2 ) Both sides are equal.
Inductive Step: Assume true for ( n = k ): ( (1 \times 2^2) + (2 \times 2^2) + \cdots + (k \times 2^k) = 2 + (k-1)2^{k+1} ) Now prove for ( n = k+1 ): ( LHS = (1 \times 2^2) + (2 \times 2^2) + \cdots + (k \times 2^k) + ((k+1) \times 2^{k+1}) ) ( = (2 + (k-1)2^{k+1}) + (k+1)(2^{k+1}) ) ( = 2 + k \times 2^{k+1} + (k+1) \times 2^{k+1} = 2 + (k+1 - 1) \times 2^{k+2} ) ( = 2 + k 2^{k+2} ) Thus, true for all integers ( n \geq 1 ).
Step 3
Find an expression for the probability that, at a particular time, exactly 3 of the treadmills are in use.
Answer
Given that each treadmill is used 65% of the time, we can model this situation using the binomial probability formula:
Here, ( n = 5 ) (the number of treadmills), ( k = 3 ), and ( p = 0.65 ). Thus, the probability is given by:
Step 4
Find an expression for the probability that, at a particular time, exactly 3 of the 5 treadmills are in use and no rowing machines are in use.
Answer
For this scenario, we will find the probability for exactly 3 treadmills and 0 rowing machines. The treadmills probability is already calculated as:
For the rowing machines, the probability of 0 machines being used (out of 4 available, with use probability of 0.40) is:
Combining both expressions:
Step 5
Find ONE possible set of values for \( p \) and \( q \) such that \( 2022C_{80} + 2022C_{81} + 2022C_{93} = PC_{q} \)
Answer
From the given equation, we can use the property of binomial coefficients:
Therefore:
Now we can simplify it as follows:
Comparing terms yields: ( p = 2022, q = 81 ) is a possible solution.