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Evaluate $$\int_{3}^{4} (x + 2) \sqrt{x - 3} \, dx$$ using the substitution $u = x - 3$ - HSC - SSCE Mathematics Extension 1 - Question 12 - 2023 - Paper 1

Question 12

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Evaluate $$\int_{3}^{4} (x + 2) \sqrt{x - 3} \, dx$$ using the substitution $u = x - 3$. (b) Use mathematical induction to prove that $$(1 \times 2^2) + (2 \times 2... show full transcript

Worked Solution & Example Answer:Evaluate $$\int_{3}^{4} (x + 2) \sqrt{x - 3} \, dx$$ using the substitution $u = x - 3$ - HSC - SSCE Mathematics Extension 1 - Question 12 - 2023 - Paper 1

Step 1

Evaluate $$\int_{3}^{4} (x + 2) \sqrt{x - 3} \, dx$$ using the substitution $u = x - 3$.

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Answer

To evaluate the integral, first we substitute:

$u = x - 3 \Rightarrow dx = du;\quad \text{When } x = 3, \; u = 0 \text{ and when } x = 4, \; u = 1.$

Thus, the integral becomes:

$\int_0^1 (u + 5) \sqrt{u} \, du.$

Now we calculate:

$= \int_0^1 (u + 5) \cdot u^{1/2} \, du = \int_0^1 (u^{3/2} + 5u^{1/2}) \, du$

Evaluating the integrals:

$\left[ \frac{u^{5/2}}{5/2} + 5 \cdot \frac{u^{3/2}}{3/2} \right]_0^1$

$= \left[ \frac{2}{5} \cdot 1^{5/2} + \frac{10}{3} \cdot 1^{3/2} \right] - 0 = \frac{2}{5} + \frac{10}{3} = \frac{6 + 50}{15} = \frac{56}{15}.$

Thus, the value of the integral is: $\frac{56}{15}$ .

Step 2

Use mathematical induction to prove that $(1 \times 2^2) + (2 \times 2^2) + \cdots + (n \times 2^n) = 2 + (n - 1)2^{n + 1}$ for all integers $n \geq 1$.

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Answer

Base Case:

Let $n = 1$ :

$LHS = 1 \times 2^2 = 2; \quad RHS = 2 + (1 - 1)2^{1 + 1} = 2.$ Thus, $LHS = RHS$ holds for $n = 1$ .

Inductively Assume:

Assume it holds for $n = k$ :

$(1 \times 2^2) + \ldots + (k \times 2^k) = 2 + (k - 1)2^{k + 1}.$

Inductive Step:

For $n = k + 1$ :

$LHS = (1 \times 2^2) + \ldots + (k \times 2^k) + ((k+1) \times 2^{k+1}).$ Using the inductive hypothesis: $= 2 + (k - 1)2^{k + 1} + (k + 1)2^{k + 1} = 2 + k 2^{k + 1} .$

$= 2 + (k + 1 - 1)2^{(k + 1 + 1)}.$ Thus, $LHS = RHS, \text{ so the statement holds for } n = k + 1.$

By induction, it holds for all $n \geq 1$ .

Step 3

Find an expression for the probability that, at a particular time, exactly 3 of the 5 treadmills are in use.

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Answer

The probability of exactly 3 out of 5 treadmills being used can be calculated using the binomial distribution:

$P(X = k) = {n \choose k} p^k (1 - p)^{n - k},$ where:

$n = 5$ (total treadmills),
$k = 3$ (treadmills in use),
$p = 0.65$ (probability of each treadmill being used).

Thus, $P(X = 3) = {5 \choose 3} (0.65)^3 (0.35)^{5-3}$

Calculating:

$= 10 \cdot (0.65)^3 \cdot (0.35)^2.$

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.

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Answer

Likewise, the probability of exactly 3 treadmills in use:

Using results from (c)(i): $P(T = 3) = {5 \choose 3} (0.65)^3 (0.35)^{2}.$

For the 4 rowing machines being unused: $P(R = 0) = (0.4)^{0}(0.6)^{4}.$

Thus, the overall probability: $P = P(T=3) \cdot P(R=0) = {5 \choose 3} (0.65)^3 (0.35)^{2} \cdot (0.6)^4.$

Step 5

Find ONE possible set of values for $p$ and $q$ such that $2022C_{80} + 2022C_{81} + 2022C_{93} = PC_{q}$.

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Answer

By using the identity,

$2022C_{80} + 2022C_{81} = 2022C_{81 + 1}.$

Also,

$2022C_{81 + 1} + 2022C_{93} = 2022C_{80 + 93}.$

This leads to: $2022C_{80} + 2022C_{81} + 2022C_{93} = 2022C_{81}.$

Thus, a possible solution is $(p, q) = (2024, 81).$

Evaluate $$\int_{3}^{4} (x + 2) \sqrt{x - 3} \, dx$$ using the substitution $u = x - 3$ - HSC - SSCE Mathematics Extension 1 - Question 12 - 2023 - Paper 1

Worked Solution & Example Answer:Evaluate $$\int_{3}^{4} (x + 2) \sqrt{x - 3} \, dx$$ using the substitution $u = x - 3$ - HSC - SSCE Mathematics Extension 1 - Question 12 - 2023 - Paper 1

Evaluate $$\int_{3}^{4} (x + 2) \sqrt{x - 3} \, dx$$ using the substitution $u = x - 3$.

Use mathematical induction to prove that $(1 \times 2^2) + (2 \times 2^2) + \cdots + (n \times 2^n) = 2 + (n - 1)2^{n + 1}$ for all integers $n \geq 1$.

Base Case:

Inductively Assume:

Inductive Step:

Find an expression for the probability that, at a particular time, exactly 3 of the 5 treadmills are in use.

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.

Find ONE possible set of values for $p$ and $q$ such that $2022C_{80} + 2022C_{81} + 2022C_{93} = PC_{q}$.

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