SSCE HSC Mathematics Extension 1 Combinations: Evaluate $$ \int

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 \time... 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, we first make the substitution:

Let $u = x - 3$ , then $du = dx$ . When $x = 3$ , $u = 0$ , and when $x = 4$ , $u = 1$ .

Thus, we can rewrite the integral as:

\int_{0}^{1} (u + 5) \sqrt{u} \, du

This simplifies to:

\int_{0}^{1} \left(u^{3/2} + 5u^{1/2}\right) \, du = \left[\frac{2}{5}u^{5/2} + \frac{10}{3}u^{3/2}\right]_{0}^{1} = \frac{2}{5} + \frac{10}{3} = \frac{6 + 50}{15} = \frac{56}{15}.

Step 2

Use mathematical induction to prove that $(1 \times 2^2) + (2 \times 2^2) + (3 \times 2^2) + \cdots + (n \times 2^n) = 2 + (n - 1)2^{n + 1}$ for all integers $n \geq 1$.

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Answer

For the base case $n=1$ :

LHS = $1 \times 2^2 = 2$ .

RHS = $2 + (1 - 1)2^{2} = 2$ .

Thus, LHS = RHS when $n=1$ . Now we assume it holds for $n=k$ :

Then, $ext{LHS} = (1 \times 2^2) + \cdots + (k \times 2^k)$ Assuming it equals $2 + (k - 1)2^{k + 1}$ .

For $n = k + 1$ :

LHS becomes: $(1 \times 2^2) + \cdots + (k \times 2^k) + ((k + 1) \times 2^{k + 1})$ By the induction hypothesis: $2 + (k - 1)2^{k + 1} + (k + 1)2^{k + 1} = 2 + k \cdot 2^{k + 1} + 2^{k + 1} = 2 + (k + 1)2^{k + 1}$ Thus, the statement is 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.

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Answer

Using the binomial probability formula: $inom{5}{3} (0.65)^{3} (0.35)^{2}$ This captures the probability of exactly 3 treadmills in use.

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

The probability for this scenario is: $inom{5}{3} (0.65)^{3} (0.35)^{2} \times (0.4)^{0} = \binom{5}{3} (0.65)^{3} (0.35)^{2}$

Step 5

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

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117 rated

Answer

Using the identity: $C_{r}^{n} = C_{n-1}^{r-1} + C_{n-1}^{r}$ We can deduce that: $2022 C_{80} + 2022 C_{81} = 2022 C_{81} + 2022 C_{93}$ means that $p = 2022, q = 81$ is a possibility.

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 $(1 \times 2^2) + (2 \times 2^2) + (3 \times 2^2) + \cdots + (n \times 2^n) = 2 + (n - 1)2^{n + 1}$ for all integers $n \geq 1$.

Find an expression for the probability that, at a particular time, exactly 3 of the 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 $2022 C_{80} + 2022 C_{81} + 2022 C_{93} = PC_{q}$.

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