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 \times 2^2... 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

Substituting $u = x - 3$ , the limits change from $x = 3$ to $u = 0$ and from $x = 4$ to $u = 1$ .

Thus, we have: $\int_{0}^{1} (u + 5)\sqrt{u} \ du$ .

Now, we can rewrite this integral as: $\int_{0}^{1} (u + 5)u^{1/2} \ du = \int_{0}^{1} (u^{3/2} + 5u^{1/2}) \ du$ .

Calculating each integral:

For $u^{3/2}$ : $\int u^{3/2} \ du = \frac{u^{5/2}}{5/2} = \frac{2}{5}u^{5/2}$ evaluated from 0 to 1 gives: $\frac{2}{5}.$
For $5u^{1/2}$ : $5 \int u^{1/2} \ du = 5 \cdot \frac{u^{3/2}}{3/2} = \frac{10}{3}u^{3/2}$ evaluated from 0 to 1 gives: $\frac{10}{3}.$

Adding both results together: $\frac{2}{5} + \frac{10}{3} = \frac{6 + 50}{15} = \frac{56}{15}.$

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

Base Case: For $n=1$ , the left-hand side (LHS) is: $1 \times 2^2 = 2,$
while the right-hand side (RHS) is: $2 + (1 - 1)2^{1 + 1} = 2.$
Thus, LHS = RHS for $n=1$ .

Inductive Step: Assume true for $n=k$ : LHS = $(1 \times 2^2) + (2 \times 2^2) + \cdots + (k \times 2^k) = 2 + (k - 1)2^{k + 1}$ .

For $n=k+1$ , we have: $\text{LHS} = (1 \times 2^2) + (2 \times 2^2) + \cdots + (k \times 2^k) + ((k + 1) \times 2^{k + 1})$ Substituting the inductive assumption: $= (2 + (k - 1)2^{k + 1}) + ((k + 1) \times 2^{k + 1}) = 2 + (k + 1 - 1)2^{k + 2} = 2 + k2^{k + 2}.$
Thus, true for $n=k+1$ . Therefore, by induction, 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 5 treadmills are in use.

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Answer

Let $X$ be the number of treadmills in use. Since each treadmill is used independently with a probability of $0.65$ , the expression becomes: $P(X=3) = {5 \choose 3} (0.65)^3 (0.35)^{2}.$

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

The expression changes to incorporate the fact that all rowing machines are not in use, which occurs with a probability of $(0.4)^0$ . Thus the expression becomes: $P(X=3, Y=0) = {5 \choose 3} (0.65)^3 (0.35)^{2} \times (0.4)^0 = {5 \choose 3} (0.65)^3 (0.35)^{2}.$

Step 5

Find ONE possible set of values for $p$ and $q$ such that $$2022C_{80} + 2022C_{81} + 2022C_{193} = PC_q.$$

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Answer

Using the identity, we can express: $2022C_{80} + 2022C_{81} = 2022C_{81} + 2022C_{80} + 2022C_{193}.$

By simplification, we find: $2022C_{81} + 2022C_{80} + 2022C_{193} = 2022C_{80}.$
Thus, taking $p=2024$ and $q=81$ gives a valid solution.

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 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 $$2022C_{80} + 2022C_{81} + 2022C_{193} = PC_q.$$

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