SSCE HSC Mathematics Extension 1 Definite integrals and substitution: 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... 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

Using the substitution $u = x - 3$ gives:

$x = u + 3$ , and thus when $x = 3$ , $u = 0$ and when $x = 4$ , $u = 1$ .

Now we rewrite the integral as:

$\int_{0}^{1} [(u + 3) + 2] \sqrt{u} \, du = \int_{0}^{1} (u + 5) \sqrt{u} \, du$

This simplifies to:

$\int_{0}^{1} (u^{3/2} + 5u^{1/2}) \, du$

Calculating the integral:

$= \left[ \frac{u^{5/2}}{5/2} + 5 \cdot \frac{u^{3/2}}{3/2} \right]_{0}^{1} \ = \left[ \frac{2}{5} + \frac{10}{3} \right]$

Combine the results:

$= \frac{6}{15} + \frac{50}{15} = \frac{56}{15}$

Step 2

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

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Answer

To prove by induction:

Base case: For $n = 1$ , LHS = $1 \times 2^2 = 2$ , RHS = $2 + (1 - 1)2^{1 + 1} = 2$ .

Both sides are equal.

Inductive step: Assume true for $n = k$ :

$LHS = (1 \times 2^2) + \cdots + (k \times 2^k) = 2 + (k - 1)2^{k + 1}$

Now for $n = k + 1$ :

$LHS = (1 \times 2^2) + \cdots + (k \times 2^k) + ((k + 1) \times 2^{k + 1})$

Substituting from the assumption:

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

Combining like terms leads to: $= 2 + k \times 2^{k + 1}$ This matches the RHS for $n = k + 1$ . Therefore, by induction, the statement holds 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:

$P(X = k) = C_n^k p^k (1 - p)^{n - k}$

For our case:

$n = 5$ (treadmills)
$k = 3$ (in use)
$p = 0.65$ (used time)
Therefore, we have:

$P(3) = C_5^3 (0.65)^3 (0.35)^{2}$

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

Using the binomial formulas for both treadmills and rowing machines:

Treadmills (exactly 3 in use):

$P(3) = C_5^3 (0.65)^3 (0.35)^2$

Rowing machines (0 in use):

$P(0) = C_4^0 (0.4)^0 (0.6)^{4} = (0.6)^{4}$

Combine results:

$P(3, 0) = P(3) \times P(0) = C_5^3 (0.65)^3 (0.35)^{2} \times (0.6)^{4}$

Step 5

Find ONE possible set of values for $p$ and $q$ such that $2022C_{30} + 2022C_{31} + 2022C_{32} = PC_q$

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Answer

From the given identity:

$2022C_{30} + 2022C_{31} + 2022C_{32} = 2022C_{32} + 2022C_{31} + 2022C_{30} = 2023C_{32}$

By matching coefficients:

Let $p = 2023$ and $q = 32$ . Thus a possible solution is:

$p = 2023, \, q = 32$

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 $$(1 \times 2^2) + (2 \times 2^2) + \cdots + (n \times 2^n) = 2 + (n - 1)2^{n + 1}$$

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 $2022C_{30} + 2022C_{31} + 2022C_{32} = PC_q$

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