Use the principle of mathematical induction to show that for all integers $n \geq 1$, $$1 \times 2 + 2 \times 5 + 3 \times 8 + \cdots + n(3n - 1) = n^2(n + 1).$$ When a particular biased coin is tossed, the probability of obtaining a head is $\frac{3}{5}$. This coin is tossed 100 times. Let $X$ be the random variable representing the number of heads obtained. This random variable will have a binomial distribution. (i) Find the expected value, $E(X)$. (ii) By finding the variance, $Var(X)$, show that the standard deviation of $X$ is approximately 5. (iii) By using a normal approximation, find the approximate probability that $X$ is between 55 and 65. To complete a course, a student must choose and pass exactly three topics. There are eight topics from which to choose. Last year 400 students completed the course. Explain, using the pigeonhole principle, why at least eight students passed exactly the same three topics. Find $$\int_0^{\pi}\cos 5x \sin 3x \, dx.$$ Find the curve which satisfies the differential equation $$\frac{dy}{dx} = -x - y$$ and passes through the point (1, 0).

SSCE HSC Mathematics Extension 1 Integrals involving trigonometric substitution: Use the principle of mathematica

Use the principle of mathematical induction to show that for all integers $n \geq 1$, $$1 \times 2 + 2 \times 5 + 3 \times 8 + \cdots + n(3n - 1) = n^2(n + 1).$$ When a particular biased coin is tossed, the probability of obtaining a head is $\frac{3}{5}$ - HSC - SSCE Mathematics Extension 1 - Question 12 - 2020 - Paper 1

Question 12

$Use-the-principle-of-mathematical-induction-to-show-that-for-all-integers-$n-\geq-1$,---$$1-\times-2-+-2-\times-5-+-3-\times-8-+-\cdots-+-n(3n---1)-=-n^2(n-+-1).$$--When-a-particular-biased-coin-is-tossed,-the-probability-of-obtaining-a-head-is-$\frac{3}{5}$-HSC-SSCE Mathematics Extension 1-Question 12-2020-Paper 1.png$

Use the principle of mathematical induction to show that for all integers $n \geq 1$, $$1 \times 2 + 2 \times 5 + 3 \times 8 + \cdots + n(3n - 1) = n^2(n + 1).$$ ... show full transcript

Worked Solution & Example Answer:Use the principle of mathematical induction to show that for all integers $n \geq 1$, $$1 \times 2 + 2 \times 5 + 3 \times 8 + \cdots + n(3n - 1) = n^2(n + 1).$$ When a particular biased coin is tossed, the probability of obtaining a head is $\frac{3}{5}$ - HSC - SSCE Mathematics Extension 1 - Question 12 - 2020 - Paper 1

Step 1

Use the principle of mathematical induction to show that for all integers $n \geq 1$, $$1 \times 2 + 2 \times 5 + 3 \times 8 + \cdots + n(3n - 1) = n^2(n + 1)$$.

96%

114 rated

Answer

To prove this by induction, we first verify the base case when $n = 1$ :

LHS: $1 \times 2 = 2$

RHS: $1^2(1 + 1) = 2$ .

Thus, the statement holds true for $n = 1$ . Now, assume it holds for $n = k$ :

LHS: $1 \times 2 + 2 \times 5 + \ldots + k(3k - 1) = k^2(k + 1)$ .

For $n = k + 1$ , the expression becomes:

$LHS = k^2(k + 1) + (k + 1)(3(k + 1) - 1).$

Upon simplifying:

$= k^2(k + 1) + (k + 1)(3k + 2) = (k + 1)(k^2 + 3k + 2) = (k + 1)^2(k + 2).$

Thus, the statement holds for $n = k + 1$ . By induction, it holds for all integers $n \geq 1$ .

Step 2

Find the expected value, $E(X)$.

99%

104 rated

Answer

The expected value for a binomial distribution is given by:

$E(X) = n \cdot p,$

where $n$ is the number of trials and $p$ is the probability of success. In this case:

$E(X) = 100 \cdot \frac{3}{5} = 60.$

Step 3

By finding the variance, $Var(X)$, show that the standard deviation of $X$ is approximately 5.

96%

101 rated

Answer

The variance of a binomial distribution is given by:

$Var(X) = n \cdot p \cdot (1 - p).$

For our case:

$Var(X) = 100 \cdot \frac{3}{5} \cdot \frac{2}{5} = 24.$

Thus, the standard deviation $\sigma$ is:

$\sigma = \sqrt{Var(X)} = \sqrt{24} \approx 4.9 \approx 5.$

Step 4

By using a normal approximation, find the approximate probability that $X$ is between 55 and 65.

98%

120 rated

Answer

Using the normal approximation:

We standardize to find:

$P(55 < X < 65) = P\left(\frac{55 - 60}{\sqrt{24}} < Z < \frac{65 - 60}{\sqrt{24}}\right).$

Calculating the Z-scores:

$P\left(-1.02 < Z < 1.02\right) \approx 0.68 \text{ (using Z-table)}.$

Step 5

Explain, using the pigeonhole principle, why at least eight students passed exactly the same three topics.

97%

117 rated

Answer

The number of ways to choose 3 topics from 8 is given by:

$\binom{8}{3} = \frac{8!}{3!(8-3)!} = 56.$

Since 400 students completed the course, and the maximum number of students who can choose the same set of 3 topics without exceeding the maximum for any topic combination is 392, by the pigeonhole principle, at least one combination must have at least 8 students.

Step 6

Find $$\int_0^{\pi}\cos 5x \sin 3x \, dx.$$

97%

121 rated

Answer

Using the product-to-sum identities, we rewrite:

$\cos A \sin B = \frac{1}{2} [\sin(A + B) - \sin(A - B)].$

So,

$\cos 5x \sin 3x = \frac{1}{2} [\sin(8x) - \sin(2x)].$

Thus, we compute the integral:

$\int_0^{\pi}\cos 5x \sin 3x \, dx = \frac{1}{2} \left[ \int_0^{\pi}\sin(8x) \, dx - \int_0^{\pi}\sin(2x) \, dx \right].$

Evaluating:

$= \frac{1}{2} \left[ \frac{-1}{8}\cos(8x) \bigg|_0^{\pi} - \frac{-1}{2}\cos(2x) \bigg|_0^{\pi} \right] = \frac{1}{2} \left[0 - 0 - (0 - 1)\right]= \frac{1}{2}(1) = \frac{1}{2}.$

Step 7

Find the curve which satisfies the differential equation $$\frac{dy}{dx} = -x - y$$ and passes through the point (1, 0).

96%

114 rated

Answer

We can rewrite the equation as:

$\frac{dy}{dx} + y = -x.$

This is a first-order linear differential equation. Using an integrating factor, we have:

$e^{\int 1 \, dx} = e^x.$

Multiplying through by the integrating factor:

$e^x \frac{dy}{dx} + e^xy = -xe^x.$

Integrating both sides gives us:

$y e^x = -\int xe^x \, dx = -xe^x + e^x + C.$

Solving for $y$ gives:

$y = -x + 1 + Ce^{-x}.$

Applying the initial condition $(1, 0)$ :

=> C = e.$$ The solution is: $$y = -x + 1 + ee^{-x}.$$

Use the principle of mathematical induction to show that for all integers $n \geq 1$, $$1 \times 2 + 2 \times 5 + 3 \times 8 + \cdots + n(3n - 1) = n^2(n + 1)$$.

Find the expected value, $E(X)$.

By finding the variance, $Var(X)$, show that the standard deviation of $X$ is approximately 5.

By using a normal approximation, find the approximate probability that $X$ is between 55 and 65.

Explain, using the pigeonhole principle, why at least eight students passed exactly the same three topics.

Find $$\int_0^{\pi}\cos 5x \sin 3x \, dx.$$

Find the curve which satisfies the differential equation $$\frac{dy}{dx} = -x - y$$ and passes through the point (1, 0).

Join the SSCE students using SimpleStudy...