SSCE HSC Mathematics Extension 1 Volumes of solids of revolution: Three different points A, B and

Question

Three different points A, B and C are chosen on a circle centred at O.

Let a = $ar{OA}$, b = $ar{OB}$ and c = $ar{OC}$. Let h = a + b + c and let H be the point such that OH = h, as shown in the diagram.

Show that $ar{BH}$ and $ar{CA}$ are perpendicular.

A solid of revolution is to be found by rotating the region bounded by the x-axis
and the curve y = (k + 1) sin(kx), where k > 0, between x = 0 and x = $rac{	ext{π}}{2k}$ about the x-axis.

Find the value of k for which the volume is $	ext{π}^{2}$.

The function f is defined by f(x) = sin(x) for all real numbers x. Let g be the function defined on [-1, 1] by g(x) = arccos(x).

Is g the inverse of f$^{2}$? Justify your answer.

The monic polynomial, P, has degree 3 and roots a, b, $	ext{γ}$.

It is given that a$^{2}$ + b$^{2}$ + $	ext{γ}^{2}$ = 85 and P'($a$) + P'($b$) + P'($	ext{γ}$) = 87.

Find a$b + b$	ext{γ} + $	ext{γ}a$.

You may use the information on page 18 to answer this question.

A chocolate factory sells 150-gram chocolate bars. There has been a complaint that the bars actually weigh less than 150 grams. A team of inspectors was sent to the factory to check. They randomly selected 16 bars, weighed them and noted that 8 bars weighed less than 150 grams.

The factory manager claims 80% of the chocolate bars produced by the factory weigh 150 grams or more.

(i) The inspectors used the normal approximation to the binomial distribution to calculate the probability, P$_{0}$, of having at least 8 bars weighing less than 150 grams in a random sample of 16, assuming the factory manager's claim is correct.

Calculate the value of P$_{0}$.

(ii) The factory manager disagrees with the method used by the inspectors as described in part (i).

Explain why the method used by the inspectors might not be valid.

Three different points A, B and C are chosen on a circle centred at O - HSC - SSCE Mathematics Extension 1 - Question 13 - 2022 - Paper 1

Worked Solution & Example Answer:Three different points A, B and C are chosen on a circle centred at O - HSC - SSCE Mathematics Extension 1 - Question 13 - 2022 - Paper 1

Show that \( \bar{BH} \) and \( \bar{CA} \) are perpendicular.

Find the value of k for which the volume is \( \text{π}^{2} \).

Is g the inverse of f$^{2}$? Justify your answer.

Find \( a b + b \text{γ} + \text{γ} a \).

Calculate the value of P$_{0}$.

Explain why the method used by the inspectors might not be valid.

Join the SSCE students using SimpleStudy...