SSCE HSC Mathematics Extension 1 Introduction to vectors: 5. (a) Find the exact value of t

Question

5. (a) Find the exact value of the volume of the solid of revolution formed when the region bounded by the curve $y=\sin 2x$, the $x$-axis and the line $x=\frac{\pi}{8}$ is rotated about the $x$-axis.

(b) Two chords of a circle, $AB$ and $CD$, intersect at $E$. The perpendiculars to $AB$ at $A$ and $CD$ at $D$ intersect at $P$. The line $PE$ meets $BC$ at $Q$, as shown in the diagram.

(i) Explain why $DPAE$ is a cyclic quadrilateral.

(ii) Prove that $\angle PAE = \angle ABC$.

(iii) Deduce that $PQ$ is perpendicular to $BC$.

(c) A particle moves in a straight line and its position at time $t$ is given by

$x = 5 + \sqrt{3}\sin 3t - \cos 3t$, where $t$ is in radians.

(i) Express $\sqrt{3}\sin 3t - \cos 3t$ in the form $R \sin(3t - \alpha)$, where $R$ is in radians.

(ii) The particle executes simple harmonic motion. Find the amplitude and the centre of motion.

(iii) When does the particle first reach its maximum speed although some were unsure with their answer given that $v(0) = 0$. A common error was to state that maximum speed occurs when $v = 0.$

5. (a) Find the exact value of the volume of the solid of revolution formed when the region bounded by the curve $y=\sin 2x$, the $x$-axis and the line $x=\frac{\pi}{8}$ is rotated about the $x$-axis - HSC - SSCE Mathematics Extension 1 - Question 5 - 2005 - Paper 1

Worked Solution & Example Answer:5. (a) Find the exact value of the volume of the solid of revolution formed when the region bounded by the curve $y=\sin 2x$, the $x$-axis and the line $x=\frac{\pi}{8}$ is rotated about the $x$-axis - HSC - SSCE Mathematics Extension 1 - Question 5 - 2005 - Paper 1

When does the particle first reach its maximum speed although some were unsure with their answer given that $v(0) = 0$.

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