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The diagram shows two circles $ ext{C}_1$ and $ ext{C}_2$ - HSC - SSCE Mathematics Extension 2 - Question 16 - 2014 - Paper 1
Question 16
The diagram shows two circles $ ext{C}_1$ and $ ext{C}_2$. The point $P$ is one of their points of intersection. The tangent to $ ext{C}_2$ at $P$ meets $ ext{C}_1$ ... show full transcript
Worked Solution & Example Answer:The diagram shows two circles $ ext{C}_1$ and $ ext{C}_2$ - HSC - SSCE Mathematics Extension 2 - Question 16 - 2014 - Paper 1
Step 1
Show that $ riangle APX = riangle LDQ$
Answer
To prove that is similar to , we can use the theorem which states that the angles in the alternate segment are equal. Here, we observe that and share the angle at point and both have their respective opposite angles equal due to the tangents. Therefore, by the AA criterion for similarity, we can conclude that .
Step 2
Show that $A$, $P$ and $C$ are collinear.
Answer
To establish the collinearity of points , , and , we note that since and are diameters of their respective circles and both are parallel, it follows that line segment must intersect line segment at point , confirming that , , and are indeed collinear.
Step 3
Show that $ABCD$ is a cyclic quadrilateral.
Answer
To show that quadrilateral is cyclic, we must prove that opposite angles sum to degrees. Since is perpendicular to (as both are diameters), this implies that the angle . Thus, by the property of cyclic quadrilaterals regarding angles subtended by the same arc, must be cyclic.
Step 4
Show that $ -2^n rac{1}{1+x^2} ext{ } - ext{ } igg( -1^{n} + x^4 - x^6 + extit{...} + (-1)^{n-1}(-2n-2) igg) imes rac{1}{x^{2}} x^{2n} ext{ } ext{ is less than or equal to } -x^{2n}.
Answer
To prove the inequality, we can start by analyzing the series on the left-hand side. We'll sum the series and show that the resulting expression is less than or equal to using induction or by comparing terms directly.
Step 5
Use integration to deduce that $ - rac{1}{2n+1} ext{ is less than or equal to } rac{ rac{ ext{π}}{4}}{1 - rac{1}{3} - rac{1}{5} - ext{...}} ext{ is less than or equal to } rac{1}{2n+1}.$
Answer
Using the established inequality, integrating both sides will show how respective limits constrain the overall relationship, allowing us to derive the necessary bounds for the integral of the series concerning rac{ ext{π}}{4}.
Step 6
Explain why $ rac{ ext{π}}{4} = 1 - rac{1}{3} + rac{1}{5} - ext{...}.
Answer
This series represents the Taylor expansion of the arctangent function around . Specifically, the series converges to rac{ ext{π}}{4}, demonstrating the fundamental relationship between integral calculus and series summation in trigonometric functions.
Step 7