In the diagram AB is the diameter of the circle - HSC - SSCE Mathematics Extension 2 - Question 5 - 2009 - Paper 1
Question 5
In the diagram AB is the diameter of the circle. The chords AC and BD intersect at X. The point Y lies on AB such that XY is perpendicular to AB. The point K is the ... show full transcript
Worked Solution & Example Answer:In the diagram AB is the diameter of the circle - HSC - SSCE Mathematics Extension 2 - Question 5 - 2009 - Paper 1
Step 1
Show that ∠AKY = ∠ABD.
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Answer
According to the Inscribed Angle Theorem, since AB is the diameter, angle ADB is a right angle. Therefore, angle ADB and angle AKY, which intercept the same arc AB, are equal. Thus, we can conclude that ∠AKY = ∠ABD.
Step 2
Show that CKDX is a cyclic quadrilateral.
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Answer
To show that CKDX is cyclic, we need to establish that the opposite angles are supplementary. Since ∠AKY = ∠ABD, and using the exterior angle theorem, it follows that ∠KCA + ∠KDX = 180^ ext{o}, confirming CKDX is a cyclic quadrilateral.
Step 3
Show that B, C and K are collinear.
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Answer
Using the fact that angles ∠AKB and ∠CKB are on a straight line, if we can show that ∠AKB + ∠CKB = 180^ ext{o}, it implies that points B, C, and K are collinear. Given the geometric properties established, this criterion holds true.
Step 4
Show that for n ≥ 1, I_n = \frac{e}{2} - nI_{n-1}.
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Answer
Using integration by parts on the integral I_n, let u = x^n and dv = e^x dx. Then, du = n x^{n-1} dx and v = e^x. Thus, we can express I_n as follows:
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Answer
Starting from the established formula, I_1 can be calculated first using integration parts:
For I_1, repeat the process of integration by parts to find its value, and substitute this into the result to calculate I_2.
Step 6
Show that f'(x) > 0 for all x > 0.
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Answer
Differentiate f(x):
f′(x)=et−e−ximes(−e−x)−1.
Analyzing the non-negativity of terms, and given the decay of e^{-x} for x > 0, implies f'(x) is indeed greater than zero.
Step 7
Hence, or otherwise, show that f'(x) > 0 for all x > 0.
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Answer
Using the result of f'(x) above and evaluating the function, it indicates that as x increases, f'(x) does not diminish and thus remains positive for all x > 0.
Step 8
Hence, show that e^{x} - e^{-x} > 2 for all x > 0.
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Answer
Utilizing the results from the derivative calculations, and properties of exponential functions, it follows that since both exponential terms grow significantly faster than linear increases for larger x-values, the established inequality must hold true.
