The triangle ABC is right-angled at A and has sides with lengths a, b and c, as shown in the diagram - HSC - SSCE Mathematics Extension 2 - Question 5 - 2005 - Paper 1
Question 5
The triangle ABC is right-angled at A and has sides with lengths a, b and c, as shown in the diagram.
By considering areas, or otherwise, show that $b^2 + c^2 = d^2(... show full transcript
Worked Solution & Example Answer:The triangle ABC is right-angled at A and has sides with lengths a, b and c, as shown in the diagram - HSC - SSCE Mathematics Extension 2 - Question 5 - 2005 - Paper 1
Step 1
Show that $b^2 + c^2 = d^2(b^2 + c^2)$
96%
114 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
To show that ( b^2 + c^2 = d^2(b^2 + c^2) ), we can consider the area of triangle ABC. The area can be expressed in two ways: as ( \frac{1}{2}ab ) and as the sum of areas of rectangles formed by the heights. Setting these equal allows us to derive the relationship needed.
Step 2
Show that \( \tan \gamma = \tan \alpha + \tan \beta \)
99%
104 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
Using the relationships established in part (i), we can apply trigonometric identities. From triangle properties, we find ( \tan \gamma ) in terms of ( \tan \alpha ) and ( \tan \beta ) to complete the proof.
Step 3
Explain why there are five different ways
96%
101 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
Ferdinand can score only one goal in five different positions in the sequence of goals scored. The initial M can appear in various configurations according to the arrangements of the other initials following specific constraints of scoring.
Step 4
In how many different ways could the outcome of this competition be recorded?
98%
120 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
The number of outcomes can be calculated using combinatorial methods. For three goals, we can model it with combinatorial coefficients, yielding a total of 10 unique arrangements.
Step 5
Explain why \( \int_{0}^{a} f(x) dx = ab - \int_{0}^{b} f^{-1}(y) dy \)
97%
117 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
This result follows from the Fundamental Theorem of Calculus and properties of increasing functions, relating areas under curves for function f and its inverse.
Step 6
Find the value of \( \int_{0}^{\frac{\pi}{4}} \sin^{-1}(x) dx \)
97%
121 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
Applying integration by parts leads to this integral simplifying to a manageable form, ultimately yielding a numerical solution through careful substitution.
Step 7
Show that the area of ABCD is given by \( \frac{2x}{\sqrt{27 - 3x^2}} \)
96%
114 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
Using the geometry of the wedge and finding dimensions in terms of x gives the desired area expression through straightforward calculations.
Step 8
Find the volume of the wedge.
99%
104 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
The volume can be evaluated by integrating the area found in the previous step across the height of the wedge's region, yielding a final calculated volume.
