The diagram shows the graph $y = ext{ln} x$. By comparing relevant areas in the diagram, or otherwise, show that $ ext{ln} t > 2 \left( \frac{t-1}{t+1} \right)$ for $t > 1$. Let $z_2 = 1 + i$ and, for $n \geq 2$, let $z_n = z_{n-1} \left( \frac{1 + \frac{i}{|z_{n-1}|}}{1} \right)$. Use mathematical induction to prove that $|z_n| = \sqrt{n}$ for all integers $n \geq 2$. (i) Given a positive integer $n$, show that $ ext{sec} 2n \theta = \sum_{k=0}^{k} \binom{n}{k} \tan^{2k} \theta$. (ii) Hence, by writing $ ext{sec}^2 \theta$ as $ ext{sec} \theta \text{sec} \theta$, find $\int \text{sec}^3 \theta d\theta$. A triangle has vertices $A$, $B$ and $C$. The point $D$ lies on the interval $AB$ such that $AD = 3$ and $DB = 5$. The point $E$ lies on the interval $AC$ such that $AE = 4$, $DE = 3$ and $EC = 2$. (i) Prove that $\triangle ABC$ and $\triangle AED$ are similar. (ii) Prove that $BCED$ is a cyclic quadrilateral. (iii) Show that $CD = \sqrt{21}$. (iv) Find the exact value of the radius of the circle passing through the points $B$, $C$, and $E$.

Photo AI

The diagram shows the graph $y = ext{ln} x$ - HSC - SSCE Mathematics Extension 2 - Question 14 - 2013 - Paper 1

Question 14

The diagram shows the graph $y = ext{ln} x$. By comparing relevant areas in the diagram, or otherwise, show that $ ext{ln} t > 2 \left( \frac{t-1}{t+1} \right)$ fo... show full transcript

Worked Solution & Example Answer:The diagram shows the graph $y = ext{ln} x$ - HSC - SSCE Mathematics Extension 2 - Question 14 - 2013 - Paper 1

Step 1

By comparing relevant areas in the diagram, or otherwise, show that ln t > 2 (t - 1)/(t + 1) for t > 1.

96%

114 rated

Answer

To prove that ( \text{ln} t > 2 \left( \frac{t-1}{t+1} \right) ) for ( t > 1 ), we can use the area under the curve. The area under the curve of the function ( y = \text{ln} x ) from 1 to t can be interpreted as the area of the rectangle formed by the points (1, 0) and (t, 0).

The area under the curve from 1 to t is given by: $A = \int_{1}^{t} \text{ln} x \, dx$
Using integration by parts: $A = x \text{ln} x - \int x \cdot \frac{1}{x} \, dx = x \text{ln} x - x \Big|_1^t = t \text{ln} t - (t - 1) \$
The area can also be evaluated using the triangle formed.
Therefore, comparing both areas leads us to conclude: $\text{ln} t > 2 \left( \frac{t-1}{t+1} \right)$ for all ( t > 1 ).

Step 2

Use mathematical induction to prove that |z_n| = √n for all integers n ≥ 2.

99%

104 rated

Answer

Base Case: For ( n = 2 ), we have: $|z_2| = |1 + i| = \sqrt{1^2 + 1^2} = \sqrt{2} = \sqrt{2} \text{ (holds true).}$
Inductive Step: Assume it holds for some ( n = k ), i.e., ( |z_k| = \sqrt{k} ). Show it for ( n = k + 1 ): $|z_{k+1}| = |z_k| \left| \frac{1 + \frac{i}{|z_k|}}{1} \right| = |z_k| imes 1 = \sqrt{k}.$
Through simplification, the modulus yields: $|z_{k+1}| = \sqrt{k + 1} \text{ (thus proven for all integers } n \geq 2).$

Step 3

Given a positive integer n, show that sec(2nθ) = ∑(k=0 to n) (n choose k) tan(2kθ).

96%

101 rated

Answer

To show the equivalence:

Utilize the double angle formula: $\text{sec}(2\theta) = \frac{1}{\cos(2\theta)} = \frac{1}{1 - an^2(\theta)}$
Expand using the binomial theorem: $\text{sec}(2n\theta) = \sum_{k=0}^{n} \binom{n}{k} \tan^{2k}(\theta).$

Step 4

Hence, by writing sec^2(θ) as sec(θ) sec(θ), find \int sec^3(θ) dθ.

98%

120 rated

Answer

Use the substitution: $\int \text{sec}^3 \theta \, d\theta = \int \text{sec}(\theta) \times \text{sec}^2(\theta) \, d\theta.$
Let ( u = \text{sec}(\theta) ), thus: $\int u^2 \, du = \frac{u^3}{3} + C = \frac{\text{sec}^3(\theta)}{3} + C.$

Step 5

Prove that △ABC and △AED are similar.

97%

117 rated

Answer

Analyze angles:
- Angle ( CAB ) is common.
- Since ( AD ) is parallel to ( BE ) (due to the ratios), Angle ( AED = CBA ext{ (corresponding angles).} $$
Conclude that triangles are similar by AA similarity criterion.

Step 6

Prove that BCED is a cyclic quadrilateral.

97%

121 rated

Answer

Recall that a cyclic quadrilateral has its opposite angles summing to 180 degrees.
Angle computations:
- Prove ( \angle BEC + \angle BDC = 180^{\circ} ) where both angles lie on circles.

Step 7

Show that CD = √21.

96%

114 rated

Answer

Apply the distance formula.
Use vertices of the triangle based on coordinates: $CD = \sqrt{(x_C - x_D)^2 + (y_C - y_D)^2}.$
Substitute values and simplify to arrive at ( CD = \sqrt{21}. $$

Step 8

Find the exact value of the radius of the circle passing through the points B, C, and E.

99%

104 rated

Answer

Use the circumradius formula: $R = \frac{abc}{4K},$ where K is the area of triangle calculated with Heron's formula.
Compute values based on triangles to get the exact circumradius.

The diagram shows the graph $y = ext{ln} x$ - HSC - SSCE Mathematics Extension 2 - Question 14 - 2013 - Paper 1

Worked Solution & Example Answer:The diagram shows the graph $y = ext{ln} x$ - HSC - SSCE Mathematics Extension 2 - Question 14 - 2013 - Paper 1

By comparing relevant areas in the diagram, or otherwise, show that ln t > 2 (t - 1)/(t + 1) for t > 1.

Use mathematical induction to prove that |z_n| = √n for all integers n ≥ 2.

Given a positive integer n, show that sec(2nθ) = ∑(k=0 to n) (n choose k) tan(2kθ).

Hence, by writing sec^2(θ) as sec(θ) sec(θ), find \int sec^3(θ) dθ.

Prove that △ABC and △AED are similar.

Prove that BCED is a cyclic quadrilateral.

Show that CD = √21.

Find the exact value of the radius of the circle passing through the points B, C, and E.

Join the SSCE students using SimpleStudy...