Find $ \int_0^{\frac{\pi}{2}} \sin^2 x \, dx $. (a) (i) By considering $ f(x) = 3 \log x - x $, show that the curve $ y = 3 \log x $ and the line $ y = x $ meet at a point $ P $ whose x-coordinate is between 1.5 and 2. (ii) Use one application of Newton's method, starting at $ x = 1.5 $, to find an approximation to the x-coordinate of $ P $. Give your answer correct to two decimal places. (c) Sophie has five coloured blocks: one red, one blue, one green, one yellow and one white. She stacks two, three or four blocks on top of one another to form a vertical tower. (i) How many different towers are there that she could form that are three blocks high? (ii) How many different towers can she form in total? (d) The points $ P, Q $ and $ T $ lie on a circle. The line $ MN $ is tangent to the circle at $ T $ with $ M $ chosen so that $ QM $ is perpendicular to $ MN $. The point $ K $ on $ PQ $ is chosen so that $ TK $ is perpendicular to $ PQ $ as shown in the diagram. (i) Show that $ QKTN $ is a cyclic quadrilateral. (ii) Show that $ \angle KMT = \angle KQT $. (iii) Hence, or otherwise, show that $ MK $ is parallel to $ TP $.

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Find $ \int_0^{\frac{\pi}{2}} \sin^2 x \, dx $ - HSC - SSCE Mathematics Extension 1 - Question 3 - 2006 - Paper 1

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Find $ \int_0^{\frac{\pi}{2}} \sin^2 x \, dx $. (a) (i) By considering $ f(x) = 3 \log x - x $, show that the curve $ y = 3 \log x $ and the line $ y = x $ ... show full transcript

Worked Solution & Example Answer:Find $ \int_0^{\frac{\pi}{2}} \sin^2 x \, dx $ - HSC - SSCE Mathematics Extension 1 - Question 3 - 2006 - Paper 1

Step 1

Find $ \int_0^{\frac{\pi}{2}} \sin^2 x \, dx $

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Answer

To find ( \int_0^{\frac{\pi}{2}} \sin^2 x , dx ), we can use the identity ( \sin^2 x = \frac{1 - \cos(2x)}{2} ):

\int_0^{\frac{\pi}{2}} \sin^2 x \, dx = \int_0^{\frac{\pi}{2}} \frac{1 - \cos(2x)}{2} \, dx = \frac{1}{2} \int_0^{\frac{\pi}{2}} 1 \, dx - \frac{1}{2} \int_0^{\frac{\pi}{2}} \cos(2x) \, dx.

The first integral evaluates to ( \frac{\pi}{4} ), and the second integral is calculated as follows:

\int \cos(2x) \, dx = \frac{1}{2} \sin(2x) + C.

Evaluating ( \int_0^{\frac{\pi}{2}} \cos(2x) , dx ):

\left[ \frac{1}{2} \sin(2x) \right]_0^{\frac{\pi}{2}} = 0 - 0 = 0.

Thus,

\int_0^{\frac{\pi}{2}} \sin^2 x \, dx = \frac{1}{2} \left( \frac{\pi}{4} - 0 \right) = \frac{\pi}{8}.

Step 2

Show that the curve $ y = 3 \log x $ and the line $ y = x $ meet at a point $ P $

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Answer

To find the intersection of ( y = 3 \log x ) and ( y = x ), we set:

$3 \log x = x.$

We know that for ( x = 1.5 ), ( 3 \log(1.5) ) gives approximately ( 1.39 ), which is less than 1.5, and for ( x = 2 ), ( 3 \log(2) ) gives approximately ( 2.08 ), which is greater than 2. Hence, by the Intermediate Value Theorem there is a solution in the interval (1.5, 2).

Step 3

Use one application of Newton's method, starting at $ x = 1.5 $, to find an approximation to the x-coordinate of $ P $

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Answer

Let ( f(x) = 3 \log x - x ). The derivative ( f'(x) = \frac{3}{x} - 1 ).

Using Newton's method:

Start with ( x_0 = 1.5 ).
Compute ( x_1 = x_0 - \frac{f(x_0)}{f'(x_0)} ).
Calculating:
- ( f(1.5) = 3 \log(1.5) - 1.5 ) and ( f'(1.5) = \frac{3}{1.5} - 1 ).
- This gives ( x_1 \approx 1.72 ).
Repeat using ( x_1 ) to find ( x_2 \approx 1.72 ) rounded to two decimal places.

Step 4

How many different towers are there that she could form that are three blocks high?

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Answer

To find the number of towers that are three blocks high, we can choose 3 blocks from 5, which can be arranged in:

$5P3 = \frac{5!}{(5-3)!} = \frac{5!}{2!} = 5 \times 4 \times 3 = 60.$

So, there are 60 different towers of height 3.

Step 5

How many different towers can she form in total?

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Answer

To find the total number of towers of varying heights:

For 2 blocks: ( 5P2 = 5 \times 4 = 20 ).
For 3 blocks: ( 5P3 = 60 ).
For 4 blocks: ( 5P4 = 5 \times 4 \times 3 \times 2 = 120. )

Thus, total:

$20 + 60 + 120 = 200.$

So, there are 200 different towers in total.

Step 6

Show that $ QKTN $ is a cyclic quadrilateral.

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Answer

To show that ( QKTN ) is cyclic, we need to show that the opposite angles sum to 180 degrees. This can often be done by showing that they subtend the same arc in the circle.

Step 7

Show that $ \angle KMT = \angle KQT $.

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Answer

Using properties of tangents and angles, since ( MN ) is tangent at ( T ), it follows that ( \angle KMT ) is equal to the angle subtended at the circumference, giving us ( \angle KQT ).

Step 8

Hence, or otherwise, show that $ MK $ is parallel to $ TP $.

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Answer

Since ( \angle KMT = \angle KQT ), by alternate angles, it follows that ( MK ) is parallel to ( TP ) as they are transversals of parallel lines.

Find \( \int_0^{\frac{\pi}{2}} \sin^2 x \, dx \) - HSC - SSCE Mathematics Extension 1 - Question 3 - 2006 - Paper 1

Worked Solution & Example Answer:Find \( \int_0^{\frac{\pi}{2}} \sin^2 x \, dx \) - HSC - SSCE Mathematics Extension 1 - Question 3 - 2006 - Paper 1

Find \( \int_0^{\frac{\pi}{2}} \sin^2 x \, dx \)

Show that the curve \( y = 3 \log x \) and the line \( y = x \) meet at a point \( P \)

Use one application of Newton's method, starting at \( x = 1.5 \), to find an approximation to the x-coordinate of \( P \)

How many different towers are there that she could form that are three blocks high?

How many different towers can she form in total?

Show that \( QKTN \) is a cyclic quadrilateral.

Show that \( \angle KMT = \angle KQT \).

Hence, or otherwise, show that \( MK \) is parallel to \( TP \).

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