Find $ \int_0^{\frac{\pi}{2}} \sin^2 x \, dx $. (a) 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. (b) (i) 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, four or five 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 $ QKT M $ is a cyclic quadrilateral. (ii) Show that $ \angle KMT = \angle KQT $. (iii) Hence, or otherwise, show that $ MK $ is parallel to $ TP $.

SSCE HSC Mathematics Extension 1 Combinations: Find \( \int

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

Question 3

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

Calculating the integral gives:

= \frac{1}{2} \left[ x - \frac{1}{2} \sin(2x) \right]_0^{\frac{\pi}{2}} = \frac{1}{2} \left( \frac{\pi}{2} - 0 \right) = \frac{\pi}{4}.

Step 2

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.

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Answer

To show that the curve and line intersect, we must find roots of the equation:

3 \log x - x = 0.

We can evaluate this at both ends of the interval:

For ( x = 1.5 ): $3 \log(1.5) - 1.5 \approx 3(0.405) - 1.5 \approx 1.215 - 1.5 < 0.$
For ( x = 2 ): $3 \log(2) - 2 \approx 3(0.693) - 2 \approx 2.079 - 2 > 0.$

Since the function changes signs between 1.5 and 2, by the Intermediate Value Theorem, there exists at least one root in that interval.

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

Newton's method uses the formula:

x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}.

We compute:

( f'(x) = \frac{3}{x} - 1 ).

Starting with ( x_0 = 1.5 ):

Calculate ( f(1.5) ) and ( f'(1.5) ).
After one iteration, we find ( x_1 \approx 1.75 ). Thus, the x-coordinate of ( P ) is approximately ( 1.75 ).

Step 4

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

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Answer

For a tower that is three blocks high using five different colored blocks: The number of different permutations for selecting and arranging 3 blocks from 5 is:

5P3 = \frac{5!}{(5-3)!} = \frac{5!}{2!} = \frac{5 \times 4 \times 3}{1} = 60.

Step 5

How many different towers can she form in total?

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Answer

To find the total number of towers of height 2, 3, 4, or 5:

For height 2: ( 5P2 = 20 ).
For height 3: ( 5P3 = 60 ).
For height 4: ( 5P4 = 120 ).
For height 5: ( 5P5 = 120 ). Adding these gives:\n $20 + 60 + 120 + 120 = 320.$ So, the total number of different towers Sophie can create is 320.

Step 6

Show that $ QKT M $ is a cyclic quadrilateral.

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Answer

For ( QKT M ) to be cyclic, we need to show that the sum of opposite angles equal 180 degrees. Angles ( \angle QKM ) and ( \angle QTM ) subtend the same arc ( QM ). Thus, ( \angle QKM + \angle QTM = 180^{\circ} ).

Step 7

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

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Answer

Since ( KM ) is a tangent to the circle at ( T ), we have that ( \angle KMT ) is equal to the angle subtended by the arc ( QT ), which is ( \angle KQT ) (alternate segment theorem). Thus, ( \angle KMT = \angle KQT. )

Step 8

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

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Answer

From the previous part, we found that ( \angle KMT = \angle KQT ). If two lines are cut by a transversal (which is ( TP )), and the corresponding angles are equal, then by the converse of the Alternate Interior Angles Theorem, the lines ( MK ) and ( TP ) are parallel.

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

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.

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 \( QKT M \) is a cyclic quadrilateral.

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

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

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