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

Photo AI

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

Question 3

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

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

96%

114 rated

Answer

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

Thus, our integral becomes:

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

Calculating the first integral, we get:

$\int_0^{\frac{\pi}{2}} 1 \, dx = \left[ x \right]_0^{\frac{\pi}{2}} = \frac{\pi}{2}$

The second integral can be calculated as:

$\int_0^{\frac{\pi}{2}} \cos(2x) \, dx = \left[ \frac{\sin(2x)}{2} \right]_0^{\frac{\pi}{2}} = \frac{\sin(\pi) - \sin(0)}{2} = 0.$

Thus,

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

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.

99%

104 rated

Answer

To find the intersection points of ( f(x) = 3 \log x - x ), we first evaluate this function at the endpoints.

For ( x = 1.5 ): [ f(1.5) = 3 \log(1.5) - 1.5 \approx 3(0.405) - 1.5 \approx 1.215 - 1.5 = -0.285. ]
For ( x = 2 ): [ f(2) = 3 \log(2) - 2 \approx 3(0.693) - 2 \approx 2.079 - 2 = 0.079. ]

Thus, since ( f(1.5) < 0 ) and ( f(2) > 0 ), by the Intermediate Value Theorem, there exists a point ( P ) in the interval ( (1.5, 2) ) where ( f(P) = 0 ).

Step 3

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

96%

101 rated

Answer

Newton's method is given by:

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

We first calculate the derivative of ( f(x) ):

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

Now, we start at ( x_0 = 1.5 ):

Calculate ( f(1.5) ) and ( f'(1.5) ): [ f(1.5) \approx -0.285, ; f'(1.5) \approx \frac{3}{1.5} - 1 \approx 1.0. ]
Using Newton's formula: [ x_1 = 1.5 - \frac{-0.285}{1.0} = 1.785. ]
Repeat for ( x_1 = 1.785 ): [ f(1.785) \approx 0.030, ; f'(1.785) \approx 0.678. ] [ x_2 = 1.785 - \frac{0.030}{0.678} \approx 1.782. ]
Continue until convergence: The answer can be approximated at ( x \approx 1.78 ) after calculus up to two decimal places.

Step 4

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

98%

120 rated

Answer

To find the total combinations for stacking three blocks from five different colored blocks, we can use the formula:

$C(n, r) = \frac{n!}{(n - r)! r!}.$

For each arrangement, we choose 3 blocks from 5,

$C(5, 3) = \frac{5!}{(5 - 3)! 3!} = 10.$

However, since the order in which the blocks are stacked matters, there are ( 3! = 6 ) ways to arrange those blocks. Thus,

Total arrangements = ( 10 \times 6 = 60 ).

Step 5

How many different towers can she form in total?

97%

117 rated

Answer

To calculate the total number of towers she can form with 2, 3, 4, or 5 blocks high, we calculate:

For two blocks: ( C(5, 2) \times 2! = 10 \times 2 = 20 )
For three blocks: ( C(5, 3) \times 3! = 10 \times 6 = 60 )
For four blocks: ( C(5, 4) \times 4! = 5 \times 24 = 120 )
For five blocks: ( C(5, 5) \times 5! = 1 \times 120 = 120 )

Adding these together:

Total = ( 20 + 60 + 120 + 120 = 320 ).

Step 6

Show that $ QKTN $ is a cyclic quadrilateral.

97%

121 rated

Answer

For quadrilateral ( QKTN ) to be cyclic, the opposite angles must sum to ( 180^\circ ). We prove:

By alternate segment theorem, ( \angle QKN + \angle QTN = 180^\circ ) since they subtend the same arc.

Thus, ( QKTN ) is cyclic.

Step 7

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

96%

114 rated

Answer

From the tangent property at point ( T ): ( \angle KMT ) is equal to the angle subtended by arc ( KT ).

Since ( KQT ) subtends the same arc, we have: ( \angle KMT = \angle KQT. )

Step 8

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

99%

104 rated

Answer

From the previous angle relationships:

If ( \angle KMT = \angle KQT ), it follows that lines ( MK ) and ( TP ) are parallel by the converse of the alternate interior angles theorem.

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 \( QKTN \) is a cyclic quadrilateral.

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

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

Join the SSCE students using SimpleStudy...