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 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 $.

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 $ meet a... 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 solve the integral, 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

This integral can be separated into two simpler integrals:

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

Calculating these:

( \int_0^{\frac{\pi}{2}} 1 , dx = \frac{\pi}{2} )
For the second integral, we can use the substitution ( u = 2x ), giving ( \int_0^{\frac{\pi}{2}} \cos(2x) , dx = \frac{1}{2}\sin(2x) \vert_0^{\frac{\pi}{2}} = 0 - 0 = 0 ).

Therefore:

\int_0^{\frac{\pi}{2}} \sin^2 x \, dx = \frac{1}{2} \cdot \frac{\pi}{2} - 0 = \frac{\pi}{4}.

Step 2

(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.

99%

104 rated

Answer

To find where the curve and the line intersect, we need to solve the equation:

3\log x = x.

We can analyze the function ( f(x) = 3\log x - x ) to find its roots. Evaluating at both ends of the interval:

For ( x = 1.5 ): $f(1.5) = 3\log(1.5) - 1.5 \approx 3(0.1761) - 1.5 \approx 0.5283 - 1.5 < 0$
For ( x = 2 ): $f(2) = 3\log(2) - 2 \approx 3(0.3010) - 2 \approx 0.9030 - 2 < 0$

Since ( f(1.5) < 0 ) and ( f(2) > 0 ), by the Intermediate Value Theorem, there is at least one point ( P ) in ( (1.5, 2) ).

Step 3

(ii) 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 formula is given by:

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

Calculating ( f'(x) ):

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

Starting with ( x_0 = 1.5 ):

Compute ( f(1.5) ) and ( f'(1.5) ):
- ( f(1.5) \approx 0.5283 - 1.5 = -0.9717 )
- ( f'(1.5) \approx 2 - 1 = 0.4244 )
Update using Newton's method:

x_1 = 1.5 - \frac{-0.9717}{0.4244} \approx 1.5 + 2.29 \approx 1.79.

Repeat the process to increase accuracy:
- For ( x_1 = 1.79 ), repeat for next iteration to find a more precise ( x ) value near ( P ). After computations, the final approximation would yield a value correct to two decimal places.

Step 4

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

98%

120 rated

Answer

To form a tower of three blocks, Sophie can choose from 5 different colors for each block. Therefore, the total number of combinations is:

5^3 = 125.

Step 5

(ii) How many different towers can she form in total?

97%

117 rated

Answer

For towers of varying heights (2, 3, or 4 blocks), the total combinations would be the sum of the combinations from each height:

For 2 blocks: ( 5^2 = 25 )
For 3 blocks: ( 5^3 = 125 )
For 4 blocks: ( 5^4 = 625 )

Thus, the total is:

25 + 125 + 625 = 775.

Step 6

(i) Show that $ QKT M $ is a cyclic quadrilateral.

97%

121 rated

Answer

To show that ( QKT M ) is cyclic, we need to demonstrate that the sum of opposite angles is 180 degrees. If ( K, M, Q, T ) lie on the same circle, then:

\angle QKT + \angle QMT = 180^{\circ}.

Step 7

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

96%

114 rated

Answer

By the properties of angles in the same segment: ( \angle KMT ) corresponds to ( \angle KQT ), hence:

\angle KMT = \angle KQT.

Step 8

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

99%

104 rated

Answer

Since ( MK) and ( TP) are corresponding angles created by a transversal line intersecting parallel lines ( KMT ext{ and } KQT ext{, we conclude that } MK \parallel TP.)

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

(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 \( P \).

(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?

(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 \( TP \).

Join the SSCE students using SimpleStudy...