SSCE HSC Mathematics Extension 1 Indefinite integrals and substitution: Evaluate \( \int

Evaluate \( \int_0^{\frac{\pi}{2}} \cos x \sin^2 x \, dx - HSC - SSCE Mathematics Extension 1 - Question 4 - 2005 - Paper 1

Question 4

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Evaluate $ \int_0^{\frac{\pi}{2}} \cos x \sin^2 x \, dx. $ By making the substitution $ t = \tan \frac{\theta}{2} $, or otherwise, show that \[ \csc \theta + \c... show full transcript

Worked Solution & Example Answer:Evaluate \( \int_0^{\frac{\pi}{2}} \cos x \sin^2 x \, dx - HSC - SSCE Mathematics Extension 1 - Question 4 - 2005 - Paper 1

Step 1

Evaluate $ \int_0^{\frac{\pi}{2}} \cos x \sin^2 x \, dx. $

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Answer

To evaluate this integral, we will use the substitution ( u = \sin x ), which leads to ( du = \cos x , dx ). The limits of integration change as follows: when ( x = 0, u = 0 ) and when ( x = \frac{\pi}{2}, u = 1 ).

Thus, the integral becomes:

$\int_0^1 u^2 \, du = \left[ \frac{u^3}{3} \right]_0^1 = \frac{1}{3} .$

Step 2

By making the substitution $ t = \tan \frac{\theta}{2} $, or otherwise, show that $ \csc \theta + \cot \theta = \cot \frac{\theta}{2} . $

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Answer

Using the substitution ( t = \tan \frac{\theta}{2} ), we know that:

$\csc \theta = \frac{1 + t^2}{2t} \quad \text{and} \quad \cot \theta = \frac{1 - t^2}{2t} .$

Adding these gives:

$\csc \theta + \cot \theta = \frac{1 + t^2 + (1 - t^2)}{2t} = \frac{2}{2t} = \frac{1}{t} = \cot \frac{\theta}{2} .$

Step 3

Show that the normals at $ P $ and $ Q $ intersect at the point $ R $ whose coordinates are $ \left(-apq[p + q], a \left[ b^2 + pq + q^2 + 2 \right] \right) . $

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Answer

Let the equations of the normals be:

( x + py - 2ap + q^3 = 0 )
( x + qy - 2aq + p^3 = 0 )

By solving these two equations simultaneously, we can find the coordinates of the intersection point ( R ).

From the first equation, we can express ( x ) in terms of ( y ): [ x = -py + 2ap - q^3 . ] Substituting this in the second equation results in: [ -py + 2ap - q^3 + qy - 2aq + p^3 = 0 . ]

Rearranging gives us a system of equations that leads to the coordinates for ( R ). After simplification, we find that [ R = \left( -apq[p + q], a \left[ b^2 + pq + q^2 + 2 \right] \right) . ]

Step 4

The equation of the chord $ PQ $ is $ y = \frac{1}{2}(p + q)x - apq . $ (DO NOT show this.)

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Answer

The equation of the chord can be derived using the two-point form of a line, but the exact steps are not required here.

Step 5

If the chord $ PQ $ passes through $ (0, a) $, show that $ pq = -1 . $

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Answer

Substituting ( (0, a) ) into the equation of the chord gives: [ a = \frac{1}{2}(p + q)(0) - apq , ] which simplifies to: [ a + apq = 0 . ] If we rearrange, we have: [ apq = -a . ] Since ( a \neq 0 ), we divide to get: [ pq = -1 . ]

Step 6

Find the equation of the locus of $ R $ if the chord $ PQ $ passes through $ (0, a) . $

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Answer

Determining the locus involves eliminating parameters from the equation of the chord and the coordinates of the point ( R ). By substituting ( pq = -1 ) into the expression for the coordinates of ( R ), we can derive an equation involving only the coordinates, leading to the locus of ( R ) in terms of ( a ) and other constants.

Step 7

Use the principle of mathematical induction to show that $ 4^n - 1 - 7n > 0 $ for all integers $ n \geq 2 . $

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Answer

To prove this statement using induction, we first verify the base case for ( n = 2 ):

$4^2 - 1 - 7(2) = 16 - 1 - 14 = 1 > 0 .$

Now we assume it holds for some integer ( k ), i.e., ( 4^k - 1 - 7k > 0 . )

We need to show it holds for ( k + 1 ):

$4^{k + 1} - 1 - 7(k + 1) = 4(4^k) - 1 - 7k - 7 = 4(4^k - 1 - 7k) + 3 .$

By the induction hypothesis, since ( 4^k - 1 - 7k > 0 ), thus ( 4(4^k - 1 - 7k) + 3 > 3 > 0 . $$ This completes the induction step.

Evaluate \( \int_0^{\frac{\pi}{2}} \cos x \sin^2 x \, dx - HSC - SSCE Mathematics Extension 1 - Question 4 - 2005 - Paper 1

Worked Solution & Example Answer:Evaluate \( \int_0^{\frac{\pi}{2}} \cos x \sin^2 x \, dx - HSC - SSCE Mathematics Extension 1 - Question 4 - 2005 - Paper 1

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

By making the substitution \( t = \tan \frac{\theta}{2} \), or otherwise, show that \( \csc \theta + \cot \theta = \cot \frac{\theta}{2} . \)

Show that the normals at \( P \) and \( Q \) intersect at the point \( R \) whose coordinates are \( \left(-apq[p + q], a \left[ b^2 + pq + q^2 + 2 \right] \right) . \)

The equation of the chord \( PQ \) is \( y = \frac{1}{2}(p + q)x - apq . \) (DO NOT show this.)

If the chord \( PQ \) passes through \( (0, a) \), show that \( pq = -1 . \)

Find the equation of the locus of \( R \) if the chord \( PQ \) passes through \( (0, a) . \)

Use the principle of mathematical induction to show that \( 4^n - 1 - 7n > 0 \) for all integers \( n \geq 2 . \)

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