<![CDATA[Math Is Fun Forum / Level of Difficulty - III Questions.]]> 2010-05-22T12:34:14Z FluxBB https://www.mathisfunforum.com/viewtopic.php?id=11756 <![CDATA[Re: Level of Difficulty - III Questions.]]> Well done,
irspow!!!

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https://www.mathisfunforum.com/profile.php?id=682 2010-05-22T12:34:14Z https://www.mathisfunforum.com/viewtopic.php?pid=140683#p140683
<![CDATA[Re: Level of Difficulty - III Questions.]]>

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https://www.mathisfunforum.com/profile.php?id=2041 2010-05-21T15:03:22Z https://www.mathisfunforum.com/viewtopic.php?pid=140617#p140617
<![CDATA[Re: Level of Difficulty - III Questions.]]> 3. tan θ=1/2

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https://www.mathisfunforum.com/profile.php?id=64625 2010-05-17T07:40:23Z https://www.mathisfunforum.com/viewtopic.php?pid=139344#p139344
<![CDATA[Re: Level of Difficulty - III Questions.]]> 1.from sine rule we get sinA+sinC=2sinB, or cos([A-C]/2)=2cos(B/2)
tan(A/2)+tan(c/2)=cos(B/2)/[cos (A/2) cos (C/2)]=2cos(B/2)/[sin(B/2)+2cos(B/2)]=2/[1+cot (B/2)]

Then the diameter of the square=2a/3
area=(2a/3 sin45)^2=a^2/9

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<![CDATA[Re: Level of Difficulty - III Questions.]]> Hi bitus,

For #4 I think you mean

doesn't work when I plug it in.

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https://www.mathisfunforum.com/profile.php?id=33790 2009-04-21T10:44:35Z https://www.mathisfunforum.com/viewtopic.php?pid=110427#p110427
<![CDATA[Re: Level of Difficulty - III Questions.]]> #6 : cosC=1/2⇒b² -(b/2)²=c² -(a-b/2)² ⇒ab=a²+b²-c² ⇒3ab=(a+b)² -c²=(a+b+c)(a+b-c)=(a+b+c)(a+b+c-2c)=
(a+b+c)²-2c(a+b+c)⇒3ab+2c(a+b+c)=(a+b+c)²⇒3ab+3c(a+b+c)=(a+b+c)²+c(a+b+c)⇒
3(a+c)(b+c)=(a+b+c)(b+c+a+c)⇒[(b+c)+(a+c)]/(a+c)(b+c)=3/(a+b+c)⇒1/(a+c)+1/(b+c)=
3/(a+b+c).

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https://www.mathisfunforum.com/profile.php?id=16452 2009-04-21T04:56:34Z https://www.mathisfunforum.com/viewtopic.php?pid=110413#p110413
<![CDATA[Re: Level of Difficulty - III Questions.]]> #1: tan(A/2) +tan(C/2) = (2/3)cot(B/2)

#4: x=pi/8 or 3pi/8

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https://www.mathisfunforum.com/profile.php?id=16452 2009-04-21T03:35:26Z https://www.mathisfunforum.com/viewtopic.php?pid=110411#p110411
<![CDATA[Re: Level of Difficulty - III Questions.]]> Hi Ganesh;
For #9

Solve for x and y in terms of m to understand how x and y behave when m varies:

After a lot of algebraic thrashing and some trial and error:

These values of m satisfy the constraints x>0 and y>0

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https://www.mathisfunforum.com/profile.php?id=33790 2009-04-13T18:25:37Z https://www.mathisfunforum.com/viewtopic.php?pid=110014#p110014
<![CDATA[Re: Level of Difficulty - III Questions.]]> 9. For what values of m does the system of equations
3x + my = m and
2x - 5y = 20
have solutions satisfying the condtions x>0, y>0?

10. Let -1 ≤  p ≤  1. Show that the equation 4x[sup]3[/sup] - 3x - p = 0 has a unique root in the interval [½, 1] and identify it.

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https://www.mathisfunforum.com/profile.php?id=682 2009-03-16T15:06:18Z https://www.mathisfunforum.com/viewtopic.php?pid=107693#p107693
<![CDATA[Re: Level of Difficulty - III Questions.]]> Answer to 7:-
Correct, JaneFairfax! ]]>
https://www.mathisfunforum.com/profile.php?id=682 2009-03-15T18:13:59Z https://www.mathisfunforum.com/viewtopic.php?pid=107554#p107554
<![CDATA[Re: Level of Difficulty - III Questions.]]>

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https://www.mathisfunforum.com/profile.php?id=6777 2009-03-15T16:42:32Z https://www.mathisfunforum.com/viewtopic.php?pid=107541#p107541
<![CDATA[Re: Level of Difficulty - III Questions.]]> 5. Find the area bounded by the curve x[sup]2[/sup]=4y and the
straight line x = 4y - 2.

6. If a, b, c are the three sides of a triangle and C = 60°, prove that

7. If a>0, b>0, and c>0, prove that

.

8.Find the equations of straight lines passing through (-2, -7) and having an interept of length 3 between the striaght lines
4x+3y = 12 and 4x+3y = 3.

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https://www.mathisfunforum.com/profile.php?id=682 2009-03-15T12:43:26Z https://www.mathisfunforum.com/viewtopic.php?pid=107496#p107496
<![CDATA[Level of Difficulty - III Questions.]]> 1. If the sides of a triangle are in Arithmetic Progression, then find the value of

in terms of
.

2. A circle is inscribed in an equilateral triangle of side a. Find the area of the square inscribed in this circle.

3. Find the general value of θ satisfying the equation
tan[sup]2[/sup]θ + sec2θ = 1.

4. Solve the equation
sin x - 3sin 2x + sin 3x = cos x - 3cos 2x + cos 3x.

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https://www.mathisfunforum.com/profile.php?id=682 2009-03-13T06:32:28Z https://www.mathisfunforum.com/viewtopic.php?pid=107259#p107259