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#1 2009-03-12 18:32:28

ganesh
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Registered: 2005-06-28
Posts: 14,865

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.


Character is who you are when no one is looking.

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#2 2009-03-15 00:43:26

ganesh
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Registered: 2005-06-28
Posts: 14,865

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.


Character is who you are when no one is looking.

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#3 2009-03-15 04:42:32

JaneFairfax
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Registered: 2007-02-23
Posts: 6,868

Re: Level of Difficulty - III Questions.


Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

A: Click here for answer.

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#4 2009-03-15 06:13:59

ganesh
Moderator
Registered: 2005-06-28
Posts: 14,865

Re: Level of Difficulty - III Questions.

Answer to 7:-
Correct, JaneFairfax!
grtjb6.jpg


Character is who you are when no one is looking.

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#5 2009-03-16 03:06:18

ganesh
Moderator
Registered: 2005-06-28
Posts: 14,865

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.


Character is who you are when no one is looking.

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#6 2009-04-13 06:25:37

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 89,048

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

Last edited by bobbym (2009-04-13 06:38:13)


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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#7 2009-04-20 15:35:26

bitus
Member
Registered: 2008-10-12
Posts: 20

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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#8 2009-04-20 16:56:34

bitus
Member
Registered: 2008-10-12
Posts: 20

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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#9 2009-04-20 22:44:35

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 89,048

Re: Level of Difficulty - III Questions.

Hi bitus,

For #4 I think you mean


doesn't work when I plug it in.


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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#10 2010-05-16 19:26:06

BO
Member
Registered: 2010-05-16
Posts: 9

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)]

2.the radius of the circle=a*tan30=a/3
Then the diameter of the square=2a/3
area=(2a/3 sin45)^2=a^2/9

Last edited by BO (2010-05-16 19:32:48)

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#11 2010-05-16 19:40:23

BO
Member
Registered: 2010-05-16
Posts: 9

Re: Level of Difficulty - III Questions.

3. tan θ=1/2

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#12 2010-05-21 03:03:22

irspow
Member
Registered: 2005-11-24
Posts: 455

Re: Level of Difficulty - III Questions.

Last edited by irspow (2010-05-21 03:04:34)

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#13 2010-05-22 00:34:14

ganesh
Moderator
Registered: 2005-06-28
Posts: 14,865

Re: Level of Difficulty - III Questions.

Well done,
irspow!!!


Character is who you are when no one is looking.

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