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#1 2009-03-13 17:32:28
- ganesh
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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
tan2θ + 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.
#2 2009-03-15 23:43:26
- ganesh
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Re: Level of Difficulty - III Questions.
5. Find the area bounded by the curve x2=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.
#3 2009-03-16 03:42:32
#4 2009-03-16 05:13:59
- ganesh
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Re: Level of Difficulty - III Questions.
Answer to 7:-
Correct, JaneFairfax!
Character is who you are when no one is looking.
#5 2009-03-17 02:06:18
- ganesh
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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 4x3 - 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.
#6 2009-04-14 04:25:37
- bobbym
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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 (2009-04-14 04:38:13)
In mathematics, you don't understand things. You just get used to them.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.
#7 2009-04-21 13:35:26
- bitus
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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
#8 2009-04-21 14:56:34
- bitus
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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).
#9 2009-04-21 20:44:35
- bobbym
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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.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.
#10 2010-05-17 17:26:06
- BO
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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 (2010-05-17 17:32:48)
#12 2010-05-22 01:03:22
- irspow
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Re: Level of Difficulty - III Questions.
Last edited by irspow (2010-05-22 01:04:34)
#13 2010-05-22 22:34:14
- ganesh
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Re: Level of Difficulty - III Questions.
Well done,
irspow!!!
Character is who you are when no one is looking.