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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.
90% of mathematicians do not understand 90% of currently published mathematics.
I am willing to wager that over 75% of the new words that appeared were nothing more than spelling errors that caught on.
#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.
90% of mathematicians do not understand 90% of currently published mathematics.
I am willing to wager that over 75% of the new words that appeared were nothing more than spelling errors that caught on.
#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.