Level of Difficulty - III Questions.

Jai Ganesh · 2009-03-12 18:32:28

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.

Jai Ganesh · 2009-03-15 00:43:26

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.

JaneFairfax · 2009-03-15 04:42:32

Jai Ganesh · 2009-03-15 06:13:59

Answer to 7:-
Correct, JaneFairfax!

Jai Ganesh · 2009-03-16 03:06:18

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.

bobbym · 2009-04-13 06:25:37

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)

bitus · 2009-04-20 15:35:26

#1: tan(A/2) +tan(C/2) = (2/3)cot(B/2)

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

bitus · 2009-04-20 16:56:34

#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).

bobbym · 2009-04-20 22:44:35

Hi bitus,

For #4 I think you mean

doesn't work when I plug it in.

BO · 2010-05-16 19:26:06

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)

BO · 2010-05-16 19:40:23

3. tan θ=1/2

irspow · 2010-05-21 03:03:22

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

Jai Ganesh · 2010-05-22 00:34:14

Well done,
irspow!!!

Math Is Fun Forum

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

Level of Difficulty - III Questions.

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

Re: Level of Difficulty - III Questions.

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

Re: Level of Difficulty - III Questions.

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

Re: Level of Difficulty - III Questions.

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

Re: Level of Difficulty - III Questions.

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

Re: Level of Difficulty - III Questions.

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

Re: Level of Difficulty - III Questions.

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

Re: Level of Difficulty - III Questions.

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

Re: Level of Difficulty - III Questions.

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

Re: Level of Difficulty - III Questions.

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

Re: Level of Difficulty - III Questions.

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

Re: Level of Difficulty - III Questions.

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

Re: Level of Difficulty - III Questions.

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