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**Ikcelaks****Member**- Registered: 2006-03-13
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**Ikcelaks****Member**- Registered: 2006-03-13
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**ganesh****Moderator**- Registered: 2005-06-28
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**Excellent, Ikcelaks! Your solution to Problem (11) is correct! I shall solve problem 8 and let you know whether you are correct!**

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**ganesh****Moderator**- Registered: 2005-06-28
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***12

What is the sum of all the coefficients of

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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**mathsyperson****Moderator**- Registered: 2005-06-22
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Why did the vector cross the road?

It wanted to be normal.

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**ganesh****Moderator**- Registered: 2005-06-28
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**mathsyperson,**

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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**ganesh****Moderator**- Registered: 2005-06-28
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***13. If the mth, nth and rth term of an Arithmetic Progression are in Geometric Progression, and m, n, and r are in Harmonic Progression, what is a/d of the Arithmetic Progression?

***14 What is the coefficient of

in(1+x)(2+x)(3+x)(4+x)..............(99+x)(100+x)?

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**ganesh****Moderator**- Registered: 2005-06-28
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***15 The points (1,3) and (5,1) are the two opoosite vertices of a rectangle. The other two vertices lie on the line y=2x+c. Find c and the remaining vertices.

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**rm****Member**- Registered: 2006-03-14
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***15

The diagonals of a parallelogram bisect each other. So y=2x+c must pass through middle of the opposite vertices - (3,2). On substituting, we get c = -4.

Let us say that vertices are (x,y) the angle between two lines from (x,y) to the other 2 vertices should be 90. Which is true if and only if m1 X m2 = -1. Using this and y=2x-4, we get the other vertices as (2,0) and (4,4)

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**ganesh****Moderator**- Registered: 2005-06-28
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**ganesh****Moderator**- Registered: 2005-06-28
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***16 Find the incentre of a triangle whose vertices are (-4,6), (-2,-5), and (2,3).

***17 Given that x=cy+bz, y=az+cx, and z=bx+ay, where x,y,z≠0,

prove that a²+b²+c²+2abc=1.

***18 Prove that

is divisible by a-b.***19 Prove that the complex numberx z1, z2 and the origin form an equilateral triangle only if z1²+z2²2z1z2=0

***20 The angles of a triangle are in the ratio 1:2:3. Show that the sides are in the ratio 1:√3:2.

***21 Prove that the maximum value of

is obtained when x=e.Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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**rm****Member**- Registered: 2006-03-14
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**ganesh****Moderator**- Registered: 2005-06-28
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**Very well done, rm! Your answers other than ***16 are correct!**

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**ganesh****Moderator**- Registered: 2005-06-28
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***22. In the Cartesian plane, four points have coordiantes (1,1), (4,2), (4,4),and (1,4). What is the are of the quadrilateral formed by joining the four points?

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**rm****Member**- Registered: 2006-03-14
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**ganesh****Moderator**- Registered: 2005-06-28
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**Excellent, rm! You are correct! **

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**ganesh****Moderator**- Registered: 2005-06-28
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***23 Find the sum of all numbers from 1 to 100 which are not divisible by 3 and 5.

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**Ikcelaks****Member**- Registered: 2006-03-13
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**ganesh****Moderator**- Registered: 2005-06-28
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**Excellent, Ikcelaks! **

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**krassi_holmz****Real Member**- Registered: 2005-12-02
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***14

12582075

IPBLE: Increasing Performance By Lowering Expectations.

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**ganesh****Moderator**- Registered: 2005-06-28
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**OUTSTANDING, krassi_holmz! **

This is how the problem is solved:-

The coefficient of

in the product would be the sum of the products of 1,2,3,...100 taking two at a time.We know

where S would be the sum of the products of 1,2,3,4...100 taken two at a time.

Therefore, the required number

The summation formulas for n and n² have been used above.

Therefore, the coeficient of would be

12582075.

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**ganesh****Moderator**- Registered: 2005-06-28
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***24. AB is the hypotenuse of a right-angled triangle ABC. If BC=x and AB+AC=y, what is the value of SinA?

***25. If

what is the value of tan(A)?

***26. If

where a+b+c≠0 and abc≠0, what is the value of

(a+b)(b+c)(c+a)?

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**ganesh****Moderator**- Registered: 2005-06-28
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***27 What is the remainder obtained when

is divided by 78?

***28 What is the remainder obtained when

is divided by 29?Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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**krassi_holmz****Real Member**- Registered: 2005-12-02
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26.

I think you've already gave that.

so:

*Last edited by krassi_holmz (2006-04-23 04:44:06)*

IPBLE: Increasing Performance By Lowering Expectations.

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**krassi_holmz****Real Member**- Registered: 2005-12-02
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27.

IPBLE: Increasing Performance By Lowering Expectations.

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