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**Ikcelaks****Member**- Registered: 2006-03-13
- Posts: 8

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**Ikcelaks****Member**- Registered: 2006-03-13
- Posts: 8

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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 19,194

**Excellent, Ikcelaks! Your solution to Problem (11) is correct! I shall solve problem 8 and let you know whether you are correct!**

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
- Posts: 19,194

***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)?

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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***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.

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

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

***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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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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***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!**

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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***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?

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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**Excellent, rm! You are correct! **

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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***23 Find the sum of all numbers from 1 to 100 which are not divisible by 3 and 5.

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

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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! **

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
- Posts: 1,905

***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.

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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***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)?

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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***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
- Posts: 1,905

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
- Posts: 1,905

27.

IPBLE: Increasing Performance By Lowering Expectations.

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