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#51 2006-03-15 04:24:56

ganesh
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Registered: 2005-06-28
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Re: *** Problems

***11

Find the value of a such that the vectors


are coplanar.


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#52 2006-03-15 06:16:52

Ikcelaks
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Re: *** Problems

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#53 2006-03-15 08:12:45

Ikcelaks
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Re: *** Problems

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#54 2006-03-15 16:15:35

ganesh
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Re: *** Problems

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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#55 2006-03-15 16:17:01

ganesh
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Re: *** Problems

***12

What is the sum of all the coefficients of

?


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#56 2006-03-16 03:59:51

mathsyperson
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Registered: 2005-06-22
Posts: 4,900

Re: *** Problems


Why did the vector cross the road?
It wanted to be normal.

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#57 2006-03-16 04:32:20

ganesh
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Re: *** Problems

mathsyperson,
greatworksmall.jpg


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#58 2006-03-18 16:42:20

ganesh
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Re: *** Problems

***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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#59 2006-03-19 15:49:43

ganesh
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Re: *** Problems

***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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#60 2006-03-20 03:41:25

rm
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Registered: 2006-03-14
Posts: 14

Re: *** Problems

***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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#61 2006-03-20 05:46:13

ganesh
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Re: *** Problems

e4.gif


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#62 2006-03-20 15:41:32

ganesh
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Re: *** Problems

***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.


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#63 2006-03-21 15:24:45

rm
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Registered: 2006-03-14
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Re: *** Problems

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#64 2006-03-21 18:05:24

ganesh
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Re: *** Problems

Very well done, rm! Your answers other than ***16 are correct!


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#65 2006-03-23 15:33:57

ganesh
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Registered: 2005-06-28
Posts: 14,391

Re: *** Problems

***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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#66 2006-03-23 16:41:34

rm
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Re: *** Problems

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#67 2006-03-23 17:24:18

ganesh
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Re: *** Problems

Excellent, rm! You are correct! smile


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#68 2006-03-24 00:33:56

ganesh
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Re: *** Problems

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


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#69 2006-03-24 01:56:34

Ikcelaks
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Registered: 2006-03-13
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Re: *** Problems

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#70 2006-03-24 03:36:31

ganesh
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Registered: 2005-06-28
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Re: *** Problems

Excellent, Ikcelaks!


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#71 2006-03-31 07:50:48

krassi_holmz
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Registered: 2005-12-02
Posts: 1,908

Re: *** Problems

***14
12582075


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#72 2006-04-14 17:57:08

ganesh
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Registered: 2005-06-28
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Re: *** Problems

OUTSTANDING, krassi_holmz! smile

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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#73 2006-04-16 16:54:44

ganesh
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Registered: 2005-06-28
Posts: 14,391

Re: *** Problems

***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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#74 2006-04-17 16:19:17

ganesh
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Registered: 2005-06-28
Posts: 14,391

Re: *** Problems

***27 What is the remainder obtained when


is divided by 78?

***28 What is the remainder obtained when

is divided by 29?


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#75 2006-04-23 04:36:13

krassi_holmz
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Registered: 2005-12-02
Posts: 1,908

Re: *** Problems

26.
I think you've already gave that.


so:

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


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