Hi!
I have got a few problems here:

1. What is the remainder when 1³ +2³ +3³ +...+100³ is divided by 7?

2. What is the number of strictly positive roots of the equation sin x= x/200?
3. For what values of x is x² < |2x-8|?

4. If f(x) =px⁷ +qx³ +rx-4 and f(-7)=3, what is the value of f(7)?

5. The x-coordinate of the foot of the perpendicular from the point (1,9) to the line y=x is ?

Can anyone show me how to solve these problems?
Thanks a lot!

I'll just give you some hints for now.

1. For this one you will need some modular arithmetic. Have you come across this before?

2. Draw a picture! (Consider simpler cases first like sin x = x/10.)

3. Find all x for which x² < 2x-8, and all x for which x² > -2x+8. Then consider which x will satisfy both at once.

4. Let g(x)=f(x)+4. Then what is g(-x) in terms of g(x)?

5. You can do this all sorts of ways; try doing a diagram and applying some trigonometry. Alternatively, if you know about the dot products of vectors, try using these.

Have fun!


David

for 1,
do you know the summation formula of

For (1), another way is to notice that (a + b) divides (a³ + b³ ). You can therefore group 1³ and 6³ (which is divisible by 1 + 6 = 7), 2³ and 5³ (divisible by 2 + 5 = 7), 3³ and 4³ , and leave 7³ on it's own. Since each of these little 'groups' are multiples of 7, you know when you add them together, you'll get a multiple of 7, so the remainder from these terms upon division by 7 is 0. You can continue this up to 98³ , but you have to consider 99³ and 100³ seperately. Then, 99 = 7x14 + 1, therefore 99³ = (7x14 + 1)³ = (multiples of 7) + 1, so this term leaves 1 upon division by 1. You can do the same for 100³ ( = [7x14 + 2]³ ).

Regards,
Olof

neat proof OLOF.

love arun