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
|
100 å r=1 | r3 |
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