harder mathematical questions ( pure mathematics)?

Question

Please help me by answering any questions you can from the following.....the more the better.
1) reffered to a fixed point origin O, the positional vector of points P and Q are (-2i+3j)  and ( ki + lj ) respectively, where k and l are constants. The point M is the midpoint of PQ. given the position vector of M relative to O is (2i+4j)
   a) find the values of k and l 
the point R is such that PQ : QR = 1:2
   b)find, in terms of i and j, the expression for OR 


2) solve the equation sin2x = sin 120, giving all solutions for 0<x<360

3) A can is made out of thin sheet metal and is in shape of a right circular cylinder closed at both ends. the radius of the circular cross section of the can is r cm. the height of the can is h cm and the total surface area is A cm^2 . given that the capacity of the can is 31.25*pie cm^3. 
   a) show that r^2 *h = 31.25
   b)find an expression of A in terms of pie and r only
   c) calculate , to one decimal place , the value of dA/dr when r=3
   d) show that A has a stationary value when r=2.5 
   e) find, giving your answer as a multiple of pie , the stationary value of A establishing whether this is a maximum or minimum value.

4) find the set of values for x which   2x(x-2)<(x+1)(x-2)

5) Sn= nSIGMA NOTATION (SUMMATION ) SYMBOL r=1           ( 50-4r)
a) write down the first three terms of the series 
b)Calculate the value of S20
c) find n such that Sn=0

ii) the sum of the first four terms of a geometric series of positive terms is 78.336 and the sum to infinity of the series is 90. calculate 
a) the common ratio of the series 
b) the first term of the series 
c) the difference, to 3 significant figures, between the sum to infinity of the series and the sum of the first thirty terms. Give your answer in standard form 

Thanks in advance :-)

Iggy Rocko · Accepted Answer

4) 2x(x - 2) < (x + 1)(x - 2)

2x(x - 2) - (x + 1)(x - 2) < 0

(2x - (x + 1))(x - 2) < 0

(x + 1)(x - 2) < 0

{x + 1 < 0 and x - 2 > 0} or {x + 1 > 0 and x - 2 < 0}

{x < -1 and x > 2} or {x > -1 and x < 2}

Discard {x < -1 and x > 2} because it has no solutions, leaving just

x > -1 and x < 2

-1 < x < 2

One question at a time.

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harder mathematical questions ( pure mathematics)?

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