Advanced Mathematics

Advanced Mathematics

Table of Contents
MAIN BODY...................................................................................................................................1
1(a) Given the function, f(x) = x2-2x, evaluate or simplify....................................................1
(b) Given f(x) = x2 -2x and g(x) =3-x. Solve the equation:...................................................1
2 (a) Given the matrices, A= 1 0 and B = -2 1 find:.......................................................2
(b) Given the matrices, A= 3 2 find...................................................................................3
3 (a) Given u= < 5, 2, 7> , v =< 3, 2, 4> and w = <4, -1, 7>..................................................3
(b) Compute the angle between vectors p =2i+3j and q= 4i+ 6j............................................4
(c) Show the vectors <2 3> and <-1 4> are linearly independent..........................................4
5(a) Given ∫k 1/(x-2) dx = In 7 find the value of k..................................................................5
PART –B..........................................................................................................................................5
1(a) Given that f(x) =2x-1......................................................................................................5
(b) The demand function of register is given by....................................................................6
P = 12- ½ (x) for 0≤ x ≤ 20.....................................................................................................6
where x is number of unit sold. Figure 1 shows price decreases as more units are sold........6
2 (a) figure 2 shows a closed cylindrical can of radius r cm and height h cm.......................7
(b) Figure 3 shows a sketch of the curve C, with equation y = 3x – x2. It passes through the
origin O and the point B on the x-axis...................................................................................8
References:.....................................................................................................................................10

MAIN BODY
1(a) Given the function, f(x) = x2-2x, evaluate or simplify.
i. f(0)
Solution:
f(x) = x2-2x
= (0)2 – 2*0
= 0
ii. f(x+h) – f(x)
h
Solution:
f(x+h) = (x+h)2 – 2 (x+h)
f(x) = x2 – 2x
= (x+h)2 – 2 (x+h) – (x2 – 2x)
h
= 2xh+h2-2h
h
= 2x+h-2
(b) Given f(x) = x2 -2x and g(x) =3-x. Solve the equation:
i. (fog) (x) =0
Solution:
(fog) (x) = f(g(x))
= f(3-x)
f(3-x)= (3-x)2 – 2(3-x)
0 = (9-6x+x2) – (6-2x)
0 =( 3-4x+x2)
0 = (3 -3x -x+x2)
1

0 = (3-x) (1-x)
x=3 and x=1
ii. (gof) (x)+ x2 +5 =0
Solution:
(gof) (x) = g(f(x))
g(f(x)) = g(x2 -2x)
= 3- (x2 -2x)
(gof) (x)+ x2 +5 =0
3- (x2 -2x) + x2+5 = 0
2x+8 = 0
x = -4
2 (a) Given the matrices, A= 1 0 and B = -2 1 find:
3 -5 4 -3
i. 2A-3B
Solution:
= 2 1 0 - 3 -2 1
3 -5 4 -3
= 2 0 - -6 3
6 -10 12 -9
= 8 -3
-6 -4
ii. AB
Solution:
AB = 1 0 -2 1
3 -5 4 -3
= 1(-2)+0(4) 1(1)+0(-3)
3(-2)+0(4) -5(1)+0(-3)
2