Download 1. Express as a single fraction: (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l

Basic Skills, Differentiation, Quadratic Theory & Remainder Theorem
1.
Express as a single fraction:
(a)
2.
3
7
+
a+2 a
3
5
+
w+2 w
(c)
5
8
+
e−6 e
(d)
4
7
−
m m +1
(e)
2
7
+
r r −9
(f)
6
9
−
v v−3
(g)
2
3
+
b +1 b + 2
(h)
5
2
+
n−4 n+6
(i)
4
7
−
s+5 s+8
(j)
2
5
−
t −7 t +9
(k)
1
7
+
x − 5 x −1
(l)
8
5
−
y −9 y −6
(c)
f ( x) =
Find the derivative of each function:
(a)
3.
(b)
3
f ( x) = 4 x −
x
2
(b)
6x 3 − x 2 + 5
f ( x) =
x
x( x − 5)
x
For each of the following functions:
(i) Find the points where the curve intercepts both axes.
(ii) Find the stationary points and justify their nature.
(iii) Sketch the curve and label each important point.
(a) y = x(x – 3)2
(b) y = 2x2 – x4
4.
For what values of p does each equation have real roots?
(a) 2x2 – 3x + p = 0
5.
(b) x2 + (p – 3)x + p = 0
Find c such that the line y = x + c is a tangent to the curve y = x2 – 3x.
What are the coordinates of the point of contact?
Unit 2: Mathematics 2
Higher
6.
Solve the following inequalities:
(a) 4x2 – 12x + 5 ≥ 0
(b) 12 + x – x2 > 0
(c) 2x2 + 4x + 1 > 0
7.
Use synthetic division to find the quotient and remainder in
each of the following:
(a) 2x2 + 3x + 4 divided by x – 1
(b) x3 + 4x2 – 3x – 11 divided by x + 4
(c) x4 – x2 + 7 divided by x + 1
(d) 2x3 + x2 + x + 10 divided by 2x + 3
8.
Factorise x3 –12 x – 16
9.
Find the remainder when 3x3 – 7x2 –15 is divided by x + 3
10.
n = 2 is a solution of 2n3 – n2 – 5n – 2 = 0.
Find the other two solutions.
11.
If x – 2 is a factor of 3x3 + 2x2 + kx + 6,
find k and then factorise this expression fully.
12.
Show that x = –1 is the only real root of the equation
x3 – x2 + 3x + 5 = 0
Unit 1: Mathematics 1
Higher