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

Survey

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts

Elementary algebra wikipedia, lookup

Homogeneous coordinates wikipedia, lookup

System of polynomial equations wikipedia, lookup

History of algebra wikipedia, lookup

Quartic function wikipedia, lookup

Cubic function wikipedia, lookup

Fundamental theorem of algebra wikipedia, lookup

Equation wikipedia, lookup

Transcript
```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
```
Related documents