Selkirk High School
Mathematics
Department
Higher Assessment
Booklet 2
Practice Questions
FORMULAE LIST
Circle:
The equation x 2  y 2  2 gx  2 fy  c  0 represents a circle centre (  g ,  f ) and radius
The equation ( x  a )2  ( y  b)2  r 2 represents a circle centre ( a , b ) and radius r.
a.b  a b cos , where 
Scalar Product:
or
is the angle between a and b
 a1 
 b1 


a.b  a1b1  a2b2  a3b3 where a  a2 and b   b2 
 
 
a 
b 
 3
 3
Trigonometric Formulae:
sin( A  B )  sin A cos B  cos A sin B
cos( A  B )  cos A cos B sin A sin B
sin 2 A  2sin A cos A
cos 2 A  cos 2 A  sin 2 A
 2 cos2 A  1
 1  2sin 2 A
Table of standard derivatives:
Table of standard integrals:
f ( x)
f '( x )
sin ax
cos ax
a cos ax
f ( x)
 f ( x) dx
sin ax
1
 cos ax  C
a
cos ax
1
sin ax  C
a
a sin ax
g2  f 2  c .
Expressions and Functions 1.1 – Logarithms & Exponentials
1.
2.
3.
4.
5.
6.
7.
Simplify loga 12 + loga 2
Solve 4x = 9
Simplify loga 21 – loga 3
Simplify 3log4 2 + log4 5
Solve ex = 7
Solve ex = 20
Simplify 2log4 6 – log4 2
Relationships and Calculus 1.1 – Polynomials & Quadratic Theory
3
2
8. (a) Show that  x  2 is a factor of f ( x )  2 x  x  13x  6
and
hence factorise f ( x ) fully.
(5)
(b) Hence solve the equation 2 x  x  12 x  10  x  4.
3
2
9. A function is defined by the formula 𝑓(𝑥) = 𝑥 3 − 4𝑥 2 + 𝑥 + 6
number.
(2)
b) Hence factorise f(x) fully.
(2)
Hence solve f(x) = 0
2
The graph of the function f ( x )  x  6 x  p
What is the value of p ?
11.
𝑤here x is a real
a) Show that ( x – 3 ) is a factor of f(x)
c)
10.
(#2.1, 1)
(#2.1 ,1)
crosses the x-axis in one distinct place.
(3)
The graph of the function 𝑓(𝑥) = 𝑘𝑥 2 − 6𝑥 − 3 does not cross or touch the x-axis,
what are the possible values for k.
(3)
Relationships and Calculus 1.3 - Differentiation
1
dy
3
12. Given y  2  2 x 2 , x  0, find
.
x
dx
(3)
dy
4 x5  2 x
, x  0, find
13. Given y 
.
2
dx
x
(3)
1
dy
14. (a) Given y  cos x find
.
3
dx
(1)
(b) Differentiate 4sin x
with respect to x.
(c) Find the differential of
(d) If
y  2 cos x
Find
1
sin x
5
(1)
dy
.
dx
(1)
15. A sketch of the curve with equation
y  x 2  4 x is shown in the diagram.
A tangent has been drawn at the point P(3,-3).
(a) Find the equation of the tangent at P. (3)
(b) What can you say about the tangent
to the curve at (2, 4)
(1)
(#2.2)