Selkirk High School
Mathematics
Department
Higher Assessment
Booklet 1
Practice Questions
“The best way to eat an elephant is
one bite at a time!”
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  is the angle between a and b
Scalar Product:
or
 b1 
 a1 
 


a.b  a1b1  a2b2  a3b3 where a  a2 and b   b2 
 
b 
a 
 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
a cos ax
cos ax
a sin ax
f ( x)
 f ( x) dx
sin ax
1
 cos ax  C
a
cos ax
1
sin ax  C
a
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g2  f 2  c .
Vectors – E&F 1.4
1. The points A, B and C have coordinates (3, -1, 0), (5, 2, 5), (9, 8, 15) respectively.
(a) Write down the components of AC .
(b) Hence show that A, B and C are collinear.
2. The points P, Q and R have coordinates (0, 1, 2), (1, 3, -1), (d, 11, -13) respectively.
(a) Write down the components of PQ .
(b) P, Q and R are collinear. Find the value of d.
3. The point T divides SU in the ratio 2:1 as shown in the diagram.
U(7, -6, 4)
Find the coordinates of T.
T
S(-2, 0, 1)
4. The point L divides KM in the ratio 2:3 as shown in the diagram.
M(-1, 8, -15)
Find the coordinates of L.
L
K(4, -2, 0)
5. The diagram shows vectors AB and AC where
1
 
AB   2 
 0
 
 3
 
AC   5 
2
 
C
A
(a) Find AB.AC
B
(b) Hence find the size of BAC
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6. The diagram shows vectors TS and TV where
3
 
TS   1 
2
 
S
 0
 
AC   7 
 6
 
T
(a) Find TS.TV
V
(b) Hence find the size of STV
7. T, PQRS is a right pyramid. It has square base, PQRS.
 4
 
PS   2 
 4 
 
 2 
 
RS   4 
 4 
 
 1 
 
RT   5 
1 
 
Write down the components of PT .
8. ABCD is a tetrahedron as shown below. M is the midpoint of BC.
1
3
 
 
AB   2  and BC   2 
2
 1 
 
 
Find AM .
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Functions – E&F 1.3
1. The graph of a cubic function y = f(x) is shown in the diagram.
(-3,7)
On separate diagrams make sketches of:
y = f(x)
(i) y = f(x + 3)
(ii) y = f(x - 1)
-5
(iii) y = f(-x)
1
-1
(iv) y = -f(x) + 1
-3
y  ax
y  3x
2. The graphs with equations y  3x
and y  a x are shown in the diagram.
If the graph with equation y  a x
passes through the point (1, 6),
find the value of a.
(2, 9)
(1, 6)
y = 8x
3. The graphs of y = 8x and its inverse function
are shown in the diagram.
Write down the equation of the inverse function.
1
1
4. For the following functions write down f 1 ( x ) :
(i) f (x )  x  3
(ii) f (x )  2x  7
(iii) f (x ) 
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x
7
3
(iv) f (x )  3x 3  5
5. Functions f and g are defined on suitable domains by f (x )  x 2 and g(x )  2x  1 .
Obtain an expression for f ( g(x )) and g(f (x )) .
6. Functions f and g are defined on suitable domains by f (x )  x 3 and g(x )  2x  1 .
Obtain an expression for f ( g(x )) and g(f (x )) .
7. Functions f and g are defined on suitable domains by f (x )  3x  2 and g(x )  x 2 .
Obtain an expression for f ( g(x )) and g(f (x )) .
 
8. For 0  x   a sine curve has a single maximum value at  ,1  and a single minimum
4 
 3

value at 
, 5  .
 4

(a) Write down an equation for this curve.
(b) Sketch the curve given by the equation in part (a).
9. For 0  x   a cosine curve has a single maximum value at  0,6  and a single minimum
 
value at  ,2  .
2 
(a) Write down an equation for this curve.
(b) Sketch the curve given by the equation in part (a).
10. For 0  x   a cosine curve has a single maximum value at  0,2  and a single minimum


value at  , 6  .
3

(a) Write down an equation for this curve.
(b) Sketch the curve given by the equation in part (a).
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Trig Equations – R&C 1.2
1. Solve the equation sin2x 
1
for 0  x  
2
2. Solve the equation tan2x  3 for 0  x  90
3. Solve the equation cos2x 
3
for 0  x  
2
4. (a) Express cos x cos21  sin x sin21 in the form cos(x  a)
(b) Hence solve cos x cos21  sin x sin21 
1
for 0  x  360
7
5. Solve the equation sin x cos31  cos x sin31 
8
for 0  x  360
11
6. Solve the equation cos x cos35  sin x sin35 
2
for 0  x  360
5
7. Solve the equation 3sin2x  3cos x for 0  x  360
8. Solve the equation 2cos2x  3cos x  4  3 for 0  x  360
Trig Formulae – E&F 1.2
1. The diagram below shows two right-angled triangles.
8
y
x
6
15
(a) Write down the exact values of cos x and cos y .
(b) Show that the exact value of cos(x  y ) is
77
.
85
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2. The diagram below shows two right-angled triangles.
7
3
x
y
4
24
(a) Write down the exact values of sin x and sin y .
(b) Show that the exact value of sin(x  y ) is
4
.
5
3. The diagram below shows two right-angled triangles.
8
7
y
x
24
6
(a) Write down the exact values of sin x and cos y .
(b) Show that the exact value of sin(x  y ) is
3
.
5
4. Express 5sin x  2cos x in the form k sin(x  a ) where k > 0 and 0  a  360
5. Express 4sin x  cos x in the form k sin(x  a ) where k > 0 and 0  a  360
6. Express 3cos x  sin x in the form k cos(x  a ) where k > 0 and 0  a  360
7. Prove that
1  cos2x
 tan x
sin2x
9. Prove that cosxtanx = sinx
8. Prove that (cosx + sinx)2 = 2sinxcosx + 1
10. Prove that cos2xtan2x = 1 – cos2x
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