Advanced Higher Prelim Practice

ADVANCED HIGHER PRELIM PRACTICE PAPER
1) Find the coefficient of u 4 v 12
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

in the expansion of  3u 2 
6
5
 .
v3 
2) A curve C is defined in terms of the parameter t by the equations
x  t 5  9t 3 , y  3t 2 .
Find the equations of the tangents to the curve C at the point 0,27.
3) Use Gaussian elimination to solve the system of equations below
2x - y - 13z = 1
x - y + 2z = -3
- x + 2y - 3z = 2
4) (a)
(b)
Factorise  x3  2 x 2  x  2.
Hence evaluate

x3  3x 2  2 x  10
dx.
 x3  2 x 2  x  2
3
2
a
b
Express your answer in the form ln    c, where a, b and c are constants.
5)
a +10 , a + 5 , a + 2 are the first three terms of a geometric sequence.
Find:
6)
(a)
the value of the first term and the common ratio of the sequence
(b)
the value of the sixth term of the sequence
(c)
the sum to infinity of the geometric series (a + 10) + (a + 5) + (a + 2) +…
Given f(x) = tan 3x, find f ( x ) and f ( x) .
Hence show that
f ( x)
 kf ( x) , stating the value of k.
f ( x)
ADVANCED HIGHER PRELIM PRACTICE PAPER
SELKIRK HIGH SCHOOL
7) A curve is defined by the equation xy 2  3x 2 y  4 for x > 0 and y > 0
dy
dx
a)
Use implicit differentiation to find
b)
Hence find an equation of the tangent to the curve where x = 1

6
8) Use the substitution u = 1 + sinθ to show that

0
4 cos 
d = p q  r
1  sin 
where p , q and r are integers.
9) Find the volume of revolution when the area between the line y = 2 and the curve y = 3x² is rotated about
the y axis.
10) Shown is part of the graph of f x  
x2  5
, x  2.
x2
0
Determine algebraically the range of the function f x  
x2  5
, x  2.
x2
ADVANCED HIGHER PRELIM PRACTICE PAPER
11(a)
(b)
1
Given f ( x)  6 tan
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x , where x > 0, obtain f (x) and simplify your answer.
Given y  x x 2 , where x > 2, use logarithmic differentiation to obtain
dy
in terms of x.
dx
12) z1  2i and z 2  1  i .
(a)
(b)
(b)
z1
in the form a + bi (where a and b are real numbers).
z2
z
Hence express z1 z2  1 in the form a +bi
z2
Express
z 
Find arg  1  .
 z2 
13) Use integration by parts to evaluate
2
 2 tan
1
xdx
1
14) A curve is defined by the parametric equations
x  t 2  2t , y  1  t 4 .
Find the equation of the tangent to the curve at the point where t = -1.
15)
Express the improper rational function f ( x) 
x 3  3x 2  8 x  2
x 2  2x  1
in the form
f ( x )  g ( x )  h( x ) ,
where g(x) is a polynomial function and h(x) is a proper rational function expressed in partial
fractions.
ADVANCED HIGHER PRELIM PRACTICE PAPER
SELKIRK HIGH SCHOOL
16) Find the solution of the differential equation
dy sec y

given that y  when t = 0.

3t
dt
6
4e

6
17)
Use the substitution u = 1 + sinθ to show that

0
4 cos 
d = p q  r
1  sin 
Where p , q and r are integers.
18)
By using the substitution t = 1 + tan x

sec 2 x
0 1  tan x dx
4
19) (a)
(b)
Calculate the sum of all the two digit natural numbers which are divisible by 3.
Find the value of  , 0   

2
, such that:
1  sin 2   sin 4   sin 6   ...  2 .
20) A scientist constructs the differential equation
dy
 e x y
dx
to describe the relationship between two quantities x and y.
(a) Find the general solution of the differential equation.
(b) Given that y = 0 when x = 1, find the particular solution, expressing y in
terms of x.
ADVANCED HIGHER PRELIM PRACTICE PAPER
SELKIRK HIGH SCHOOL
21) Two complex numbers, z1 and z 2 , are given by z1  3  2i and z 2  6  ki,
where k is a real number.
a) Write z1 z2 in the form a+ib.
b) Given z12  3z 2 is a purely real number, find the value of k.
22) The function f is defined by f ( x) 
(a)
x2  3
, x  1, x  R.
x 1
(i)
Write down the equation of the vertical asymptote of f.
(ii)
Show that f has a non-vertical asymptote and obtain its equation.
(iii)
Find the point(s) of intersection with the x- and y- axes.
(b)
Find the coordinates and nature of the stationary points of f.
(c)
Sketch the graph of y = f(x), indicating the features found in (a) and (b).
ADVANCED HIGHER PRELIM PRACTICE PAPER
SELKIRK HIGH SCHOOL
ADVANCED HIGHER PRELIM PRACTICE PAPER
 6  2 6 r   5 
  3u
 3 

v 
r 0  r 
12
2 r
6
 6  6 r
r u
  3  5 v 3r
1) r 0  r 
 
6
r
r4
6 2
4
 3  5
4
 
84,375
2)
6
5t  27t
t 5  9t 3  0  t  3,0,3
3
3t 2  27  t  3,3

1
1
or
9
9
9 y   x  243 or 9 y  x  243
9 y  x  243 or 9 y   x  243
3)
 2 1  1 


 1  1 2  3
 1 2  3 2 


2  3
 1 1


 0 1   4 7  (or equivalent)
 1 2
 3 2 

2
 3
1 1


 0 1   4 7  (or equivalent)
0 1
 1  1 

2
 3
1 1


 0 1   4 7  (or equivalent)
 0 0    3  8


SELKIRK HIGH SCHOOL
ADVANCED HIGHER PRELIM PRACTICE PAPER
inconsistent (accept “no solution” or “no answer”)
4)  x  1x  1x  2 (or equivalent)
  x 3  2 x 2  x  2 x 3  3x 2  2 x  10
 5 x 2  x  12
 1
2  x  2x 2  x 3



5 x 2  x  12 
Ax  1x  2  Bx  1x  2  C x  1x  1
A  1, B  4, C  2
3
1
4
2 
2   1  x  1  x  1  x  2 dx (or equivalent)
  x  ln x  1  4 ln x  1  2 ln x  2

 3  ln 2  4 ln 4  2 ln 5 
 2  ln 1  4 ln 3  2 ln 4
 200 
 1
 81 
 ln 
5)
a5 a2

a  10 a  5
 a
5
2
5
5
2

5
 10
2
3
r
5
5 3
  
2 5

243
1250
5
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7
3
1
x   ,y   ,z  
2
2
2
ADVANCED HIGHER PRELIM PRACTICE PAPER
5
 2
3
1
5

25
4
SELKIRK HIGH SCHOOL

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