C3
A Booster Course
Workbook
1.
a) Sketch, on the same set of axis the graphs of y = x and y = 2x − 3 .
(3)
b) Hence, or otherwise, solve the equation
x = 2x − 3
(4)
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2.
The function f is defined by
f:x→
x+2
, x ∈ℜ
x −1
a) Show that for all values of x, ff (x) = x.
-1
(3)
b) Hence, write down an expression for f (x).
(1)
The function g is defined by
g : x → 2x − 3 , x ∈ℜ
c) Solve the equation gf (x) = 0.
(4)
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3.
The diagram shows the graph of y = f (x) which meets the x-axis at the point
and the y-axis at the point ( 0, −3) .
( 94 , 0 )
a) Sketch on separate diagrams the graphs of
i) y = f ( x )
ii) y = f −1 ( x )
(4)
Given that f (x) is of the form f ( x ) ≡ ax 2 + b , x ≥ 0 ,
1
b) Find the values of the constants a and b.
c) Find an expression for f
−1
( x) .
(3)
(3)
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4.
The functions f and g are defined by
f : x → kx + 2 , x ∈ℜ
g : x → x − 3k , x ∈ℜ
where k is a constant.
a) Find expressions in terms of k for
i) f −1 ( x )
ii) fg ( x )
(4)
Given that fg (7) = 4,
b) Find the values of k.
(1)
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5.
Figure 1 shows the graphs of y = x and y = x − 2 +1 . The point P is the minimum
point of y = x − 2 +1 , and Q is the point of intersection of the two graphs.
Figure 1
a) Find the coordinates of P.
(1)
b) Show that the y coordinate of Q is
3
2
.
(4)
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6.
The function f is defined as
f:x
x +1
,
x −1
x ∈ℜ
By considering ff (x), show that the function f has the line of symmetry y = x.
(5)
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7.
The functions f is defined by
f:x
a) Show that f ( x ) =
b) Find f
−1
3( x +1)
1
−
,
2
2x + 7x − 4 x + 4
1
2x −1
(4)
( x)
c) Find the domain of f
−1
x ∈ℜ
(3)
( x)
(1)
Given that the function g is defined by
g : x  ln ( x +1)
1
d) Find the solution of fg ( x ) = .
7
(4)
(Taken from Jan 2012 paper)
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8.
a) Solve the inequality 3x − 4 < 7 .
(3)
b) Find, using algebra, the values of x for which
x 2x + 5 − 3 = 0
(3)
c) Sketch the graphs of y = x + 3 and y = x − 5 . Use algebra the coordinates of
where these lines meet.
(3)
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© BlueStar Mathematics Workshops (2011)
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9.
The functions f is defined by
f : x  x −1 − 3 , x ∈ℜ
a) Solve the equation f ( x ) = 4 .
(2)
The function g is defined by
g : x  x 2 − 4x +18 , x ≥ 0 .
b) Find the range of g.
(3)
c) Evaluate gf (-4).
(3)
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10.
The functions f and g are defined by
f : x  cos x ,
π
g:x  x+ ,
2
x ∈ℜ
x≥0
a) State the range of f (x).
(2)
b) Find the domain of fg (x).
(3)
c) Determine the range of fg (x).
(2)
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11.
Find the solutions to the following equation to 3 decimal places.
2e x + 3e − x = 7
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(5)
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12.
Solve the following simultaneous equations, giving your values to 4 significant
figures.
e y + 5 − 9x = 0
y − ln ( x + 4) = 2
(7)
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13.
At time t = 0, there are 800 bacteria present in a culture. The number of bacteria
present at time t hours is modeled by the continuous variable N and the relationship
N = ae bt
where a and b are constants.
a) State the value of a.
(1)
Given that when t = 2, N = 7200,
b) Find the value of b in the form ln k.
(3)
c) Find, to the nearest minute, the time taken for the number of bacteria present to
double.
(4)
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© BlueStar Mathematics Workshops (2011)
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13.
A bead is projected vertically upwards in a jar of liquid with a velocity of 13 ms-1. Its
velocity, v ms-1, at time t seconds after projection, is given by
v = ce − kt − 2
a) Find the value of c.
Given that the bead has a velocity of 7 ms-1 after 5.1 seconds,
(1)
b) Find the value of k correct to 4 decimal places.
c) Find the time taken for its velocity to decrease from 10 ms-1 to 4 ms-1.
(3)
(4)
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f ( x ) ≡ e 5−2 x − x 5
14.
Show that the equation f (x) = 0
a) has a root in the interval (1.4, 1.5),
(2)
b) can be written as x = e
1−kx
, stating the value of k.
(2)
c) Using the iteration formula xn+1 = e
, with x0 = 1.5 and the value of k found in
b), find x1, x2 and x3. Give the value of x3 correct to 3 decimal places.
(4)
1−kxn
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15.
The diagram shows part of the curve with the equation y = 3x + ln x − x 2 and the line
y = x . Given that the curve and the line intersect at the points A and B, show that
a) The x coordinates of A and B are the solutions of the equation x = e x
2
−2 x
(2)
b) The x coordinate of A lies in the interval (0.4, 0.5),
(1)
c) The x coordinate of B lies in the interval (2.3, 2.4).
d) Use the iteration formula xn+1 = e
A correct to decimal places.
xn2 −2 xn
(1)
, with x0 = 0.5 , to find the x coordinate of
(3)
e) Justify your answer of part d).
(2)
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© BlueStar Mathematics Workshops (2011)
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16.
a) Prove that, for cos x ≠ 0 ,
sin 2x − tan x ≡ tan x cos2x
(5)
b) Hence, or otherwise, solve the equation.
sin 2x − tan x = 2 cos2x ,
for x in the interval 0 ≤ x ≤ 180° .
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(4)
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© BlueStar Mathematics Workshops (2011)
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17.
a) Use the identities of sin ( A + B ) and sin ( A − B ) to prove that
⎛ P + Q⎞ ⎛ P − Q⎞
sin P − sinQ ≡ 2 cos ⎜
sin
⎝ 2 ⎟⎠ ⎜⎝ 2 ⎟⎠
(4)
b) Hence, or otherwise, solve the equation.
sin 4x = sin 2x ,
for x in the interval 0 ≤ x ≤ 180° .
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(6)
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18.
a) Express 2 cos x° + 5sin x° in the form R cos ( x − α ) ° where R > 0 and 0 < α < 90
giving your values to 3 significant figures.
(4)
b) Hence, or otherwise, solve the equation.
2 cos x + 5sin x = 3 ,
for x in the interval 0 ≤ x ≤ 360° , giving your answers to 1 decimal place.
(4)
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19.
a) Find the exact values of R and α , where R > 0 and 0 < α <
π
, for which
2
cos x − sin x ≡ R cos ( x + α ) .
(4)
b) Use the identity
⎛ X +Y ⎞
⎛ X −Y ⎞
cos X + cosY = 2 cos ⎜
cos ⎜
⎟
⎝ 2 ⎠
⎝ 2 ⎟⎠
or otherwise, find in terms of π , the values of x in the interval 0 < x < 2π , for
which
π⎞
⎛
cos x + 2 cos ⎜ 3x − ⎟ = sin x
⎝
4⎠
(8)
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20.
a) Prove that for all values of x
cos ( x + 30 ) ° + sin x° ≡ cos ( x − 30 ) °
(4)
b) Hence, find the exact value of cos 75° − cos15° , giving your answer in the form
k 2.
(3)
c) Solve the equation
3cos ( x + 30 ) ° + sin x° = 3cos ( x − 30 ) ° +1 ,
for x in the interval −180° ≤ x ≤ 180° .
(6)
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21.
a) Express 4sin x° − cos x° in the form Rsin ( x − α ) ° , where R > 0 and 0 < α < 90° .
Give the values of R and α to 3 significant figures.
(4)
b) Show that the equation
2cosec x° − cot x° + 4 = 0
can be written in the form
4sin x° − cos x° + 2 = 0 .
(2)
c) Hence, or otherwise, solve the equation
2cosec x° − cot x° + 4 = 0
for the values of x in the interval 0 < x < 360° .
(4)
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22.
a) Express 3cosθ + 4sin θ in the form R cos ( x − α ) , where R > 0 and 0 < α <
π
.
2
(4)
b) Given that the function f is defined by
f (θ ) ≡ 1− 3cos2θ − 4sin 2θ , 0 ≤ θ ≤ π ,
state the range of f (θ ) and solve the equation f (θ ) = 0 .
(6)
c) Fine the coordinates of the turning points of the curve with the equation
y=
2
3cos x + 4sin x
for the values of x in the interval 0 < x < 2π .
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(3)
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© BlueStar Mathematics Workshops (2011)
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23.
a) Prove the identity
1 − cos x
x
≡ tan 2
1 + cos x
2
(4)
π
b) Use the above identity to find the value of tan 2
in the form a + b 3 , where a
12
and b are integers.
(3)
c) Hence, or otherwise, solve the equation
1 − cos x
x
= 1 − sec ,
1 + cos x
2
for the values of x in the interval 0 < x < 2π , giving your values in terms of π .
(5)
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24.
a) Use the identities of cos ( A + B ) and cos ( A − B ) to prove that
sin Asin B ≡
1
⎡cos ( A − B ) − cos ( A + B )⎤⎦
2⎣
(3)
b) Hence, or otherwise, find the values of x in the interval 0 ≤ x ≤ π for which
π⎞
π⎞
⎛
⎛
4sin ⎜ x + ⎟ = cosec ⎜ x − ⎟
⎝
⎝
3⎠
6⎠
giving your answers as exact multiplies of π .
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(7)
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25.
a) For values of θ in the interval 0 ≤ θ ≤ 360° , solve the equation.
2sin (θ + 30°) = sin (θ − 30°)
(6)
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26.
a) Use the identity
cos ( A + B ) ≡ cos A cos B − sin Asin B
to prove
cos x ≡ 2 cos2
x
−1
2
(3)
b) Solve the equation
sin x
x
= 3cot ,
1 + cos x
2
for the values of x in the interval 0 ≤ x ≤ 360° .
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(7)
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27.
a) Prove the identity
cosec θ − sin θ ≡ cosθ cot θ
b) Find the values of x in the interval 0 ≤ x ≤ 2π for which
(3)
2sec x + tan x = 2 cos x ,
giving your answers in terms of π .
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(6)
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28.
a) Use the identities of cos ( A + B ) and cos ( A − B ) to prove that
⎛ P + Q⎞
⎛ P − Q⎞
cos P + cosQ = 2 cos ⎜
cos ⎜
⎝ 2 ⎟⎠
⎝ 2 ⎟⎠
(4)
b) Hence, or otherwise, find the values of x in the interval 0 ≤ x ≤ 2π for which
cos x + cos2x + cos3x = 0
(7)
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29.
a) By writing 3θ = ( 2θ + θ ) , show that
sin 3θ = 3sin θ − 4sin 3 θ .
(4)
b) Hence, or otherwise, solve
For 0 < θ < π .
28sin 3θ − 21sin θ + 5 = 0 ,
(5)
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30.
A curve has the equation x = tan 2 y .
a) Show that
dy
1
.
=
dx 2 x ( x +1)
π
b) Find the equation of the normal to the curve when y = .
4
(5)
(3)
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31.
Differentiate the following with respect to x
a)
( 4x −1)5
(2)
b)
e
3x
(1)
c) Hence, or otherwise, find
dy
given that the curve y,
dx
y = e3( 4 x−1)
5
(3)
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32.
Differentiate the following with respect to x.
3x + 5
a)
3
b)
ex
2x +1
(3)
(4)
c) Hence, or otherwise, show that the curve
1
⎡ ⎛ ex ⎞
⎤3
y = ⎢3 ⎜
+
5
⎥
⎟
⎣ ⎝ 2x +1⎠
⎦
differentiates to
dy
=
dx
e x ( 2x −1)
( 2x +1)2 3 (3( 2ex+1 ) + 5)
x
2
(4)
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33.
The curve with the equation y =
1 2
x − 3ln x , x > 0, has a stationary point at A.
2
a) Find the exact x coordinate of A.
(3)
b) Determine the nature of this stationary point.
(2)
c) Show that the y coordinate of A is
3
(1 − ln 3)
2
(2)
d) Find the equation of the tangent to the curve at the point where x = 1, giving your
answer in the form ax + by = c , where a, b and c are integers.
(3)
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34.
a) Use the derivatives of sin x and cos x to prove that
d
(cot x ) = −cosec 2 x
dx
(4)
b) Show that the curve with the equation
y = e x cot x
has no turning points.
(5)
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© BlueStar Mathematics Workshops (2011)
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35.
Given that
show that
f ( x ) = e 2 x cos3x ,
f ' ( x ) = Re 2 x cos (3x + α )
where R and α are constants to be found.
(5)
END
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