1 1.
In the real number number system, the inverse of addition is represented by (A) (B) (C)
x + 0 = x
(D)
x ( y + z ) = xy + xz
5.
If x + b = x ; x, b ∈ N , then the value of x in terms of b is
x + (− x) = 0 0 + x = x + 0
(A)
−b
(B)
−
(C) 2.
(A)
∑
∑
r =2
r =1 n
(B)
∑ r =1
n
∑ r =1
n
(C)
∑(2 +
2
r =1 n
(D)
6.
r
2
n
r
) = 2 + ∑ r 2 r =1
n
∑ =∑ r
2
r =1
2
r
7.
The ( k + 1) th term in
∑
r (r − 1) is
r =1
4.
The polynomial P( x) = 2 x3 + x2 − 13 x + 6, when divided by ( x − 1), gives a remainder of (A)
−4
(B)
0
(C)
6
(D)
18
r = 0
n
3.
2
r
r =1
2 r =
b
(D)
n
2
2 b
Which of the following followin g statements stateme nts is true? n
b
2
(4 x)3 − ( 4 y )3 can be expressed in the form
(A)
( 4 x − 4 y ) (16 x 2 − 16 y 2 )
(B)
( 4 x − 4 y ) (16 x 2 + 16 y 2 )
(A)
k
(B)
k + 1
(C)
( 4 x − 4 y ) (16 x 2 − 16 xy − 16 y 2 )
(C)
k (k + 1)
(D)
( 4 x − 4 y ) (16 x 2 + 16 xy + 16 y 2 )
(D)
(k + 1) 2
The basic wage W b and the overtime wage W o of a shop attendant never differ by more than $100. An inequality representing this statement is (A)
Wo − W b ≤ 100
(B)
Wo − W b < 100
(C)
Wo − W b ≥ 100
(D)
Wo − W b > 100
8.
If α and β represent roots of the equation x 2 − px + q = 0, then the value of α 2 + β 2 is 2
(A) (B)
p − q
(C)
p 2 − 2q
(D)
p 2 + 2q
p
CAPE Unit 1 P1 2008 ROR
2
9.
25 The exact value of 16 (A) (B) (C) (D)
10.
11.
−
1 2
is
12.
Which of the following followin g mapping diagrams does NOT represent a function? function?
2 5
(A)
y
4 5 5
x
4 5 2
Rationalising
2
−
1
2
+
1
(A)
1− 2 2
(B)
1+
(C)
3+2 2
(D)
3−2 2
2 3
The expression written as
gives (B)
y
2
x
2 − 4 x + 3 x 2 can be
(C)
y
2
(A)
2 3 3 x − − 3 2
(B)
2 2 3 x − − 3 3
(C)
3 2 3 x − + 2 3
x
2
2
(D)
y
2
(D)
2 2 3 x − + 3 3 x
CAPE Unit 1 P1 2008 ROR
3
Item 13 refers to the diagram below.
15.
The sketch below shows a function y = f ( x ).
The function y = f ( x ) is represented by (A)
13.
The function f ( x ) is decreasing for the range
14.
(A)
x < 3
(B)
x ≥ 5
(C)
3 ≤ x < 4
(D)
4 ≤ x < 5
The general quadratic equation with roots α and α and β may be written as (A)
0 x 2 − (α + β ) x − αβ =
(B)
0 x 2 + (α + β ) x − αβ =
(C)
0 x 2 − (α + β ) x + αβ =
(D)
0 x 2 + (α + β ) x + αβ =
(B)
(C)
(D)
CAPE Unit 1 P1 2008 ROR
4
16.
If
b = λ i + 5 j are
a = 5i + j and
parallel vectors, then the value value of λ is (A)
−5
(B)
1
(C)
5
(D)
25
20.
Which of the following sketches BEST represents the curve y = cos
1 2
x, (0 ≤ x ≤ 2π ) ?
(A)
Items 17 – 18 refer to a circle with general equation 2 2 x − 2 x + y + 4 y − 11 = 0.
17.
18.
The coordinates of the centre of the circle are (A)
(− 1, 1, − 2)
(B)
(−1, 2) 2)
(C)
(1, − 2)
(D)
(1, 2)
(B)
The radius of the circle is (C)
19.
(A)
2 units
(B)
4 units
(C)
6 units
(D)
11 units
The
function functi on
sin x +
π
can be
2
(D)
simplified to (A)
− cos x
(B)
− sin x
(C)
cos x
(D)
sin x
CAPE Unit 1 P1 2008 ROR
5
21.
Sin (3 (30o − A) is equal to
(A)
(B)
(C)
(D)
1 2 1 2
cos A −
3 2
cos A + 3
2 3 2
cos A +
cos A −
3 2 1 2 1 2
24.
If 2 cos θ + 9 sin θ = r cos(θ − α ), where π
r > 0 and 0 < α <
, then the maximum 2 value of the expression is
sin A
sin A
sin A
(A)
11
(B)
85
(C)
11
(D)
85
sin A
Item 25 refers to t o the following diagram. diagram.
22.
If β is an acute angle and cos β =
5 13
,
then sec β = (A) (B) (C) (D)
23.
5 13 12 13 12
Which of the following followin g equations BEST represents the graph shown above?
13
(A)
y = sin x
(B)
y = sin2 x
(C)
y = 2sin x
(D)
y = sin
13
25.
5
2sin 2sin θ cos cos φ is equivalent to
(A)
sin(θ + φ ) + sin(θ − φ )
(B)
sin(θ + φ ) − sin(θ − φ )
(C)
cos(θ + φ ) + cos(θ − φ )
(D)
cos(θ + φ ) − cos(θ − φ )
26.
1 2
x
The point A has coordinates (3, − 2).
The vector 3 OA is (A)
(3, − 6)
(B)
(9, − 2)
(C)
( −92)
(D)
( −96) CAPE Unit 1 P1 2008 ROR
6
Items 27 – 28 refer to the vectors
30.
2 x = 2t , y = t has equation
x = 3i − 8 j, y = 6i + 12 j, z = 6i + 4mj, m ∈ R.
27.
The value of the scalar product x y is
(A)
28.
−78
(B)
78
(C)
−114
(D)
114
The curve with parametric representation (A)
xy = 4
(B)
2 x y = 4
(C)
4 x = y 2
(D)
4 y = x 2
2
Item 31 refers to the diagram below.
The vectors y and z are perpendicular when m has a value of (A)
−3
(B)
−
(C) (D)
2 y = x
3 4 3 4 31.
3
In the diagram diagra m above showing showi ng the graph of y 2 = x, y is NOT defined for
29.
The line passing through the centre of the circle
( x − 3) + ( y + 2) = 25 2
2
and
parallel to the x-axis, has the equation (A)
y = − 2
(B)
y = 3
(C)
y = 2( x − 3)
(D)
y = 3( x − 2)
32.
(A)
x = 0
(B)
x < 0
(C)
x > 0
(D)
x ≥ 0
The
function
f ( x ) =
x − 2 2
x + 2
is
discontinuous for the domain value of (A)
−2
(B)
− 2
(C)
2
(D)
2
CAPE Unit 1 P1 2008 ROR
7
33.
lim
x → 3
x 2 − 9 x − 3
Item 36 refers to the diagram below which shows the curve
= is
2 2 x + y = 4, 0 ≤ x ≤ 2.
−∞
(A)
34.
(B)
0
(C)
6
(D)
∞
y
2 2 2 x + y = 4
= 1, where x is x measured in radians, then the value of
Given that lim
x → 0
lim
sin4 x
x → 0
x
(A)
4
(B) (C)
(D)
36.
is
An expression for obtaining the volume generated by rotating the bounded, shaded region through 360o about the x-axis is (A)
4 x
π
∫
2
∫
2
0
sin4 x
(B)
π
x
0
( 4 − y 2 ) d x ( 4 − x2 ) d x
2
4sin x
(C)
π
x
∫ (4 +
)
y 2 d x
0
2
(D)
π
∫ (4 +
x
2
0
35.
x
2
sin x
) d x
Given that f ( x ) = (2 x + 1)3 , then f ′ ′(2) equals (A)
37. 25
(B)
75
(C)
125
(D)
The first derivative of −
(A)
−
x − 1
is
x
(
)
2 x 2 − 1 x
(B)
(
)
2 x 2 − 1
150
2 x
(C)
(D)
1 2
( x 2 − 1) −
2
2 x
( x 2 − 1)
2
CAPE Unit 1 P1 2008 ROR
8
38.
If
d y d x
Item 41 refers to t o the diagram below.
= cos x, then
(A)
y = sin x + k
(B)
y = cos x + k
y
y = ln x
1
(C)
y = − cos x + k
(D)
y = − sin x + k
2
x
Item 39 refers to the following diagram 41. y y =
0.5
1
The gradient of the normal to the curve y = ln x at x = 2 is (A)
−2
(B)
−
x
4
x
(C) (D)
39.
The area of the finite region shaded in the diagram is (A)
40.
42.
∫
1 2 1 2 2
π
2
2co 2cos5 x d x is
0
ln(4 − 0.5)
(B)
ln(0. n(0.5 − 4)
(A)
(C)
ln 0. 0.5 − ln 4
(B)
(D)
ln 4 − ln 0. 0.5
−
2 5 2 5
(C)
−10
(D)
10
The stationary point of the function 2 y = ( x − 1) is
(A)
(0, 1)
(B)
(−1, 0)
(C)
(1, 0)
(D)
(0, (0, − 1)
CAPE Unit 1 P1 2008 ROR
9
43.
d y
= 2 x, a sketch of y versus x d x may be represented by
Given
44.
A curve is defined by the equation 2 y = − 5( 2 x − 1) .
Given that x increases at a rate of 1 unit per second when x = 1, what is the
I.
corresponding rate of change for y?
1 0
(A)
−40
(B)
−20
(C)
20
(D)
40
II. Item 45 refers to t o the following diagram. diagram.
1 0 0
1
III.
1 0
45.
IV.
(A)
I and II only
(B)
III and IV only
(C)
I, II and IV only
(D)
II, III, IV only
From the diagram above, which of the following statements are true? I.
′(1) < 0 f ′
II.
f (1) > k
III.
f ( 2) = 0
IV.
f ′(2) = k
(A)
I and II only
(B)
I and III only
(C)
II and III only
(D)
II and IV only
CAPE Unit 1 P1 2008 ROR
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