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Higher Homework
Unit 1
Multiple Choice
1.
2.
3.
4.
The line 2 x + 3 y = 12 cuts the x-axis at A and the y-axis at B. The midpoint of AB is
A.
(3, 2)
B.
(2, 3)
C.
(5, 0)
D.
(0, 5)
The value of the derivative of the function f defined as f ( x) = 3 x 2 1 at the point where x = 4 is
A.
60
B.
23
C.
24
D.
64
The gradient of the line perpendicular to the line with equation 2 x 3 y = 5 is
A.
B.
2
3
C.
D.
3
2
2
3
3
2
If A = 2r (r + h) , where h is a constant, the derivative of A with respect to r is
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A.
2 (r + h)
B.
2 (2r + h)
C.
2r
D.
6r
5.
6.
For the functions defined by f ( x) = 5 x 2 and g ( x) = 3 x 1 . f ( g ( x)) equals
A.
45 x 2 30 x + 5
B.
15 x 2 1
C.
45 x 2 5
D.
45 x 2 + 5
The vertices of a triangle are A(3, 8), B (1, 2) and C(5, 4).
Here are two statements about the triangle ABC:
I
it is isosceles
II
it is right-angled at B
Which of these statements is/are true?
7.
Both
B.
I only
C.
II only
D.
Neither
A sequence is defined by the recurrence relation U n +1 = 4U n + 5 , U 0 = 5 .
The value of
8.
A.
For f ( x) =
U 2 U1
equals
2
3x
x
A.
80
B.
40
C.
10
D.
75
, f ' ( x) equals
A.
B.
C.
3x
x
3
x
3
2 x
3
D.
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2x 2
9.
The line PQ is perpendicular to RS where P is (4, 2), Q(6, 4) , R(0, -3) and S(2, k). The value of
k is
10.
11.
A.
B.
C.
D.
The exact limit of the sequence defined by the recurrence relation U n +1 = 0 8U n 6 is
A.
7 5
B.
7 5
C.
30
D.
30
A function is defined is f ( x) = 3 x 4 4 x 3 . Here are 3 statements about the function
I
it is always increasing
II
it is never decreasing
III
it has stationary points where x = 0 and x = 1
Which of these statements is/are true for the above function?
12.
A.
I only
B.
II only
C.
III only
D.
some other combination
A function is defined is f ( x) = 3 x 4 4 x 3 . The range of values of x for which this function is
NOT increasing is
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A.
x 1
B.
x <1
C.
x 1
D.
x >1
13.
14.
15.
16.
When 2 x 2 + 8 x 3 is written in the form a ( x + b) 2 + c the values of a, b and c are
11
The gradient of the tangent to the curve y = 3 x 2 2 at the point where x = 2 is
A.
10
B.
C.
12
D.
The line y mx = 6 is parallel to the line 3 y + 2 x = 3 . The value of m is
A.
B.
C.
2
3
D.
2
3
A curve has equation y = 2 x 3 3 . Which line in the table correctly identifies the coordinates and
nature of its turning point?
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Turning point
Nature
(0, 3)
Point of inflexion
(0, 6)
Point of inflexion
(0, 0)
Minimum
(0, 3)
Minimum
17.
A(4, -2), B (-2, 5) and C (6, 3) are the three vertices of a triangle. The gradient of the median AD
is
18.
19.
A.
1
3
B.
1
5
C.
D.
1
3
A(4, -2), B (-2, 5) and C (6, 3) are the three vertices of a triangle. The gradient of the altitude BE is
A.
5
2
B.
C.
2
5
D.
2
5
5
2
The lines with equations x + 2 y = 8 and 3 x 2 y = 16 intersect at the point T. The coordinates of
T are
20.
A.
(2, 5)
B.
(6, 1)
C.
(2, 3)
D.
(2, 5)
The gradient of all lines perpendicular to the line AB where A is the point (2, 1) and B is (2, 1) is
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A.
1
2
B.
C.
D.
1
2
21.
The equation of the line through the point (0, 2) which is parallel to 2 x y = 7 is
A.
y = 2( x 1)
B.
y = 2( x + 1)
C.
y = 2( x + 1)
D.
y = 2( x 1)
22.
23.
The derivative of the function f ( x) = 2 x 3 with respect to x is
A.
6 73
x
7
B.
1 13
x
3
C.
8 13
x
3
D.
3 13
x
4
The function f is defined as f ( x) = x 3 3 x. The range of values for which this function is
increasing is
24.
A.
1 < x < 1
B.
x > 1 and x < 1
C.
x >1
D.
x < 1
When x 2 5 x + 3 is written in the form a ( x b) 2 + c the values of a, b and c are
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25
3 25
25
925
25
3 25
25
9 25
25.
The coordinates and nature of the turning point of the graph of y = 6 ( x + 5) 2 are
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Turning point
Nature
(5, 6)
Maximum
(5, 6)
Minimum
(6, 5)
Maximum
(6, 5)
Minimum
