Transcript Higher Mathematics - Prestwick Academy

Objective Questions
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7π
6
1. The exact value of tan
1
B. 
3
A.  3
C.
is:
1
3
D.
3
o
2. The period of tan3x , x є R , is:
A. 60
B. 120
C. 180
3. This diagram is most likely
to be part of the graph of:
A. cosx
o
C. 1 - sinx o
B. 1  sinx
o
D. 2cosx o  1
D. 540
2
y
1
90
180
270
answer
360
x
7π
6
1. The exact value of tan
1
B. 
3
A.  3
C.
is:
1
3
D.
3
o
2. The period of tan3x , x є R , is:
A. 60
B. 120
C. 180
3. This diagram is most likely
to be part of the graph of:
A. cosx
o
C. 1 - sinx o
B. 1  sinx
o
D. 2cosx o  1
D. 540
2
y
1
90
180
270
360
x
1.
Which of the following has (have) a negative value:
I. sin
5
12
II. sin
5
6
III. tan
5
3
IV. cos
5
4
A. Only I, II, III
B. Only I and III
C. Only III and IV
D. Some other response or combination


2. The minimum value of 1  cos x 
when x is:
A. 0
B.
π
3
π
3π
occurs
,0  x
3
2
C.
4π
3
D. π
3. Which of the following could be this graph:
1 o
A. cos x  1
2
B. 2  sin2x
C. 2cosx o
D. 1  cos2x o
o
2
y
1
90
180
270
360
answer
x
1.
Which of the following has (have) a negative value:
I. sin
5
12
II. sin
5
6
III. tan
5
3
IV. cos
5
4
A. Only I, II, III
B. Only I and III
C. Only III and IV
D. Some other response or combination


2. The minimum value of 1  cos x 
when x is:
A. 0
B.
π
3
π
3π
occurs
,0  x
3
2
C.
4π
3
D. π
3. Which of the following could be this graph:
1 o
A. cos x  1
2
B. 2  sin2x
C. 2cosx o
D. 1  cos2x o
o
2
y
1
90
180
270
360
x
1.
Which of the following is/are solution(s) of sin2x = 1, x є R:
I.

6
A. I only
II.
3
III.
4

4
B. II only
5
IV.
6
C. II & III only
D. None of I, II, III, IV
π

0

x

2

,
2sin



 has a maximum value when θ is:
2. If
6

π 
π
5
A. 0
B.
6
C.
D.
3
6
3. The line with equation y = -1 intersects the curve
y = √2sinx , at :
A. 315o
B. - 60o
C. 210o
D. 150o
√2
y
90
-√2
180
270
360
x
answer
1.
Which of the following is/are solution(s) of sin2x = 1, x є R:
I.

6
A. I only
II.

4
B. II only
3
III.
4
5
IV.
6
C. II & III only
D. None of I, II, III, IV
π

0

x

2

,
2sin



 has a maximum value when θ is:
2. If
6

π 
π
5
A. 0
B.
6
C.
D.
3
6
3. The line with equation y = -1 intersects the curve
y = √2sinx , at :
A. 315o
B. - 60o
C. 210o
D. 150o
√2
y
90
-√2
180
270
360
x
1.
The exact value of cos
A.  3
B. 
3
2
5π
6
is:
1
3
C.
2. The maximum value of 1 - sin x 

D.
3
π
 , 0  x  2π
6
occurs when x = t. What is the value of t?
3π
A.
2
π
B.
2
4π
C.
3
3. This diagram is most likely
to be part of the graph of:
A. cosx o - 2
B. 2sinx o
C. 2 - sinx o
D. cosx o - 1
5
D.
6
y
2
180
360
540
-2
answer
x
1.
The exact value of cos
A.  3
B. 
3
2
5π
6
is:
1
3
C.
2. The maximum value of 1 - sin x 

D.
3
π
 , 0  x  2π
6
occurs when x = t. What is the value of t?
3π
A.
2
π
B.
2
4π
C.
3
3. This diagram is most likely
to be part of the graph of:
A. cosx o - 2
B. 2sinx o
C. 2 - sinx o
D. cosx o - 1
5
D.
6
y
2
180
-2
360
540
x
1.
The exact value of sin (-120o) is:
A.
3
B. 
1
3
C. -
3
2
1
2
D.
π

0

x

2

,
2sin



 has a minimum value when θ is:
2. If
6

π
5π
5
A. 0
B.
6
C.
D.
6
3
3. The line with equation y = √3 intersects the curve
y = 2cosx , at :
A. 330o
B. - 60o
C. 45o
D. 420o
2
y
180
360
540
-2
answer
x
1.
The exact value of sin (-120o) is:
A.
3
B. 
1
3
C. -
3
2
1
2
D.
π

0

x

2

,
2sin



 has a minimum value when θ is:
2. If
6

π
5π
5
A. 0
B.
6
C.
D.
6
3
3. The line with equation y = √3 intersects the curve
y = 2cosx , at :
A. 330o
B. - 60o
C. 45o
D. 420o
2
y
180
-2
360
540
x
1.
The exact value of cos 135o is:
A. 
1
2
B.
1
2
C. -
1
2
D.
3
2. The largest possible domain of, f(x)  (2  x) is:
A. -2  x  2
B. x  2
3. This diagram is most likely
to be part of the graph of:
A. sin(x  45)o
B. sin(x - 45)o
C. sin(45- x)o
D. - sin(x  45)o
C. x  -2
1
D. x  2
y
90
180
270
-1
answer
360
x
1.
The exact value of cos 135o is:
A. 
1
2
B.
1
2
C. -
1
2
D.
3
2. The largest possible domain of, f(x)  (2  x) is:
A. -2  x  2
B. x  2
3. This diagram is most likely
to be part of the graph of:
A. sin(x  45)o
B. sin(x - 45)o
C. sin(45- x)o
D. - sin(x  45)o
C. x  -2
1
y
90
-1
D. x  2
180
270
360
x
y
1.
(-1,3)
Which of the following graphs
represents y = -f(x + 2):
A
B
y
(-3,2)
C
y
(-1,5)
(1,5)
(3,2)
(-3,2)
(-3,2) (3,2)
5
x
x
-5
2. The exact value of cos
A. 
1
2
B. 
3
2
3
-3
5π
3
D
y
Y = f(x)
x
(5,-2)
y
(5,4)
(3,2)
(3,2)
x
(-1,-1)
-5
1
x
(-3,-3)
is:
C. -
1
3
D.
1
2
3. Functions f and g , are given by f(x) = 3x2 + 1 and
g(x) = x2 - 4. Find an expression for f(g(x)).
A. 4x2 - 3
B. 3x4 - 3
C. 9x4  6x2  1
D. 3x4  24x2  49
answer
y
(-1,3)
1.
Which of the following graphs
represents y = -f(x + 2):
A
B
y
(-3,2)
C
y
(-1,5)
(1,5)
(3,2)
(-3,2)
(-3,2) (3,2)
5
x
x
-5
2. The exact value of cos
A. 
1
2
B. 
3
2
3
-3
5π
3
D
y
Y = f(x)
x
(5,-2)
y
(5,4)
(3,2)
(3,2)
x
(-1,-1)
-5
1
x
(-3,-3)
is:
C. -
1
3
D.
1
2
3. Functions f and g , are given by f(x) = 3x2 + 1 and
g(x) = x2 - 4. Find an expression for f(g(x)).
A. 4x2 - 3
B. 3x4 - 3
C. 9x4  6x2  1
D. 3x4  24x2  49
1.
For which real values of x is the function f : x 
defined on the set of real numbers?
1
(1 x2 )
A. All x except x  1 and x  -1
B. -1  x  1 only
C. x  1 and x  -1 only
D.
x  1 only
π

3


2. The minimum value of 0    2 , 1 - 2cos 
occurs when x is: A.

3
B.

C. 
2
D.
3. The line with equation y = 2 intersects the curve
y = 1 - 2sinx , at :
A.
C.
4
3
B.
5
6
D.
3
7
4
7
6
-1
y
180
360
answer
x

6
1.
For which real values of x is the function f : x 
defined on the set of real numbers?
1
(1 x2 )
A. All x except x  1 and x  -1
B. -1  x  1 only
C. x  1 and x  -1 only
D. x  1 only
π

3


2. The minimum value of 0    2 , 1 - 2cos 
occurs when x is: A.

3
B.

C. 
2
D.
3. The line with equation y = 2 intersects the curve
y = 1 - 2sinx , at :
A.
C.
4
3
B.
5
6
D.
3
7
4
7
6
-1
y
180
360
x

6
1.
Which of the following is/are solution(s) of 2sin2x = √3:
I.

II.
6
A. I only

III.
3
B. I & II only
2
3
IV.

4
C. II & III only
D. None of I, II, III, IV
π
π
sin
3
3
1
D.
2
2. Which of these would be the exact value of 2cos
A. -
1
2
B.
3
2
C. 0
3. Functions f and g , are given by f(x) = x2 – 2x and
g(x) = -3x. Find an expression for f(g(x)).
A. - 3x2  6x
B. - 3x2 - 2x
C. 9x2  6x
D. x2 - 5x
answer
?
1.
Which of the following is/are solution(s) of 2sin2x = √3:
I.

II.
6
A. I only

III.
3
B. I & II only
2
3
IV.

4
C. II & III only
D. None of I, II, III, IV
π
π
sin
3
3
1
D.
2
2. Which of these would be the exact value of 2cos
A. -
1
2
B.
3
2
C. 0
3. Functions f and g , are given by f(x) = x2 – 2x and
g(x) = -3x. Find an expression for f(g(x)).
A. - 3x2  6x
B. - 3x2 - 2x
C. 9x2  6x
D. x2 - 5x
?
y
1.
(-2,3)
Which of the following graphs
represents y = -2f(x) + 1:
y
A
B
0
x
(-2,-5)
x
x
1
-4
C
(1,1)
(-4,1)
(-3,6)
-5
y
Y = f(x)
D
y
y
(2,7)
(3,6)
(4,1)
(-1,1)
0
x
x
5
x3  1
, x  R , then g-1(x) equals:
2. Given that g(x) 
2
A.
2
x 3 1
B.
3
(2x  1)
3. Functions f and g, are given by
C. 23 (x  1)
f(x) 
Find an expression for f(g(x)).
A.
1
x 4 - 2x 2  3
B.
1
x 4  4x 2  3
C.
1
x2  2
D. 1 
3
2x
and g(x) = x2 - 1.
1
x 4 - 2x 2
D. x 4  2x 2  4
answer
y
1.
(-2,3)
Which of the following graphs
represents y = -2f(x) + 1:
y
A
B
0
x
(-2,-5)
x
x
1
-4
C
(1,1)
(-4,1)
(-3,6)
-5
y
Y = f(x)
D
y
y
(2,7)
(3,6)
(4,1)
(-1,1)
0
x
x
5
x3  1
, x  R , then g-1(x) equals:
2. Given that g(x) 
2
A.
2
x 3 1
B.
3
(2x  1)
3. Functions f and g, are given by
C. 23 (x  1)
f(x) 
Find an expression for f(g(x)).
A.
1
x 4 - 2x 2  3
B.
1
x 4  4x 2  3
C.
1
x2  2
D. 1 
3
2x
and g(x) = x2 - 1.
1
x 4 - 2x 2
D. x 4  2x 2  4
1.
The largest possible domain of, f(x) 2 x is:
A. x  0
B. x  0
C. x  0


2. The minimum value of 1 - 3cos x 
D. x  0
π
,0  x π
6
occurs when x = t. What is the value of t?
A. 0
B.
π
6
C.
π
2
D. π
3. The line with equation y = 1 intersects the curve
y = 4sin2x , at :
A. 150o
B. 210o
C. 45o
D. 300o
answer
1.
The largest possible domain of, f(x) 2 x is:
A. x  0
B. x  0
C. x  0


2. The minimum value of 1 - 3cos x 
D. x  0
π
,0  x π
6
occurs when x = t. What is the value of t?
A. 0
B.
π
6
C.
π
2
D. π
3. The line with equation y = 1 intersects the curve
y = 4sin2x , at :
A. 150o
B. 210o
C. 45o
D. 300o
1.
y
(-1,5)
Which of the following functions
represents the black curve:
A. y = g(-x) + 2
B. y = -g(x) - 2
C. y = 2 – g(x)
D. y = g(x – 2)
(-1,-3)
(1,3)
y = g(x)
x
(1,-1)
5x
, x  R , then h-1(x) equals:
2. Given that h(x) 
2
A.
2
x 5
B. 2x  5
C.
3. Functions f and g, are given by f(x) 
Find an expression for g(f(x)).
2 - x2
A.
1 - x2
2
B.
1 - x2
2x
5
1
1 - x2
x2
C.
1 - x2
D. 5 - 2x
and g(x) = 1 + x.
D. 2
answer
1.
y
(-1,5)
Which of the following functions
represents the black curve:
A. y = g(-x) + 2
B. y = -g(x) - 2
C. y = 2 – g(x)
D. y = g(x – 2)
(-1,-3)
(1,3)
y = g(x)
x
(1,-1)
5x
, x  R , then h-1(x) equals:
2. Given that h(x) 
2
A.
2
x 5
B. 2x  5
C.
3. Functions f and g, are given by f(x) 
Find an expression for g(f(x)).
2 - x2
A.
1 - x2
2
B.
1 - x2
2x
5
1
1 - x2
x2
C.
1 - x2
D. 5 - 2x
and g(x) = 1 + x.
D. 2
1.
For which real values of x is the function
1
9 - x2
f: x 
defined on the set of real numbers?
A. All x except x  3 and x  - 3
B. x  3 only
C. x  3 and x  - 3 only
D. - 3  x  3 only
2. The equation of the straight line through the points
(1 , -2) and (-3 , 4) is:
A. 3x + 2y = -1
B. 3x – 2y = 7
C. 2x + 3y = -4
D. None of these
3. Which of the following is/are solution(s) of √3tan2x = -1:
I.
5π
6
A. I only
II.
5π
3
B. III & IV only
III.
5π
12
C. III only
IV.
11π
12
D. II only
answer
1.
For which real values of x is the function
1
9 - x2
f: x 
defined on the set of real numbers?
A. All x except x  3 and x  - 3
B. x  3 only
C. x  3 and x  - 3 only
D. - 3  x  3 only
2. The equation of the straight line through the points
(1 , -2) and (-3 , 4) is:
A. 3x + 2y = -1
B. 3x – 2y = 7
C. 2x + 3y = -4
D. None of these
3. Which of the following is/are solution(s) of √3tan2x = -1:
I.
5π
6
A. I only
II.
5π
3
B. III & IV only
III.
5π
12
C. III only
IV.
11π
12
D. II only
1.
The gradient of a straight line parallel to the line
x + 3y + 7 = 0 is:
A. - 3
B.
1
3
C. 7
2. Functions f and g, are given by
Find an expression for f(g(x)).
x
A. 2
x 1
1
B.
x1
1
f(x) 
x
D. -
1
3
and
g(x) 
x2
D.
1  x2
C. x  1
2
3. The line with equation y = 4 intersects the curve
y = 1 - 6sinx , at :
A.
7
6
B.
4
3
C.
5
4
1
x2  1
D.
5
6
answer
1.
The gradient of a straight line parallel to the line
x + 3y + 7 = 0 is:
A. - 3
B.
1
3
C. 7
2. Functions f and g, are given by
Find an expression for f(g(x)).
x
A. 2
x 1
1
B.
x1
1
f(x) 
x
D. -
1
3
and
g(x) 
x2
D.
1  x2
C. x  1
2
3. The line with equation y = 4 intersects the curve
y = 1 - 6sinx , at :
A.
7
6
B.
4
3
C.
5
4
1
x2  1
D.
5
6
1.
The line joining the points (-2,-3) and (6, k) has gradient .
The value of k is:
A.
7
3
B.
17
3
C.
25
3
D. 9
2. Which of the following could be this graph:
A. 2cosx o  1
B. 2  sin3x o
C. 1  2sin3x o
D. 1 - 3sin2x o
4
y
180
-2
x
π

0



2

,
1

2sin


3. The minimum value of


3

occurs when x is: A. π
3
11π
B.
6
5π
C.
3
answer
5π
D.
6
1.
The line joining the points (-2,-3) and (6, k) has gradient .
The value of k is:
A.
7
3
B.
17
3
C.
25
3
D. 9
2. Which of the following could be this graph:
A. 2cosx o  1
B. 2  sin3x o
C. 1  2sin3x o
D. 1 - 3sin2x o
4
y
180
-2
x
π

0



2

,
1

2sin


3. The minimum value of


3

occurs when x is: A. π
3
11π
B.
6
5π
C.
3
5π
D.
6
1.
For which real values of x is the function f : x 
defined on the set of real numbers?
1
x  3x  5 
A. All x except x  3 and x  - 5
B. x  
C. x  0 only
D. - 5  x  3 only
2. Which of the following is the inverse of f(x) = x – 2 ,
where x є R ?
A.
1
x-2
B. x  2
C. 2x  1
D.
1
x 2
3. If the points (p , q) , (3 , -2) and (-1 , 4) are collinear, then
the relationship connecting p and q could be:
A. 2p + 3q = 13
B. 3p – 2q = 5
C. 3p + 2q = 5
D. 3p – 2q = 13
answer
1.
For which real values of x is the function f : x 
defined on the set of real numbers?
1
x  3x  5 
A. All x except x  3 and x  - 5
B. x  
C. x  0 only
D. - 5  x  3 only
2. Which of the following is the inverse of f(x) = x – 2 ,
where x є R ?
A.
1
x-2
B. x  2
C. 2x  1
D.
1
x 2
3. If the points (p , q) , (3 , -2) and (-1 , 4) are collinear, then
the relationship connecting p and q could be:
A. 2p + 3q = 13
B. 3p – 2q = 5
C. 3p + 2q = 5
D. 3p – 2q = 13
y
1.
3
Which of the following graphs
represents y = f(1 - x) :
A
y
B
(-1,3)
(1,1)
-3
x
y
C
(1,3)
(-1,1)
(2,1)
-2
y
y = f(x)
(-1,3)
(-3,1)
x
3 x
D
x
y
(-2,1)
2
x
-2
x
1 x
2. Which of the following is the equation of a line
perpendicular to the line x - 3y + 4 = 0
A. y = -3x
B. y = x
C. y = -x
3. Functions f and g, are given by f(x)  12
x
Find an expression for f(g(x)).
A. x2  2x  1
x2
B. 2
x 1
C.
and
1
x 2  2x  1
D. y = -x
g(x) 
D.
answer
1
x 1
1
x 3  x2
y
1.
3
Which of the following graphs
represents y = f(1 - x) :
A
y
B
(-1,3)
(1,1)
-3
x
y
C
(1,3)
(-1,1)
(2,1)
-2
y
y = f(x)
D (-2,1)
(-1,3)
(-3,1)
x
3 x
x
y
2
x
1 x
x
-2
2. Which of the following is the equation of a line
perpendicular to the line x - 3y + 4 = 0
A. y = -3x
B. y = x
C. y = -x
3. Functions f and g, are given by f(x)  12
x
Find an expression for f(g(x)).
A. x2  2x  1
x2
B. 2
x 1
C.
and
1
x 2  2x  1
D. y = -x
g(x) 
D.
1
x 1
1
x 3  x2
1.
The line 2y = 3x + 6 meets the y-axis at C. The gradient
of the line joining C to A (4,-3) is:
A.
9
4
B. -
2
3
C.
9
4
D. -
3
2
2. Which of these would be the exact value of 2cos
A.
4
2
B. 1
C. 0
D.
π
π
sin
4
4
1
2
3. The line with equation y = 1 intersects the curve
y = 3tan2x , at :
A.
π
3
B.
7π
6
C.
5π
6
D.
answer
π
4
?
1.
The line 2y = 3x + 6 meets the y-axis at C. The gradient
of the line joining C to A (4,-3) is:
A.
9
4
B. -
2
3
C.
9
4
D. -
3
2
2. Which of these would be the exact value of 2cos
A.
4
2
B. 1
C. 0
D.
π
π
sin
4
4
1
2
3. The line with equation y = 1 intersects the curve
y = 3tan2x , at :
A.
π
3
B.
7π
6
C.
5π
6
D.
π
4
?
1.
The straight lines with equations ay = 3x + 7 and y = 5x + 2
are perpendicular. The value of a is:
A. -
1
5
B. -
5
3
C. -
3
5
D. -15
2. Which of the following could be this graph:
1 o
A. 2 - 2sin x
2
1
B. 2  sin2x o
2
C. 2sin2x o  2
1
D. 2 - 4cos x o
2
4
y
2


3. The maximum value of 0    2 , 1  2sin 
occurs when x is: A. π
4
B.
7π
4
x
720
C.
5π
4
π

4
answer
D.
3π
4
1.
The straight lines with equations ay = 3x + 7 and y = 5x + 2
are perpendicular. The value of a is:
A. -
1
5
B. -
5
3
C. -
3
5
D. -15
2. Which of the following could be this graph:
1 o
A. 2 - 2sin x
2
1
B. 2  sin2x o
2
C. 2sin2x o  2
1
D. 2 - 4cos x o
2
4
y
2


3. The maximum value of 0    2 , 1  2sin 
occurs when x is: A. π
4
B.
7π
4
x
720
C.
5π
4
π

4
D.
3π
4
1.
R and S have coordinates (5,-7) and (-1,-3) respectively.
The perpendicular bisector of RS has a gradient of -.
What is the equation of the perpendicular bisector of RS?
A. 3y = 2x + 13
B. 3y = -2x + 19
C. 2y = -3x - 19
D. 2y = 3x - 13
2. Find the gradient of the line AB:
A. m = 1
C. m = -1
A
y
B. m = -√2
1
D. m = - 2
45o
B
x
3. What is the solution of the equation 2cosx - √3 = 0
π
5π
11π
5π
B.
C.
D.
where 3π  x  2π? A.
2
6
6
6
answer
3
1.
R and S have coordinates (5,-7) and (-1,-3) respectively.
The perpendicular bisector of RS has a gradient of -.
What is the equation of the perpendicular bisector of RS?
A. 3y = 2x + 13
B. 3y = -2x + 19
C. 2y = -3x - 19
D. 2y = 3x - 13
2. Find the gradient of the line AB:
A. m = 1
C. m = -1
A
y
B. m = -√2
1
D. m = - 2
45o
B
x
3. What is the solution of the equation 2cosx - √3 = 0
π
5π
11π
5π
B.
C.
D.
where 3π  x  2π? A.
2
6
6
6
3
1.
2.
The side of a triangle has equation y = -x – 3.
Which of these could be the equation of an altitude
passing through this side?
A. 2y + x – 3 = 0
B. 2y – 3x + 3 = 0
C. 2y + 3x – 1 = 0
D. 3y – 2x + 1 = 0
The vertices of triangle STV are S(-4,10) , T(10,3) and V(0,-10).
Which of the following is the equation of the median TM?
A. 4y = x + 2
B. y = 4x + 2
C. y = -2x + 23
D. y = 2x - 2
3. Functions f and g, are given by
Find an expression for f(g(x)).
x1
A.
x
x2
B.
x1
1
f(x) 
x
C.
1
x1
and
g(x) 
D. x  1
answer
x
x 1
1.
2.
The side of a triangle has equation y = -x – 3.
Which of these could be the equation of an altitude
passing through this side?
A. 2y + x – 3 = 0
B. 2y – 3x + 3 = 0
C. 2y + 3x – 1 = 0
D. 3y – 2x + 1 = 0
The vertices of triangle STV are S(-4,10) , T(10,3) and V(0,-10).
Which of the following is the equation of the median TM?
A. 4y = x + 2
B. y = 4x + 2
C. y = -2x + 23
D. y = 2x - 2
3. Functions f and g, are given by
Find an expression for f(g(x)).
x1
A.
x
x2
B.
x1
1
f(x) 
x
C.
1
x1
and
g(x) 
D. x  1
x
x 1
1.
If f(x)  2x
A. 
2
3
; f’(4) equals:
B. 2
C. 3
D. 6
2. If the line ax - 2y + 5 = 0 is parallel to the line
3x + y - 4 = 0, a is equal to:
A. -6
B. -
C. 
3. PQ, of length 2, is parallel to OY.
QR, of length 4, is parallel to OX.
Angle PQR = 90o. P is the point (1,2).
The line PR cuts OY at:
A. (0,)
B. (0,)
D. 
y
Q
4
R
2
P (1,2)
x
0
C. (0,-)
D. (0,-)
answer
1.
If f(x)  2x
A. 
2
3
; f’(4) equals:
B. 2
C. 3
D. 6
2. If the line ax - 2y + 5 = 0 is parallel to the line
3x + y - 4 = 0, a is equal to:
A. -6
B. -
C. 
3. PQ, of length 2, is parallel to OY.
QR, of length 4, is parallel to OX.
Angle PQR = 90o. P is the point (1,2).
The line PR cuts OY at:
A. (0,)
B. (0,)
D. 
y
Q
4
R
2
P (1,2)
x
0
C. (0,-)
D. (0,-)
1.
This diagram is most likely
y
to be part of the graph of:
1 o
A. 2 - cos x
4
C. 2cos4x o - 1
1
B. cos4x o  3
-3
1
D. cos2x o - 1
4
2. Find the gradient of the line ST:
A. m = -1
B. m = 1
A.
1
2x
y
1
D. m = 2
C. m = -√2
3. If f(x) 
S
135o
1
and x ≠ 0 then f’(x) equals:
2
x
B. -
2
x3
C. -
1
x
x
90
D. -
1
x3
T
answer
x
1.
If f(x) = x√x , x > 0 ; f’(x) equals:
1
A. 1 
2 x
B. 1 
x
3
C.
x
5
2 52
D.
x
5
2. Which of the following is/are true of the line with
equation 3x - 2y + 3 = 0?
I.
It passes through the point (-2,-3)
II. It is parallel to the line 6x + 4y + 3 = 0
III. It is perpendicular to the line 2x + 3y + 3 = 0
A. I only
B. I & III only
C. III only
D. Some other combination of responses
3.
The line with equation y = √3 intersects the curve y = 2cosx, at:
A. 330o
B. - 60o
C. 45o
D. 420o
answer
1.
The gradient of the curve y = 5x3 - 10x at the point (1,-5)
is:
A. -5
B. 5
C. 15
D. None of these
2. f and g are functions on the set of real numbers such that
f(x) = 2x – 1 and f(g(x)) = 4x + 1, g(x) equals:
A. 8x + 1
B. 8x - 3
C. 2x + 3
1
3. Functions f and g, are given by f(x) 
x
Find an expression for g(f(x)).
x1
A.
x2
x2
B.
x1
C.
1
x1
D. 2x + 1
and
g(x) 
D. x  1
answer
x
x 1
1.
The x-coordinate of the point at which the curve
y = 6 – 3x2 has gradient 12 is:
A. -6
B. -2
C. -√2
D. -1
2. The vertices of triangle ABC are A(1,-7), B(-4,7) & C(-1,3).
Which of the following is the equation of the median CM?
A. y = 6x + 4
B. y = 6x + 9
C. 2y = x + 7
D. 2y = 3x - 9
π

3. The maximum value of 0    2 , 3  2sin  
3

occurs when x is: A. 11π
6
B.
7π
6
C.
5π
6
answer
D.
5π
3
Question 27
How do you
show that
a curve is
always increasing ?
answer
Answer to Question 27
(i) Differentiate
’
(ii) show that f (x) is a
perfect square
Question 28
How do you find the
equation of a tangent
to a curve at the point
when x = a ?
answer
 (i)
(ii)
Answer to Question 28
Differentiate
’
fit a into f (x) to get
the gradient (m)
(iii) fit a into f(x) to get
the tangent point (a,b)
(iv) use y-b=m(x-a)
Question 29
For what values of a
function is the
function said to be
undefined ?
answer
Answer to Question 29
When you fit in a value
of x and you cannot get
an answer
Question 30
How do you draw
the graph of f(x-1)
given the graph of
f(x) ?
answer
Answer to Question 30
Move the graph 1 unit
to the right
Question 31
How do you find
f(g(x)) for given
functions f(x) and
g(x) ?
answer
Answer to Question 31
Fit g(x) into f(x)
i.e. each x in f(x) is
replaced by the
function g(x)
Question 32
What two things do
you require in order
to find the equation
of a straight line ?
answer
Answer to Question 32
The gradient of the line
and a point on the line
y
1
m
(a,b)
x
Question 33
How do you find the
midpoint of a line
joining two points ?
answer
Answer to Question 33
Add the coordinates
and divide by two
x+x , y+y
1
2
1
2
(
2
2
)
(x1,y1)
y
(x2,y2)
x
Question 34
What is the
gradient of a
vertical line ?
answer
Answer to Question 34
undefined
y
x
Question 35
How do you find the
median AM of
triangle ABC ?
answer
Answer to Question 35 A
 (i) find the
mid point
of BC (M)
(ii)
find the C
gradient of AM
(iii) use y-b = m(x-a)
M
B
Question 36
Which two points
does the graph
x
y = a always pass
through ?
answer
Answer to Question 36
(0,1) and (1,a)
Question 37
What is the
perpendicular
bisector of a line ?
answer
Answer to Question 37
A line which cuts the
o
given line in half at 90
Question 38
How do you draw
the graph of f(x+1)
given the graph of
f(x) ?
answer
Answer to Question 38
Move the graph 1 unit
to the left
Question 39
How do you find the
equation of a
perpendicular
bisector of a line ?
answer
 (i)
(ii)
(iii)
(iv)
Answer to Question 39
find the midpoint of the line
find the gradient of the line
find the gradient
perpendicular to the given
line
Use midpoint and gradient in
y-b = m(x-a)
M(a,b)
Question 40
For what values is this
function undefined ?
x
f(x) =
(x+2)(x-3)
answer
Answer to Question 40
-2 and 3
Question 41
What are the two
formulae used to
find the area of a
triangle ?
answer
Answer to Question 41
A = ½base x height
A = ½bcsinA
A
b
C
B a s e
height
a
c
B
Question 42
What three
processes do you go
through in order to
factorise a
quadratic ?
answer
Answer to Question 42
(i) common factor
(ii) difference of two
squares
(iii) trinomial
Question 43
What is the
equation of a
vertical line passing
through (a,b) ?
answer
Answer to Question 43
x = a
y
(a,b)
x
Question 44
What is the
Theorem of
Pythagoras ?
answer
Answer to Question 44
For ΔABC,
right-angled at A,
2
2
2
a =b +c
a
B
C
b
c
A
Question 45
What do you know
about the gradients
of two parallel
lines?
answer
Answer to Question 45
They are the same
Question 46
How do you draw
the graph of f’(x)
given the graph of
f(x) ?
answer
Answer to Question 46
 (i) plot x coords of st. points on
x-axis (SPs become roots)
(ii) look at each part of f(x)
separately:
if rising, graph of f’(x) is
above x-axis
if falling, graph
of f’(x) is
below x-axis
Question 47
How do you get the
gradient of a line
with an equation like
3x + 2y = 5 ?
answer
Answer to Question 47
Rearrange into the
form
y = mx + c
(ii) read off
gradient = m
(i)
Question 48
What is loga1
equal to ?
answer
Answer to Question 48
0
Question 49
How do you find the
length of a line
joining two points ?
answer
Answer to Question 49
√(x2 –
2
x1)
y
A(x1,y1)
+
2
(y2 –y1)
B(x2,y2)
x
Question 50
What is the
Converse of
Pythagoras ?
answer
Answer to Question 50
2
a
2
b
2
c
If
=
+
then ΔABC is
right-angled at A
a
B
C
b
c
A
Question 51
How do you find the
gradient of a line
joining two points ?
answer
Answer to Question 51
m = y2 – y1
x2 – x 1
y
A(x1,y1)
B(x2,y2)
x
Question 52
How do you find the
altitude AN of
ΔABC ?
answer
Answer to Question 52
 (i)find the gradient
of BC
(ii) find the gradient
A
of AN,
perpendicular
to BC
B
N
(iii) use y-b=m(x-a)
C
Question 53
For a curve, how do
you find the
stationary points
and their nature ?
answer
Answer to Question 53
 (i) differentiate
(ii) let f’(x) = 0
(iii) solve to find
stationary points
(iv) find y-coordinates
(v) draw nature table
Question 54
How do you draw
the graph of 3+f(x)
given the graph of
f(x) ?
answer
Answer to Question 54
move graph up 3
Question 55
How do you find
where a curve is
increasing ?
answer
Answer to Question 55
 (i)differentiate
(ii) let f’(x) = 0
(iii) solve to find stationary
points
(iv) draw nature table
(v) read values for which
graph is increasing
Question 56
How do you find
where two lines
intersect ?
answer
Answer to Question 56
Simultaneous equations
Question 57
How do you draw
the graph of 3-f(x)
given the graph of
f(x) ?
answer
Answer to Question 57
Reflect the graph in
the x-axis,
then move it up 3
Question 58
How do you draw
the graph of f(-x)
given the graph of
f(x) ?
answer
Answer to Question 58
Reflect the graph in
the y-axis
Question 59
How do you solve
equations like
100 = 0 ?
42
x
answer
Answer to Question 59
multiply by the
denominator of the
2
fraction (here x )
(ii) factorise and solve
(i)
Question 60
How do you find the
exact values of
sin(A+B), cos(A-B) etc.
given that
3
cosA = /5 and
12
sinB = /13 ?
answer
 (i)
 (ii)
 (iii)
 (iv)
Answer to Question
60
A
draw
two Δs
find
missing sides
expand
formula
fit in values
from Δs
5
3
B
13
12
Question 61
How do you solve
equations like
o
o
Cos2x - 5cosx = 2 ?
(0 ≤ x ≤ 360)
answer
Answer to Question 61
o
2
2cos x -1
fit in
for
o
cos2x
(ii) factorise
(iii) solve the equation
(i)
Question 62
What is
sin x
cos x
equal to ?
answer
Answer to Question 62
tan x
Question 63
How do you show
that x-1 is a factor
of the function
3
f(x)=x -3x+2 ?
answer
 (i)
Answer to Question 63
rewrite the function as
3
2
f(x)=x +0x -3x+2
 (ii) use synthetic division
with 1 on the outside
 (iii) show that
remainder = 0
Question 64
What is the
sine rule ?
answer
Answer to Question 64
 a
b
c
=
=
sinA sinb sinC
A
c
b
B
C
a
Question 65
Given f’(x) and a
point on the curve,
how do you find
f(x) ?
answer
Answer to Question 65
(i) integrate
(ii) fit in given point
to work out value
of C
Question 66
How do you solve
quadratic
inequations like
2
x - 5x + 6 ≤ 0 ?
answer
Answer to Question 66
(i) factorise
(ii) draw graph
(iii) read values
below x-axis
Question 67
How do you change
from radians to
degrees ?
answer
Answer to Question 67
 Divide by π and
multiply by 180
Question 68
What is the
condition for real
roots ?
answer

2
b
Answer to Question 68
– 4ac ≥ 0
Question 69
 How do you find the value of
a in the polynomial
x3+ax2+4x+3 given a factor
of the polynomial or the
remainder when the
polynomial is divided by a
number ?
answer
Answer to Question 69
 (i) do synthetic division
(ii) let the expression
= 0 or the remainder
(iii) solve the equation
Question 70
 How do you find f(x) if
 f’(x) = 5-3x2 and
 the curve passes through
the point (1,9) ?
answer
 (i)
 (ii)
Answer to Question 70
f(x) = ∫f'(x) dx
find C by replacing
point (1,9) into f(x)
 (iii) write down completed
formula for f(x)
Question 71
What is
2
2
sin x + cos x
equal to ?
answer
1
Answer to Question 71
Question 72
How do you find the
equation of the
tangent to a circle at a
particular point on the
circumference ?
answer

Answer to Question
72
y
(i) find the
centre
(a,b)
(ii) find gradient
from centre
x
to point
(iii) find perpendicular gradient
(iv) use y-b=m(x-a)
C
Question 73
How do you find
2
x + 1 dx ?
∫ √x
answer

Answer to Question 73
(i) change root to
power
(ii)
(iii)
(iv)
(v)
split up into fractions
simplify each term
integrate each term
REMEMBER +C
Question 74
How do you show
that the root of a
function lies
between two given
values ?
answer
Answer to Question 74
 fit in two values and
show one is positive
and one is negative
+ve
x
-ve
Question 75
How do you find
exact values of
sin2x and cos2x
3
given cosx = /5 ?
answer
Answer to Question 75
 (i)draw a
right-angled
triangle
(ii) find the
missing side
(iii)
expand the double
angle formula
(iv)
fit in values from Δ
5
3
A
Question 76
What is the turning
point of
2
y=2(x-a) +b ?
Max or min ?
answer
Answer to Question 76

(i) (a,b)
minimum
(a,b)
Question 77
How do you
n
integrate x ?
answer

Answer to Question 77
n+1
x
n+1
+C
Question 78
How do you solve
equations like
o
o
cos2x -5sinx = 0 ?
(0≤x≤360)
answer

(i)
Answer to Question 78
o
2
fit in 1-2sin x
o
for cos2x
(ii) factorise
(iii) solve equation
Question 79
How do you
complete the square
for functions like
2
2x + 12x + 3 ?
answer
Answer to Question 79
(i) multiply out
2
a(x+p) +q
(ii) compare with
given function
(iii) find a, p and q

Question 80
How do you solve
equations of the form
o
sin2x = 0.5 ?
(0≤x≤360)
answer
Answer to Question 80
 (i) decide on the
2 quadrants (sin is +ve)
 (ii) press INV sin to get
angle
 (iii) work out your 2 angles
 (iv) divide each by 2
Question 81
How do you solve
quadratic inequations
like
2
x +5x-6 ≥ 0 ?
answer
Answer to Question 81
(i) factorise
(ii) draw graph
(iii) read values
above x-axis

Question 82
What is the centre
and radius of a
circle with equation
2
2
2
x +y =r ?
answer
Answer to Question 82
(i) centre (0,0)
(ii) radius = r

Question 83
How do you calculate
the area under a
curve ?
answer
Answer to Question 83
(i) integrate
(ii) fit in two limits
and subtract to
find area

Question 84
How do you find the
root of an equation
between two given
values to 1 dp ?
answer
Answer to Question 84
iteration
Question 85
How do you solve
equations of the form
o
2
sin x = 0.5 ?
(0≤x≤360)
answer
Answer to Question 85
(i) rearrange to get
o
sinx = ± …
(ii) find answers in
all 4 quadrants
Question 86
How do you name the
angle between a line
and a plane ?
answer
Answer to Question 86
 (i) start at end of line (A)
 (ii) go to where line meets A
the plane (B)
C
 (iii) go to the point B
on the plane
directly under
the start of the line (C)

ABC
Question 87
What is the condition
for equal roots ?
answer
Answer to Question 87
2
b –
4ac = 0
Question 88
What is the turning
point of
2
y = b-3(x-a) ?

max or min ?
answer
Answer to Question 88
(a,b)
Maximum
(a,b)
Question 89
What is the quadratic
formula and explain
when it is used ?
answer
x =
Answer to Question 89
2
-b±√(b -4ac)
2a
It is used to find roots
of a quadratic equation
when it is difficult to
factorise.
Question 90
How do you prove that
a line is a tangent to a
circle ?
answer
Answer to Question 90
Rearrange line to make
y = or x =
Fit line into circle
Prove it has equal roots
2
using b -4ac = 0 or
repeated roots
Question 91
How do you find the
exact value of
sin (α-β),
4
given that sinα = /5
12
and cosβ = /13 ?
answer
Answer to Question 91
 (i) draw triangles
for α and β
 (ii) work out
cosα and sinβ
 (iii) expand
formula for sin(α-β)
 (iv) insert exact values
α
5
4
13
β
12
Question 92
How do you solve
equations of the form
o
cosx = - 0.8 ?
(0≤x≤360)
answer
Answer to Question 92
(i) decide on the
2 quadrants (cos is -ve)
 (ii) ignore the sign and
press INV cos to get
angle
 (iii) work out your 2 angles

Question 93
How do you change
from degrees to
radians ?
answer
Answer to Question 93
Divide by 180 and
multiply by π
Question 94
How do you find the
exact values of sin x
or tan x given
cos x = a ?
b
answer
Answer to Question 94
 (i) draw triangle
b
a
 (ii) use Pythagoras
to fill in missing side
 (iii) read values off
triangle using
SOHCAHTOA
x
Question 95
How do you factorise a
cubic expression like
3
2
x -2x -x+2 ?
answer
Answer to Question 95
Synthetic division
using factors of last
number
factor
1
-2
-1
2
Remainder=0
Question 96
What is the centre
and radius of a circle
of the form
2
2
x +y +2gx+2fy+c=0 ?
answer
Answer to Question 96
Centre (-g,-f)
2
2
Radius √(g +f -c)
Question 97
How do you remember
the exact values of
o
o
o
30 , 45 and 60 ?
answer
Answer
to
Question
97
o
 sin30 = ½
 Draw right-angled
triangle
 Complete using Pythagoras
 Do similar
o
for tan 45 =1
60o
1
2
30o
√3
45o
√2
1
1
45o
Question 98
How do you calculate
the area between two
curves ?
answer
Answer to Question 98
(i) let equations equal
each other
(ii) solve to find limits
(iii) integrate
(upper - lower)
functions between limits
Question 99
How do you solve an
equation like
3sinx+1 = 0 ?
answer
Answer to Question 99
 (i) rearrange to sinx =
 (ii) decide on 2 quadrants
 (iii) ignore any –ve and press
INV sin to get angle
 (iv) work out two answers
Question 100
What is the condition
for no real roots ?
answer
2
b
Answer to Question 100
– 4ac < 0
Question 101
How do you find
b

∫
a
3
x
dx ?
answer
Answer to Question 101

3+1
x
[ 3+1
 then
]
1
b
a
4
/4[(b )
-
4
(a )]
Question 102
How do you find where
a line and a circle
intersect ?
answer
Answer to Question 102
Rearrange line to get
x = … or y = …
Fit into circle and solve
Question 103
State the cosine rule
to find an angle
answer
Answer to Question 103
cos A =
2
b
2
c
+ 2bc
2
a
A
c
b
B
C
a
Question 104
What is the centre
and radius of a circle
of the form
2
2
2
(x-a) +(y-b) = r ?
answer
Answer to Question 104
Centre (a,b)
Radius = r
y
r
C (a,b)
x
Question 105
State the cosine rule
to find a missing side
answer
2
a
Answer to Question 105
=
2
2
b +c -2bccosA
A
c
b
B
C
a
Question 106
How do you find
n
∫ (ax + b) dx ?
answer
 (i)
 (ii)
 (iii)
 i.e.
Answer to Question 106
increase power by 1
divide by new power
divide by the
derivative of
the bracket
n+1
(ax+b)
a(n+1)
+ C
Question 107
How do you find
the coordinates of a
point which divides a
line in a ratio e.g.
3:2 ?
answer
 (i)
 (ii)
 (iii)
 (iv)
 (v)
Answer to Question 107
write in form AB = 3
BC 2
cross-multiply
write AB = (b-a)
solve to find missing
vector
rewrite as point (*,*)
A
3
B
2
C
Question 108
What is a
position vector ?
answer
Answer to Question 108
A vector which starts at
the origin
Question 109
How do you express
acosx+bsinx+c
in the form
kcos(x-α) etc?
answer
Answer to Question 109
 (i) expand brackets
and equate like terms
S A
 (ii) find k =√(a2+b2)
T C
 (iii) identify quadrant α is in
 (iv) find α , tanα = sinα
cosα
Question 110
How do you
differentiate a
bracket without
multiplying it out ?
answer
Answer to Question 110
(i) multiply by old power
(ii) decrease power by 1
(iii) multiply by
derivative of bracket
Question 111
What is
Logax – logay
equal to ?
answer
Answer to Question 111

x
loga
y
Question 112
What do you get
when you
differentiate cosx ?
answer
Answer to Question 112
-sinx
Question 113
How do you show
that two vectors are
perpendicular ?
answer
Answer to Question 113
Show that a.b=0
a
b
Question 114
How do you
integrate sin ax ?
answer
Answer to Question 114
1
- /
a
cos ax + C
Question 115
How do you draw a
graph of the form
y = acosx
or y = asinx ?
answer
Answer to Question 115
Draw y = cosx
or y = sinx graph
with a maximum of a
and a minimum of -a
Question 116
How do you find the
maximum or
minimum values of
acosx + bsinx + c ?
answer
Answer to Question 116
(i) change acosx+bsinx
into Rcos(x-a)
(ii) max is R+c
Question 117
How do you find a
unit vector parallel
to a given vector ?
answer
Answer to Question 117
(i) find the length of
the given vector
(ii) divide all the
components by
this length
Question 118
How do you
integrate cos ax ?
answer
Answer to Question 118
1
/
a
sin ax + C
Question 119
How do you draw a
graph of the form

y = cos(x+a)
 or y = sin(x+a) ?
answer
Answer to Question 119
Move the graph of
y=cosx or y=sinx
a units to the LEFT
Question 120
What is a unit
vector ?
answer
Answer to Question 120
A vector of length 1 unit
Question 121
How do you draw a
graph of the form

y = cos bx
 or y = sin bx ?
answer
Answer to Question 121
Draw the normal graph
but fit in b waves
o
o
between 0 and 360
Question 122
What is
loga x + loga y equal
to ?
answer
Answer to Question 122
Loga xy
Question 123
What do you get
when you
differentiate sin x ?
answer
Answer to Question 123
cos x
Question 124
How do you find the
angle between two
vectors ?
answer
Answer to Question 124

a.b
cos =
ab
a

b
Question 125
Given an equation
-3k
like m = moe and
an amount by which
it has been decayed,
how do you find k ?
answer
Answer to Question 125
(i) fit in m and mo
-3k
(ii) rearrange to get e =
(iii) take logs
(iv) solve
Question 126
If u = ai+bj+ck
then what is u in
component form ?
answer
Answer to Question 126
U=
a
b
c
Question 127
What do you get
when you
differentiate
cosax ?
answer
Answer to Question 127
-asinax
Question 128
How do you solve an
equation of the form
acosx + bsinx + c=0 ?
answer
Answer to Question 128
 Change acosx+bsinx
into Rcos(x- a)
Rearrange and solve
Question 129
What is loga
to ?
n
x
equal
answer
Answer to Question 129
 nloga x
Question 130
How would you
differentiate a
function like
3
y = sin x ?
answer
Answer to Question 130
3
 (i) write as (sin x)
 (ii) multiply by the power
 (iii) decrease power by one
 (iv) multiply by the derivative
the bracket
 i.e. 3cosx sin2x
of
Question 131
State the three
rules of logs ?
answer
Answer to Question 131
 (i) logaxy = logax + logay
(ii) loga
x
= logax – logay
y
n
(iii) logax = nlogax
Question 132
How do you solve
equations of the
form
x
3 = 0.155 ?
answer
Answer to Question 132
 (i) take logs of both sides
(ii) bring x down to front
(iii) solve the equation
Question 133
Given experimental
data, how do you
find an equation in
x
the form y=ab or
b
y=ax ?
answer
Answer to Question 133
 (i) take logs of both sides
(ii) rearrange to get a
straight line equation
(iii) determine type
(iv) find solution
Question 134
How would you
differentiate a
function like
y = sin ax ?
answer
Answer to Question 134

dy/
=
acos
ax
dx
Question 135
If u =
a
b
c
then what is u ?
answer
Answer to Question 135
 work
out length
2
2
2
√(a +b +c )
Question 136
How do you add or
subtract vectors ?
answer
Answer to Question 136
 add
or subtract
matching components
Question 137
What does
a.a equal ?
answer
Answer to Question 137

2
a
Question 138
How do you prove
that three 3-D
points are
collinear ?
answer
Answer to Question 138

Prove they are the
same vector multiplied
by different or the
same numbers
Question 139
Express the
n
equation y=kx in the
form of the equation
of a straight line,
Y=nX+c.
answer
Answer to Question 139

logy = nlogx + logk
Question 140
Who loves maths ?
answer
Answer to Question 140

ME !!!!!