1. Higher Mathematics
Objective Questions

2. Objective Questions 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99
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3. Set 1 The exact value of tan is: The period of tan3xo, x є R , is:
3. This diagram is most likely
to be part of the graph of: 1 2 y x 90
180
270
360
answer

4. Set 1 The exact value of tan is: The period of tan3xo, x є R , is:
3. This diagram is most likely
to be part of the graph of: 1 2 y x 90
180
270
360

5. Set 2 Which of the following has (have) a negative value:
The minimum value of occurs
when x is:
3. Which of the following could be this graph: 1 2 y x 90
180
270
360
answer

6. Set 2 Which of the following has (have) a negative value:
The minimum value of occurs
when x is:
3. Which of the following could be this graph: 1 2 y x 90
180
270
360

7. Set 3 Which of the following is/are solution(s) of sin2x = 1, x є R:
If has a maximum value when θ is:
3. The line with equation y = -1 intersects the curve
y = √2sinx , at : √2 y x 90
180
270
360
-√2
answer

8. Set 3 Which of the following is/are solution(s) of sin2x = 1, x є R:
If has a maximum value when θ is:
3. The line with equation y = -1 intersects the curve
y = √2sinx , at : √2 y x 90
180
270
360
-√2

9. Set 4 The exact value of cos is: The maximum value of
occurs when x = t. What is the value of t?
3. This diagram is most likely
to be part of the graph of: 2 y x
180
360
540 -2
answer

10. Set 4 The exact value of cos is: The maximum value of
occurs when x = t. What is the value of t?
3. This diagram is most likely
to be part of the graph of: 2 y x
180
360
540 -2

11. Set 5 The exact value of sin (-120o) is:
2. If has a minimum value when θ is:
3. The line with equation y = √3 intersects the curve
y = 2cosx , at : 2 y x
180
540
360 -2
answer

12. Set 5 The exact value of sin (-120o) is:
2. If has a minimum value when θ is:
3. The line with equation y = √3 intersects the curve
y = 2cosx , at : 2 y x
180
540
360 -2

13. Set 6 The exact value of cos 135o is:
2. The largest possible domain of, is:
3. This diagram is most likely
to be part of the graph of: 1 -1 y x 90
180
270
360
answer

14. Set 6 The exact value of cos 135o is:
2. The largest possible domain of, is:
3. This diagram is most likely
to be part of the graph of: 1 -1 y x 90
180
270
360

15. Set 7 Which of the following graphs represents y = -f(x + 2): A B C D
(-1,3) 3 y x -3
(5,-2)
Y = f(x)
Which of the following graphs
represents y = -f(x + 2):
A B C D
The exact value of cos is:
3. Functions f and g , are given by f(x) = 3x and
g(x) = x Find an expression for f(g(x)).
(-1,-1) y x
(-3,2)
(5,4)
(3,2)
(-1,5) y x
(-3,2) 5
(3,2)
(1,5) y x
(-3,2) -5
(3,2)
(-3,-3) y x -5 1
(3,2)
answer

16. Set 7 Which of the following graphs represents y = -f(x + 2): A B C D
(-1,3) 3 y x -3
(5,-2)
Y = f(x)
Set 7
Which of the following graphs
represents y = -f(x + 2):
A B C D
The exact value of cos is:
3. Functions f and g , are given by f(x) = 3x and
g(x) = x Find an expression for f(g(x)).
(-1,-1) y x
(-3,2)
(5,4)
(3,2)
(-1,5) y x
(-3,2) 5
(3,2)
(1,5) y x
(-3,2) -5
(3,2)
(-3,-3) y x -5 1
(3,2)

17. Set 8 For which real values of x is the function
defined on the set of real numbers?
The minimum value of
occurs when x is:
3. The line with equation y = 2 intersects the curve
y = 1 - 2sinx , at : 3 y x
180
360 -1
answer

18. Set 8 For which real values of x is the function
defined on the set of real numbers?
The minimum value of
occurs when x is:
3. The line with equation y = 2 intersects the curve
y = 1 - 2sinx , at : 3 y x
180
360 -1

19. Set 9 Which of the following is/are solution(s) of 2sin2x = √3:
Which of these would be the exact value of ?
3. Functions f and g , are given by f(x) = x2 – 2x and
g(x) = -3x. Find an expression for f(g(x)).
answer

20. Set 9 Which of the following is/are solution(s) of 2sin2x = √3:
Which of these would be the exact value of ?
3. Functions f and g , are given by f(x) = x2 – 2x and
g(x) = -3x. Find an expression for f(g(x)).

21. Set 10 Which of the following graphs represents y = -2f(x) + 1:
(-2,3) 1 y x -4
Y = f(x)
Which of the following graphs
represents y = -2f(x) + 1:
A B C D
2. Given that then g-1(x) equals:
3. Functions f and g, are given by and g(x) = x2 - 1.
Find an expression for f(g(x)).
(-3,6) y x -5
(1,1) y x
(-2,-5)
(-4,1)
(3,6) y x 5
(2,7) y x
(-1,1)
(4,1)
answer

22. Set 10 Which of the following graphs represents y = -2f(x) + 1:
(-2,3) 1 y x -4
Y = f(x)
Which of the following graphs
represents y = -2f(x) + 1:
A B C D
2. Given that then g-1(x) equals:
3. Functions f and g, are given by and g(x) = x2 - 1.
Find an expression for f(g(x)).
(-3,6) y x -5
(1,1) y x
(-2,-5)
(-4,1)
(3,6) y x 5
(2,7) y x
(-1,1)
(4,1)

23. Set 11 1. The largest possible domain of, is: 2. The minimum value of
occurs when x = t. What is the value of t?
3. The line with equation y = 1 intersects the curve
y = 4sin2x , at :
answer

24. Set 11 1. The largest possible domain of, is: 2. The minimum value of
occurs when x = t. What is the value of t?
3. The line with equation y = 1 intersects the curve
y = 4sin2x , at :

25. Set 12 Which of the following functions represents the black curve:
(-1,5)
(1,-1) y x
y = g(x)
(-1,-3)
(1,3)
Which of the following functions
represents the black curve:
A. y = g(-x) B. y = -g(x) - 2
C. y = 2 – g(x) D. y = g(x – 2)
2. Given that then h-1(x) equals:
3. Functions f and g, are given by and g(x) = 1 + x.
Find an expression for g(f(x)).
answer

26. Set 12 Which of the following functions represents the black curve:
(-1,5)
(1,-1) y x
y = g(x)
(-1,-3)
(1,3)
Which of the following functions
represents the black curve:
A. y = g(-x) B. y = -g(x) - 2
C. y = 2 – g(x) D. y = g(x – 2)
2. Given that then h-1(x) equals:
3. Functions f and g, are given by and g(x) = 1 + x.
Find an expression for g(f(x)).

27. Set 13 For which real values of x is the function
defined on the set of real numbers?
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:
answer

28. Set 13 For which real values of x is the function
defined on the set of real numbers?
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:

29. Set 14
The gradient of a straight line parallel to the line x + 3y = 0 is:
2. Functions f and g, are given by and
Find an expression for f(g(x)).
3. The line with equation y = 4 intersects the curve
y = 1 - 6sinx , at :
answer

30. Set 14
The gradient of a straight line parallel to the line x + 3y = 0 is:
2. Functions f and g, are given by and
Find an expression for f(g(x)).
3. The line with equation y = 4 intersects the curve
y = 1 - 6sinx , at :

31. Set 15
The line joining the points (-2,-3) and (6, k) has gradient . The value of k is:
2. Which of the following could be this graph:
3. The minimum value of
occurs when x is: y 4 x -2
180
answer

32. Set 15
The line joining the points (-2,-3) and (6, k) has gradient . The value of k is:
2. Which of the following could be this graph:
3. The minimum value of
occurs when x is: y 4 x -2
180

33. Set 16 For which real values of x is the function
defined on the set of real numbers?
Which of the following is the inverse of f(x) = x – 2 , where x є R ?
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

34. Set 16 For which real values of x is the function
defined on the set of real numbers?
Which of the following is the inverse of f(x) = x – 2 , where x є R ?
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

35. Set 17 Which of the following graphs represents y = f(1 - x) : A B C D3
(2,1) y x -2
y = f(x)
Set 17
Which of the following graphs
represents y = f(1 - x) :
A B C D
2. Which of the following is the equation of a line perpendicular to the line x - 3y = 0
A. y = -3x B. y = x C. y = -x D. y = -x
3. Functions f and g, are given by and
Find an expression for f(g(x)).
(-1,3) y x 1
(-3,1)
(-1,3) y x -3
(1,1)
(1,3) y x 3
(-1,1) 2 y x
(-2,1) -2 x
answer

36. Set 17 Which of the following graphs represents y = f(1 - x) : A B C D3
(2,1) y x -2
y = f(x)
Set 17
Which of the following graphs
represents y = f(1 - x) :
A B C D
2. Which of the following is the equation of a line perpendicular to the line x - 3y = 0
A. y = -3x B. y = x C. y = -x D. y = -x
3. Functions f and g, are given by and
Find an expression for f(g(x)).
(-1,3) y x 1
(-3,1)
(-1,3) y x -3
(1,1)
(1,3) y x 3
(-1,1) 2 y x
(-2,1) -2 x

37. Set 18
The line 2y = 3x + 6 meets the y-axis at C. The gradient of the line joining C to A (4,-3) is:
2. Which of these would be the exact value of ?
3. The line with equation y = 1 intersects the curve
y = 3tan2x , at :
answer

38. Set 18
The line 2y = 3x + 6 meets the y-axis at C. The gradient of the line joining C to A (4,-3) is:
2. Which of these would be the exact value of ?
3. The line with equation y = 1 intersects the curve
y = 3tan2x , at :

39. Set 19
The straight lines with equations ay = 3x + 7 and y = 5x + 2 are perpendicular. The value of a is:
2. Which of the following could be this graph:
3. The maximum value of
occurs when x is: 4 y x
720 2
answer

40. Set 19
The straight lines with equations ay = 3x + 7 and y = 5x + 2 are perpendicular. The value of a is:
2. Which of the following could be this graph:
3. The maximum value of
occurs when x is: 4 y x
720 2

41. Set 20 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 B. 3y = -2x + 19
C. 2y = -3x D. 2y = 3x - 13
2. Find the gradient of the line AB:
A. m = 1 B. m = -√2
C. m = -1 D. m = -
3. What is the solution of the equation 2cosx - √3 = 0 where ? y x
45o A B
answer

42. Set 20 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 B. 3y = -2x + 19
C. 2y = -3x D. 2y = 3x - 13
2. Find the gradient of the line AB:
A. m = 1 B. m = -√2
C. m = -1 D. m = -
3. What is the solution of the equation 2cosx - √3 = 0 where ? y x
45o A B

43. Set 21 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 B. y = 4x + 2
C. y = -2x D. y = 2x - 2
3. Functions f and g, are given by and
Find an expression for f(g(x)).
answer

44. Set 21 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 B. y = 4x + 2
C. y = -2x D. y = 2x - 2
3. Functions f and g, are given by and
Find an expression for f(g(x)).

45. Set 22 If f’(4) equals: A.  B. 2 C. 3 D. 6
2. If the line ax - 2y = 0 is parallel to the line
3x + y = 0, a is equal to:
A B. - C.  D. 
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,) C. (0,-) D. (0,-) y x Q R 4
P (1,2) 2
answer

46. Set 22 If f’(4) equals: A.  B. 2 C. 3 D. 6
2. If the line ax - 2y = 0 is parallel to the line
3x + y = 0, a is equal to:
A B. - C.  D. 
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,) C. (0,-) D. (0,-) y x Q R 4
P (1,2) 2

47. Set 23 1. This diagram is most likely to be part of the graph of:
2. Find the gradient of the line ST:
A. m = B. m = 1
C. m = -√2 D. m = -
3. If and x ≠ 0 then f’(x) equals: 1 y x 90 -3 y x
135o S T
answer

49. Set 24 If f(x) = x√x , x > 0 ; f’(x) equals:
Which of the following is/are true of the line with
equation 3x - 2y = 0?
I It passes through the point (-2,-3)
II. It is parallel to the line 6x + 4y = 0
III. It is perpendicular to the line 2x + 3y = 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:
answer

51. Set 25
The gradient of the curve y = 5x3 - 10x at the point (1,-5) is: A B C D. None of these
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 B. 8x C. 2x D. 2x + 1
3. Functions f and g, are given by and
Find an expression for g(f(x)).
answer

53. Set 26 The x-coordinate of the point at which the curve
y = 6 – 3x2 has gradient 12 is:
A B C. -√ 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 B. y = 6x + 9
C. 2y = x D. 2y = 3x - 9
3. The maximum value of
occurs when x is:
answer

55. Question 27
How do you
show that
a curve is
always increasing ?
answer

56. (ii) show that f’(x) is a perfect square
Answer to Question 27
(i) Differentiate
(ii) show that f’(x) is a perfect square

57. Question 28
How do you find the equation of a tangent to a curve at the point when x = a ?
answer

58. (ii) fit a into f’(x) to get the gradient (m)
Answer to Question 28
(i) Differentiate
(ii) 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)

59. For what values of a function is the function said to be undefined ?
Question 29
For what values of a function is the function said to be undefined ?
answer

60. When you fit in a value of x and you cannot get an answer
Answer to Question 29
When you fit in a value of x and you cannot get an answer

61. How do you draw the graph of f(x-1) given the graph of f(x) ?
Question 30
How do you draw the graph of f(x-1) given the graph of f(x) ?
answer

62. Move the graph 1 unit to the right
Answer to Question 30
Move the graph 1 unit to the right

63. How do you find f(g(x)) for given functions f(x) and g(x) ?
Question 31
How do you find f(g(x)) for given functions f(x) and g(x) ?
answer

64. i.e. each x in f(x) is replaced by the function g(x)
Answer to Question 31
Fit g(x) into f(x)
i.e. each x in f(x) is replaced by the function g(x)

65. Question 32
What two things do you require in order to find the equation of a straight line ?
answer

66. The gradient of the line and a point on the line
Answer to Question 32
The gradient of the line and a point on the line x y
(a,b) m 1

67. How do you find the midpoint of a line joining two points ?
Question 33
How do you find the midpoint of a line joining two points ?
answer

68. ( ) Add the coordinates and divide by two x1+ x2 , y1+ y2
Answer to Question 33
Add the coordinates and divide by two
x1+ x2 , y1+ y2
2 2 x y
(x2,y2)
(x1,y1) ( )

69. What is the gradient of a vertical line ?
Question 34
What is the gradient of a vertical line ?
answer

70. Answer to Question 34
undefined x y

71. How do you find the median AM of triangle ABC ?
Question 35
How do you find the median AM of triangle ABC ?
answer

72. Answer to Question 35 (i) find the mid point of BC (M) (ii) find the
gradient of AM
(iii) use y-b = m(x-a)

73. Which two points does the graph y = ax always pass through ?
Question 36
Which two points does the graph
y = ax always pass through ?
answer

74. Answer to Question 36
(0,1) and (1,a)

75. What is the perpendicular bisector of a line ?
Question 37
What is the perpendicular bisector of a line ?
answer

76. A line which cuts the given line in half at 90o
Answer to Question 37
A line which cuts the given line in half at 90o

77. How do you draw the graph of f(x+1) given the graph of f(x) ?
Question 38
How do you draw the graph of f(x+1) given the graph of f(x) ?
answer

78. Move the graph 1 unit to the left
Answer to Question 38
Move the graph 1 unit to the left

79. How do you find the equation of a perpendicular bisector of a line ?
Question 39
How do you find the equation of a perpendicular bisector of a line ?
answer

80. Answer to Question 39 (i) find the midpoint of the line
(ii) find the gradient of the line
(iii) find the gradient perpendicular to the given line
(iv) Use midpoint and gradient in y-b = m(x-a) M
(a,b)

81. For what values is this function undefined ? f(x) = x
Question 40
For what values is this function undefined ?
f(x) = x
(x+2)(x-3)
answer

82. Answer to Question 40
-2 and 3

83. What are the two formulae used to find the area of a triangle ?
Question 41
What are the two formulae used to find the area of a triangle ?
answer

84. A = ½base x height A = ½bcsinA Answer to Question 41 A B C a b c

85. Question 42
What three processes do you go through in order to factorise a quadratic ?
answer

86. (ii) difference of two squares (iii) trinomial
Answer to Question 42
(i) common factor
(ii) difference of two squares
(iii) trinomial

87. What is the equation of a vertical line passing through (a,b) ?
Question 43
What is the equation of a vertical line passing through (a,b) ?
answer

88. Answer to Question 43
x = a x y
(a,b)

89. What is the Theorem of Pythagoras ?
Question 44
What is the Theorem of Pythagoras ?
answer

90. For ΔABC, right-angled at A, a2 = b2 + c2 Answer to Question 44 C a b

91. What do you know about the gradients of two parallel lines?
Question 45
What do you know about the gradients of two parallel lines?
answer

92. Answer to Question 45
They are the same

93. How do you draw the graph of f’(x) given the graph of f(x) ?
Question 46
How do you draw the graph of f’(x) given the graph of f(x) ?
answer

94. 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

95. How do you get the gradient of a line with an equation like
Question 47
How do you get the gradient of a line with an equation like
3x + 2y = 5 ?
answer

96. (i) Rearrange into the form y = mx + c (ii) read off gradient = m
Answer to Question 47
(i) Rearrange into the form
y = mx + c
(ii) read off
gradient = m

97. Question 48
What is loga1
equal to ?
answer

98. Answer to Question 48

99. How do you find the length of a line joining two points ?
Question 49
How do you find the length of a line joining two points ?
answer

100. Answer to Question 49
√(x2 – x1)2 + (y2 –y1)2
A(x1,y1)
B(x2,y2) x y

101. What is the Converse of Pythagoras ?
Question 50
What is the Converse of Pythagoras ?
answer

102. If a2 = b2 + c2 then ΔABC is right-angled at A Answer to Question 50 C

103. How do you find the gradient of a line joining two points ?
Question 51
How do you find the gradient of a line joining two points ?
answer

104. Answer to Question 51
m = y2 – y1
x2 – x1
A(x1,y1)
B(x2,y2) x y

105. How do you find the altitude AN of ΔABC ?
Question 52
How do you find the altitude AN of ΔABC ?
answer

106. (i) find the gradient of BC (ii) find the gradient of AN,
Answer to Question 52
(i) find the gradient of BC
(ii) find the gradient of AN,
perpendicular
to BC
(iii) use y-b=m(x-a) A N B C

107. For a curve, how do you find the stationary points and their nature ?
Question 53
For a curve, how do you find the stationary points and their nature ?
answer

108. (iii) solve to find stationary points (iv) find y-coordinates
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

109. How do you draw the graph of 3+f(x) given the graph of f(x) ?
Question 54
How do you draw the graph of 3+f(x) given the graph of f(x) ?
answer

110. Answer to Question 54
move graph up 3

111. How do you find where a curve is increasing ?
Question 55
How do you find where a curve is increasing ?
answer

112. 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

113. How do you find where two lines intersect ?
Question 56
How do you find where two lines intersect ?
answer

114. Simultaneous equations
Answer to Question 56
Simultaneous equations

115. How do you draw the graph of 3-f(x) given the graph of f(x) ?
Question 57
How do you draw the graph of 3-f(x) given the graph of f(x) ?
answer

116. Reflect the graph in the x-axis, then move it up 3
Answer to Question 57
Reflect the graph in the x-axis,
then move it up 3

117. How do you draw the graph of f(-x) given the graph of f(x) ?
Question 58
How do you draw the graph of f(-x) given the graph of f(x) ?
answer

118. Reflect the graph in the y-axis
Answer to Question 58
Reflect the graph in the y-axis

119. How do you solve equations like 100 = 0 x2 ? 4 -
Question 59
How do you solve equations like
100 = 0 x2 ?
4 -
answer

120. (i) multiply by the denominator of the fraction (here x2)
Answer to Question 59
(i) multiply by the denominator of the fraction (here x2)
(ii) factorise and solve

121. How do you find the exact values of sin(A+B), cos(A-B) etc. given that
Question 60
How do you find the exact values of
sin(A+B), cos(A-B) etc.
given that
cosA = 3/5 and
sinB = 12/13 ?
answer

122. Answer to Question 60 (i) draw two Δs (ii) find missing sides3 5
(i) draw
two Δs
(ii) find
missing sides
(iii) expand
formula
(iv) fit in values
from Δs B 12 13

123. How do you solve equations like Cos2xo - 5cosxo = 2 ? (0 ≤ x ≤ 360)
Question 61
How do you solve equations like
Cos2xo - 5cosxo = 2 ?
(0 ≤ x ≤ 360)
answer

124. (i) fit in 2cos2xo-1 for cos2xo (ii) factorise
Answer to Question 61
(i) fit in 2cos2xo-1 for cos2xo
(ii) factorise
(iii) solve the equation

125. Question 62
What is
sin x
cos x
equal to ?
answer

126. Answer to Question 62
tan x

127. How do you show that x-1 is a factor of the function f(x)=x3-3x+2 ?
Question 63
How do you show that x-1 is a factor of the function f(x)=x3-3x+2 ?
answer

128. (i) rewrite the function as f(x)=x3+0x2-3x+2
Answer to Question 63
(i) rewrite the function as
f(x)=x3+0x2-3x+2
(ii) use synthetic division with 1 on the outside
(iii) show that
remainder = 0

129. Question 64
What is the
sine rule ?
answer

130. Answer to Question 64
a b c
sinA sinb sinC = = A B C a b c

131. Given f’(x) and a point on the curve, how do you find
Question 65
Given f’(x) and a point on the curve, how do you find
f(x) ?
answer

132. (ii) fit in given point to work out value of C
Answer to Question 65
(i) integrate
(ii) fit in given point to work out value
of C

133. How do you solve quadratic inequations like
Question 66
How do you solve quadratic inequations like
x2 - 5x + 6 ≤ 0 ?
answer

134. (iii) read values below x-axis
Answer to Question 66
(i) factorise
(ii) draw graph
(iii) read values below x-axis

135. How do you change from radians to degrees ?
Question 67
How do you change from radians to degrees ?
answer

136. Divide by π and multiply by 180
Answer to Question 67
Divide by π and multiply by 180

137. What is the condition for real roots ?
Question 68
What is the condition for real roots ?
answer

138. Answer to Question 68
b2 – 4ac ≥ 0

139. 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

140. (i) do synthetic division (ii) let the expression = 0 or the remainder
Answer to Question 69
(i) do synthetic division
(ii) let the expression = 0 or the remainder
(iii) solve the equation

141. the curve passes through the point (1,9) ?
Question 70
How do you find f(x) if
f’(x) = 5-3x2 and
the curve passes through the point (1,9) ?
answer

142. (ii) find C by replacing point (1,9) into f(x)
Answer to Question 70
(i) f(x) = ∫f'(x) dx
(ii) find C by replacing point (1,9) into f(x)
(iii) write down completed formula for f(x)

143. Question 71
What is
sin2x + cos2x
equal to ?
answer

144. Answer to Question 71 1

145. Question 72
How do you find the equation of the tangent to a circle at a particular point on the circumference ?
answer

146. Answer to Question 72 (i) find the centre (ii) find gradientx y
(a,b) C
(i) find the
centre
(ii) find gradient
from centre
to point
(iii) find perpendicular gradient
(iv) use y-b=m(x-a)

147. Question 73
How do you find
x2 + 1 √x ∫
dx ?
answer

148. (i) change root to power
Answer to Question 73
(i) change root to power
(ii) split up into fractions
(iii) simplify each term
(iv) integrate each term
(v) REMEMBER +C

149. Question 74
How do you show that the root of a function lies between two given values ?
answer

150. fit in two values and show one is positive and one is negative
Answer to Question 74
fit in two values and show one is positive and one is negative x
+ve
-ve

151. How do you find exact values of sin2x and cos2x given cosx =3/5 ?
Question 75
How do you find exact values of sin2x and cos2x given cosx =3/5 ?
answer

152. 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 A 3 5

153. What is the turning point of y=2(x-a)2+b ? Max or min ?
Question 76
What is the turning point of
y=2(x-a)2+b ?
Max or min ?
answer

154. Answer to Question 76
(i) (a,b)
minimum
(a,b)

155. Question 77
How do you integrate xn ?
answer

156. Answer to Question 77
xn+1
n+1
+ C

157. How do you solve equations like cos2xo-5sinxo = 0 ? (0≤x≤360)
Question 78
How do you solve equations like
cos2xo-5sinxo = 0 ?
(0≤x≤360)
answer

158. (ii) factorise (iii) solve equation (i) fit in 1-2sin2xo for cos2xo
Answer to Question 78
(i) fit in 1-2sin2xo for cos2xo
(ii) factorise
(iii) solve equation

159. How do you complete the square for functions like
Question 79
How do you complete the square for functions like
2x2 + 12x + 3 ?
answer

160. (ii) compare with given function (iii) find a, p and q
Answer to Question 79
(i) multiply out a(x+p)2+q
(ii) compare with given function
(iii) find a, p and q

161. How do you solve equations of the form sin2xo = 0.5 ? (0≤x≤360)
Question 80
How do you solve equations of the form
sin2xo = 0.5 ?
(0≤x≤360)
answer

162. (iv) divide each by 2 Answer to Question 80
(i) decide on the quadrants (sin is +ve)
(ii) press INV sin to get angle
(iii) work out your 2 angles
(iv) divide each by 2

163. How do you solve quadratic inequations like
Question 81
How do you solve quadratic inequations like
x2+5x-6 ≥ 0 ?
answer

164. (iii) read values above x-axis
Answer to Question 81
(i) factorise
(ii) draw graph
(iii) read values above x-axis

165. What is the centre and radius of a circle with equation x2 + y2 = r2 ?
Question 82
What is the centre and radius of a circle with equation x2 + y2 = r2 ?
answer

166. Answer to Question 82
(i) centre (0,0)
(ii) radius = r

167. How do you calculate the area under a curve ?
Question 83
How do you calculate the area under a curve ?
answer

168. (ii) fit in two limits and subtract to find area
Answer to Question 83
(i) integrate
(ii) fit in two limits and subtract to find area

169. Question 84
How do you find the root of an equation between two given values to 1 dp ?
answer

170. Answer to Question 84
iteration

171. How do you solve equations of the form sin2xo = 0.5 ? (0≤x≤360)
Question 85
How do you solve equations of the form
sin2xo = 0.5 ?
(0≤x≤360)
answer

172. (i) rearrange to get sinxo = ± … (ii) find answers in all 4 quadrants
Answer to Question 85
(i) rearrange to get sinxo = ± …
(ii) find answers in all 4 quadrants

173. How do you name the angle between a line and a plane ?
Question 86
How do you name the angle between a line and a plane ?
answer

174. Answer to Question 86 (i) start at end of line (A)
(ii) go to where line meets
the plane (B)
(iii) go to the point
on the plane
directly under
the start of the line (C)
ABC A B C

175. What is the condition for equal roots ?
Question 87
What is the condition for equal roots ?
answer

176. Answer to Question 87
b2 – 4ac = 0

177. What is the turning point of y = b-3(x-a)2 ? max or min ?
Question 88
What is the turning point of
y = b-3(x-a)2 ?
max or min ?
answer

178. Answer to Question 88
(a,b)
Maximum
(a,b)

179. What is the quadratic formula and explain when it is used ?
Question 89
What is the quadratic formula and explain when it is used ?
answer

180. Answer to Question 89
x = -b±√(b2-4ac) 2a
It is used to find roots of a quadratic equation when it is difficult to factorise.

181. How do you prove that a line is a tangent to a circle ?
Question 90
How do you prove that a line is a tangent to a circle ?
answer

182. Prove it has equal roots using b2-4ac = 0 or repeated roots
Answer to Question 90
Rearrange line to make
y = or x =
Fit line into circle
Prove it has equal roots using b2-4ac = 0 or repeated roots

183. How do you find the exact value of sin (α-β), given that sinα =4/5
Question 91
How do you find the
exact value of
sin (α-β),
given that sinα =4/5
and cosβ = 12/13 ?
answer

184. 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 4 5 12 13 β

185. How do you solve equations of the form cosxo = - 0.8 ? (0≤x≤360)
Question 92
How do you solve equations of the form
cosxo = ?
(0≤x≤360)
answer

186. (ii) ignore the sign and press INV cos to get angle
Answer to Question 92
(i) decide on the quadrants (cos is -ve)
(ii) ignore the sign and press INV cos to get angle
(iii) work out your 2 angles

187. How do you change from degrees to radians ?
Question 93
How do you change from degrees to radians ?
answer

188. Divide by 180 and multiply by π
Answer to Question 93
Divide by 180 and multiply by π

189. How do you find the exact values of sin x or tan x given
Question 94
How do you find the exact values of sin x or tan x given
cos x = a ? b
answer

190. (ii) use Pythagoras to fill in missing side
Answer to Question 94
(i) draw triangle
(ii) use Pythagoras to fill in missing side
(iii) read values off triangle using SOHCAHTOA a b x

191. How do you factorise a cubic expression like x3-2x2-x+2 ?
Question 95
How do you factorise a cubic expression like
x3-2x2-x+2 ?
answer

192. Synthetic division using factors of last number
Answer to Question 95
Synthetic division using factors of last number
Remainder=0
factor 1 -2 -1 2

193. What is the centre and radius of a circle of the form
Question 96
What is the centre and radius of a circle of the form
x2+y2+2gx+2fy+c=0 ?
answer

194. Answer to Question 96
Centre (-g,-f)
Radius √(g2+f2-c)

195. How do you remember the exact values of 30o, 45o and 60o ?
Question 97
How do you remember the exact values of 30o, 45o and 60o ?
answer

196. Complete using Pythagoras Do similar for tan 45o =1
Answer to Question 97
sin30o = ½
Draw right-angled
triangle
Complete using Pythagoras
Do similar
for tan 45o =1
30o
60o 1 2 √3
45o 1 √2

197. How do you calculate the area between two curves ?
Question 98
How do you calculate the area between two curves ?
answer

198. (i) let equations equal each other (ii) solve to find limits
Answer to Question 98
(i) let equations equal each other
(ii) solve to find limits
(iii) integrate
(upper - lower) functions between limits

199. How do you solve an equation like 3sinx+1 = 0 ?
Question 99
How do you solve an equation like
3sinx+1 = 0 ?
answer

200. (ii) decide on 2 quadrants
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

201. What is the condition for no real roots ?
Question 100
What is the condition for no real roots ?
answer

202. Answer to Question 100
b2 – 4ac < 0

203. Question 101
How do you find
∫ x3 dx ? b a
answer

204. Answer to Question 101
x3+1
3+1
then 1/4[(b4) - (a4)] [ ] b a

205. How do you find where a line and a circle intersect ?
Question 102
How do you find where a line and a circle intersect ?
answer

206. Fit into circle and solve
Answer to Question 102
Rearrange line to get
x = … or y = …
Fit into circle and solve

207. State the cosine rule to find an angle
Question 103
State the cosine rule to find an angle
answer

208. Answer to Question 103
cos A = b2 + c2 - a2
2bc A B C a b c

209. What is the centre and radius of a circle of the form
Question 104
What is the centre and radius of a circle of the form
(x-a)2+(y-b)2 = r2 ?
answer

210. Answer to Question 104
Centre (a,b)
Radius = r x y
(a,b) C r

211. State the cosine rule to find a missing side
Question 105
State the cosine rule to find a missing side
answer

212. Answer to Question 105
a2 = b2+c2-2bccosA A B C a b c

213. Question 106
How do you find
∫ (ax + b)n dx ?
answer

214. Answer to Question 106 (i) increase power by 1
(ii) divide by new power
(iii) divide by the derivative of
the bracket
i.e. (ax+b)n+1
a(n+1)
+ C

215. the coordinates of a point which divides a line in a ratio e.g. 3:2 ?
Question 107
How do you find
the coordinates of a point which divides a line in a ratio e.g. 3:2 ?
answer

216. (iv) solve to find missing vector (v) rewrite as point (*,*)
Answer to Question 107 A B C 3 2
(i) write in form AB = 3
BC 2
(ii) cross-multiply
(iii) write AB = (b-a)
(iv) solve to find missing vector
(v) rewrite as point (*,*)

217. Question 108
What is a
position vector ?
answer

218. A vector which starts at the origin
Answer to Question 108
A vector which starts at the origin

219. How do you express acosx+bsinx+c in the form kcos(x-α) etc?
Question 109
How do you express acosx+bsinx+c
in the form
kcos(x-α) etc?
answer

220. S A T C Answer to Question 109
(i) expand brackets and equate like terms
(ii) find k =√(a2+b2)
(iii) identify quadrant α is in
(iv) find α , tanα = sinα
cosα S A T C

221. How do you differentiate a bracket without multiplying it out ?
Question 110
How do you differentiate a bracket without multiplying it out ?
answer

222. (i) multiply by old power (ii) decrease power by 1
Answer to Question 110
(i) multiply by old power
(ii) decrease power by 1
(iii) multiply by derivative of bracket

223. Question 111
What is
Logax – logay
equal to ?
answer

224. Answer to Question 111 x
loga y

225. What do you get when you differentiate cosx ?
Question 112
What do you get when you differentiate cosx ?
answer

226. Answer to Question 112
-sinx

227. How do you show that two vectors are perpendicular ?
Question 113
How do you show that two vectors are perpendicular ?
answer

228. Answer to Question 113
Show that a.b=0 a b

229. How do you integrate sin ax ?
Question 114
How do you integrate sin ax ?
answer

230. Answer to Question 114
-1/a cos ax + C

231. How do you draw a graph of the form y = acosx or y = asinx ?
Question 115
How do you draw a graph of the form
y = acosx
or y = asinx ?
answer

232. with a maximum of a and a minimum of -a
Answer to Question 115
Draw y = cosx
or y = sinx graph
with a maximum of a and a minimum of -a

233. How do you find the maximum or minimum values of
Question 116
How do you find the maximum or minimum values of
acosx + bsinx + c ?
answer

234. (i) change acosx+bsinx into Rcos(x-a) (ii) max is R+c
Answer to Question 116
(i) change acosx+bsinx into Rcos(x-a)
(ii) max is R+c

235. How do you find a unit vector parallel to a given vector ?
Question 117
How do you find a unit vector parallel to a given vector ?
answer

236. (i) find the length of the given vector
Answer to Question 117
(i) find the length of the given vector
(ii) divide all the components by this length

237. How do you integrate cos ax ?
Question 118
How do you integrate cos ax ?
answer

238. Answer to Question 118
1/a sin ax + C

239. How do you draw a graph of the form y = cos(x+a) or y = sin(x+a) ?
Question 119
How do you draw a graph of the form
y = cos(x+a)
or y = sin(x+a) ?
answer

240. Move the graph of y=cosx or y=sinx a units to the LEFT
Answer to Question 119
Move the graph of y=cosx or y=sinx
a units to the LEFT

241. Question 120
What is a unit vector ?
answer

242. Answer to Question 120
A vector of length 1 unit

243. How do you draw a graph of the form y = cos bx or y = sin bx ?
Question 121
How do you draw a graph of the form
y = cos bx
or y = sin bx ?
answer

244. Draw the normal graph but fit in b waves between 0o and 360o
Answer to Question 121
Draw the normal graph but fit in b waves between 0o and 360o

245. Question 122
What is
loga x + loga y equal to ?
answer

246. Answer to Question 122
Loga xy

247. What do you get when you differentiate sin x ?
Question 123
What do you get when you differentiate sin x ?
answer

248. Answer to Question 123
cos x

249. How do you find the angle between two vectors ?
Question 124
How do you find the angle between two vectors ?
answer

250. Answer to Question 124
a.b
a b
cosq = a b q

251. Question 125
Given an equation like m = moe-3k and an amount by which it has been decayed, how do you find k ?
answer

252. (ii) rearrange to get e-3k = (iii) take logs (iv) solve
Answer to Question 125
(i) fit in m and mo
(ii) rearrange to get e-3k =
(iii) take logs
(iv) solve

253. then what is u in component form ?
Question 126
If u = ai+bj+ck
then what is u in component form ?
answer

254. Answer to Question 126 a b c
U =

255. What do you get when you differentiate
Question 127
What do you get when you differentiate
cosax ?
answer

256. Answer to Question 127
-asinax

257. How do you solve an equation of the form acosx + bsinx + c=0 ?
Question 128
How do you solve an equation of the form acosx + bsinx + c=0 ?
answer

258. Change acosx+bsinx into Rcos(x- a) Rearrange and solve
Answer to Question 128
Change acosx+bsinx into Rcos(x- a)
Rearrange and solve

259. Question 129
What is loga xn equal to ?
answer

260. Answer to Question 129
nloga x

261. How would you differentiate a function like
Question 130
How would you differentiate a function like
y = sin3 x ?
answer

262. Answer to Question 130
(i) write as (sin x)3
(ii) multiply by the power
(iii) decrease power by one
(iv) multiply by the derivative of the bracket
i.e. 3cosx sin2x

263. State the three rules of logs ?
Question 131
State the three rules of logs ?
answer

264. (i) logaxy = logax + logay (ii) loga = logax – logay
Answer to Question 131
(i) logaxy = logax + logay
(ii) loga = logax – logay
(iii) logaxn = nlogax x y

265. How do you solve equations of the form
Question 132
How do you solve equations of the form
3x = ?
answer

266. (i) take logs of both sides (ii) bring x down to front
Answer to Question 132
(i) take logs of both sides
(ii) bring x down to front
(iii) solve the equation

267. Question 133
Given experimental data, how do you find an equation in the form y=abx or y=axb ?
answer

268. (i) take logs of both sides
Answer to Question 133
(i) take logs of both sides
(ii) rearrange to get a straight line equation
(iii) determine type
(iv) find solution

269. How would you differentiate a function like
Question 134
How would you differentiate a function like
y = sin ax ?
answer

270. Answer to Question 134
dy/dx = acos ax

271. Question 135
If u =
then what is u ? a b c
answer

272. Answer to Question 135
work out length
√(a2+b2+c2)

273. How do you add or subtract vectors ?
Question 136
How do you add or subtract vectors ?
answer

274. add or subtract matching components
Answer to Question 136
add or subtract matching components

275. Question 137
What does
a.a equal ?
answer

276. Answer to Question 137 a2

277. How do you prove that three 3-D points are
Question 138
How do you prove that three 3-D points are
collinear ?
answer

278. Answer to Question 138
Prove they are the same vector multiplied by different or the same numbers

279. Question 139
Express the equation y=kxn in the form of the equation of a straight line, Y=nX+c.
answer

280. Answer to Question 139
logy = nlogx + logk

281. Question 140
Who loves maths ?
answer

282. Answer to Question 140
ME !!!!!