Higher Mathematics Objective Questions

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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140

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

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

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

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

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

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

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

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

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

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

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

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

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) = 3x2 + 1 and g(x) = x2 - 4. 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

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) = 3x2 + 1 and g(x) = x2 - 4. 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)

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

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

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

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)).

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

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)

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

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 :

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) + 2 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

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) + 2 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)).

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

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:

Set 14 The gradient of a straight line parallel to the line x + 3y + 7 = 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

Set 14 The gradient of a straight line parallel to the line x + 3y + 7 = 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 :

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

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

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

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

Set 17 Which of the following graphs represents y = f(1 - x) : A B C D 3 (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 + 4 = 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

Set 17 Which of the following graphs represents y = f(1 - x) : A B C D 3 (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 + 4 = 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

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

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 :

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

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

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 + 13 B. 3y = -2x + 19 C. 2y = -3x - 19 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

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 + 13 B. 3y = -2x + 19 C. 2y = -3x - 19 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

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 + 2 B. y = 4x + 2 C. y = -2x + 23 D. y = 2x - 2 3. Functions f and g, are given by and Find an expression for f(g(x)). answer

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 + 2 B. y = 4x + 2 C. y = -2x + 23 D. y = 2x - 2 3. Functions f and g, are given by and Find an expression for f(g(x)).

Set 22 If f’(4) equals: A.  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.  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

Set 22 If f’(4) equals: A.  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.  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

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 = -1 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

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 + 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: answer

Set 25 The gradient of the curve y = 5x3 - 10x at the point (1,-5) is: A. -5 B. 5 C. 15 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 + 1 B. 8x - 3 C. 2x + 3 D. 2x + 1 3. Functions f and g, are given by and Find an expression for g(f(x)). answer

Set 26 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 3. The maximum value of occurs when x is: answer

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

(ii) show that f’(x) is a perfect square 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

(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)

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

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

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

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

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

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)

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

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

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

( ) 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) ( )

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

Answer to Question 34 undefined x y

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

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)

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

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

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

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

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

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

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

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)

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

Answer to Question 40 -2 and 3

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

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

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

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

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

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

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

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

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

Answer to Question 45 They are the same

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

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

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

(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

Question 48 What is loga1 equal to ? answer

Answer to Question 48

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

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

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

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

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

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

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

(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

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

(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

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

Answer to Question 54 move graph up 3

How do you find where a curve is increasing ? 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

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

Simultaneous equations Answer to Question 56 Simultaneous equations

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

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

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

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

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

(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

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

Answer to Question 60 (i) draw two Δs (ii) find missing sides 3 5 (i) draw two Δs (ii) find missing sides (iii) expand formula (iv) fit in values from Δs B 12 13

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

(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

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

Answer to Question 62 tan x

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

(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

Question 64 What is the sine rule ? answer

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

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

(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

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

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

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

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

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

Answer to Question 68 b2 – 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

(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

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

(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)

Question 71 What is sin2x + cos2x equal to ? answer

Answer to Question 71 1

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 (i) find the centre (ii) find gradient x 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)

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

(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

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

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

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

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

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

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

Question 77 How do you integrate xn ? answer

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

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

(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

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

(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

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

(iv) divide each by 2 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

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

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

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

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

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

(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

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

Answer to Question 84 iteration

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

(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

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

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

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

Answer to Question 87 b2 – 4ac = 0

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

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

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

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.

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

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

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

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

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 = - 0.8 ? (0≤x≤360) answer

(ii) ignore the sign and press INV cos to get angle 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

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

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

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

(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

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

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

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

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

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

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

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

(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

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

(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

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

Answer to Question 100 b2 – 4ac < 0

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

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

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

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

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

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

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

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

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

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

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

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

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

(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 (*,*)

Question 108 What is a position vector ? answer

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

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

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

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

(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

Question 111 What is Logax – logay equal to ? answer

Answer to Question 111 x loga y

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

Answer to Question 112 -sinx

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

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

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

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

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

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

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

(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

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

(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

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

Answer to Question 118 1/a sin ax + C

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

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

Question 120 What is a unit vector ? answer

Answer to Question 120 A vector of length 1 unit

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

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

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

Answer to Question 122 Loga xy

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

Answer to Question 123 cos x

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

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

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

(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

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

Answer to Question 126 a b c U =

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

Answer to Question 127 -asinax

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

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

Question 129 What is loga xn equal to ? answer

Answer to Question 129 nloga x

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

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

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

(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

How do you solve equations of the form Question 132 How do you solve equations of the form 3x = 0.155 ? answer

(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

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

(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

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

Answer to Question 134 dy/dx = acos ax

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

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

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

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

Question 137 What does a.a equal ? answer

Answer to Question 137 a2

How do you prove that three 3-D points are 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 equation y=kxn 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 !!!!!