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## Purple questions are about PROBABILITY AND STATISTICS

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### Question 1. Calculate the distance between the points(-,3,4) and (8,-7)

Draw a diagram and use Pythagoras' Theorem or use a formula.

242=15.6(to 3 s.f.)

### Question 2. Find x if sin x = 0.5

Calculator might help.

30°

### Question 3. Expand 3(x - 4)2

Expand the brackets

3x2 − 24x + 48

### Question 4. What is the equation of the normal to the curve y = x2 when x = 2?

Differentiate and find the reciporacal of gradient and change sign.

4y + x − 18 = 0

### Question 5. Find the equation of a line parallel to the line y = x and passing through the point (3, 2)?

What is the gradient of the line y = x?

y = x - 1

38.7°

if y = xn
y1 = nxn-1

12x2

### Question 8. What is ∫ 4x dx

Add ONE to the exponent and divide by the new exponent and then add c

2x2 + c

36.9°

### Question 10. Simplify 3x2y .(2xy2)3

Expand the bracket first.

24x5y7

### Question 11. If sin x = 5/6, what is the size of angle x in degrees?

Change 5/6 into a decimal.

x = 56.4°

### Question 12. What is the y-intercept of the function f(x) = 2x2 + 3x + 5?

Put x = 0 into the function.
i.e Find f(0)

(0,5)

### Question 13. A right-angled triangle has two short sides of 4.5 cm and 5.5 cm. How long is the hypotenuse (to 2 sig. fig.)?

Use Pythagoras' Theorem

7.1cm

P(6)xP(6)

1/36

### Question 15. What type of graph is produced by the equation (x - 1)2 + y2 = 16?

Draw a sketch by putting x = 0 to find the y-intercept and then y = 0 to find the x-intercept.

A circle centre (1,0) radius 4 the x-intercept.

### Question 16. What type of graph is produced by the function y(x − 3) = 6

If desperate, plot a few points

It's a rectangle hyperbola

### Question 17. Differentiate f(x) = 3x(2x − 6)?

Expand the brackets first

f'(x)=12x − 18

### Question 18. What is the value of sin2 x + cos2 x?

This expression is called a trigonomical identity

1

### Question 19. If f(x) = 1/x find f'(x)?

Write 1/x as an index x-1

-1/x2

### Question 20. Given the lengths of all three sides of a non-right-angled triangle, which mathematical formula would you use to find one of the angles?

It's either the sine rule or the cosine rule!

The cosine rule is needed.

### Question 21. Solve the equation 2sin x = 1 for 0° < x < 90°

Divide both sides by 2

x = 30°

### Question 22. Expand 8(2x − 3)2

Square the brackets first

32x2 − 96x + 72

### Question 23. Expand (x − 1)3

Expand the two brackets first, then multiply by the third bracket

x3 − 3x2 + 3x − 1

### Question 24. If f(x) = 2x3 - 3x2 + 4x − 2 find f'(x)

Differentiate term by term

f'(x) = 6x2 − 6x + 4

### Question 25. Simplify 4(2a2b)3

Cube the brackets first

32a6b3

### Question 26. What is the solution for the equation 2x2 + 6 = 8x

Take all terms to one side of the equation and then factorise

x = 1 or 3

### Question 27. A line has a gradient of 3 and passes through the point (2, 1). Find its equation in the form ax + by + c = 0

Use the equation yy1 = m(xx1)

3xy − 5 = 0

### Question 28. Find ∫ 4x + 3 dx

Don't forget the constant!

2x2+ 3x + c

### Question 29. What is the area of a sector of a circle of radius 5 cm if the angle at the centre of the sector is 60°?

Area = 13.1 units2

### Question 30. What is the probability that when two marbles are drawn from a bag containing 6 red marbles and 5 blue marbles, both of the marbles are blue? (The marbles are drawn together.)

What is the probability that the first ball is blue AND the second ball is blue.

2/11

### Question 31. What is the upper quartile of 12, 3, 30, 7, 9, 2, 15, 19, 20, 6?

Arrange in order and then split the data into two equal parts and then split the top part of the data into havles again.

19

Use the formula
A = 0.5 r20

64cm2

### Question 33. If x = 4, what is the value of 2x7

Remember that exponents come before multiplying by 2

32768

Brackets first!

576

### Question 35. Complete the following cosine rule calculation to find xx2 = 342 + 282 − 2 x 34 x 28 cos 36°

Do exponents and multiplication before adding and subtraction.

x = 20(to 2 sig. fig.)

### Question 36. Differentiate 3x4 − 2x + 3

Differentiate term by term
Remember f(x) = x
then f'(x) = 0

12x3 −2

### Question 37. Find the length of an arc of a circle of radius 12 cm if the angle made by the arc at the centre of the circle is 60°

Use the formula 1 =r θ

Lengths of arc = 12.6(to 3 sig.fig.)

### Question 38. If the mean of a set of 10 scores is 34 and the mean of another set of 20 scores is 42, what is the mean of the combined set of 30 scores?

Work out the total number of marks scored in both tests and divide by 30

New mean = 39.3 (to 3 s. f.)

### Question 39. Solve the equation 3x2 + 4 = 22

Subtract 4 from both sides and then divide both sides by 3 and take the square root.

x = ± 2.4 (to 1 d.p.)

### Question 40. Differentiate 5/x2

Write the term in index form and then multiply by the power and lower the power by 1.

-10x−3 = -10/x3.

### Question 41. If sin x = 0.345, what is x in radians?

Change mode of calculator to RAD

x = 0.35 radians (to 2 sig. fig.)

### Question 42. A six-sided dice is thrown. What is the probability that a number greater than a 4 comes up?

How many numbers on a six sided die are greater than 4?

Probability = 1/3

x = 5 or −1

### Question 44. If cos 2x = 0.5 find x if 0° < x < 180°

Find the solutions between 0° and 360° and then divide by 2.

x = 30° or 150°

### Question 45. What is the value of x if sin2x + cos2 45° = 1? (where 0° < x < 90° )

This is an IDENTITY!

x = 45°

### Question 46. Find the minimum value of the function f(x) = 2x2 − 3x

Find f '(x) and then put f '(x)= 0
This value of x will then provide the minimum value of the function.

Minimum value of the function is −1.125

### Question 47. What is the period of the function y = 2sin x

How often does the function repeat itself?

Period is 360° or 2Π radians

### Question 48. Factorise ab − ac − 3b + 3c

Use the grouping technique to find a common factor.

(a − 3)(bc)

### Question 49. What is the amplitude of the function y = 3sin 2x?

The amplitude is the height of the graph above or below the x-axis.

Amplitude is 3

### Question 50. Find ∫ (4x - 2)2 dx

Expand the brackets and then integrate term by term and don't forget the "c"!

16x3/3 − 8x2 + 4x + c

### Question 51. What is the y-intercept of the function y = 2sin x - 2?

Does the -2 make the graph of y = 2sin x move up or down?

The y-intercept is -2

### Question 52. Solve sin x = cos xfor 0° < x < 180°

Think of where the graphs of y = sin x and y = cos x meet.

x = 45°

### Question 53. Find the area enclosed by the curve y = x2, the x-axis and the line x = 4

Sketch the graph and then use integration

64/3 units2

### Question 54. What is the interquartile range of 30, 14, 6, 12, 9, 4, 3?

The interquartile range is the difference between the upper quartile and the lower quartile

Interquartile range = 10

### Question 55. What is the y-intercept of the function y = 3e2x

This is an exponential graph.
Put x = 0 to find the y-intercept.

y-intercept is 3

### Question 56. What is the range of the function y = sin x?

The range is the set of y-values for the function.

-1 ≤ y ≤ 1

### Question 57. Simplify 2(x − 3)2 - 3(x + 1)2

Expand brackets first , then beware of negative signs.

x2 − 18x + 15

### Question 58. What is the value of sin (Π/4) in surd form?

Use the "special " triangles

1 / √2

### Question 59. What is the midpoint of the line joining the point A = (2, 5) to the point B = (−2, −5)?

Use the mid-point formula!

(0, 0)

### Question 60. If a = 6, b = 2 and c = 3 evaluate a2bc

Work out a2 first

216

### Question 61. Write cos 30° in surd form

Use the "special" triangles.

√3/ 2

### Question 62. If y = 2x(x − 6) find y '

Expand the bracket and then differentiate!

y ' = 4x - 12

### Question 63. What is the period of the function y = sin 3x?

The period of the graph is the length of the interval over which it repeats itself.

360° / 3 = 120° or 2Π/ 3

### Question 64. Find ∫ (x − 3)(3x + 2) dx?

Expand the brackets first.

x3 − 3.5x2 − 6x + c

### Question 65. Solve the equation 3sin x = 2 for 0° < x < 90°

Divide both sides by 3 first.

x = 41.8°

### Question 66. What are the coordinates of the centre of the circle (x − 3)2 + (y + 1)2 = 9?

For x2 + y2 = 9 the centre would be (0, 0)

Minimum value of the function is −1.125

### Question 67. What is the mean of the numbers 3, 5, 10, 12, 20?

The mean is the SUM of all of the values divided by the NUMBER of values

mean = 50/5 = 10

### Question 68. What is the length of the diagonal of a rectangle 4 cm by 5 cm?

Use Pythagoras' Theorem

The diagonal is 6.4 cm long (to 2 sig.fig.).

### Question 69. What is the probability that two balls chosen at random (without replacement) from a bag of 12 balls will be blue, if the bag contains 8 blue balls and 4 red balls?

Find each probability, (remember WITHOUT replacement) and then MULTIPLY them together.

14 / 33

### Question 70. A right-angled triangle has two sides 6 cm in length. What is the length of the other side?

Draw a diagram and split the triangle into two equal parts.

The other side is 8.49 (to 3 s.f.) cm long.

### Question 71. Differentiate 4 / x3

Write 4 / x3 as 4x-3

The y-intercept is -2

Use y = mx + c

y = 2x + 3

### Question 73. What is the maximum value of the function y = 2sin x?

Think of the shape of the graph.
What effect does the 2 have on the function?

Maximum value is 2

### Question 74. What is the equation of the linear function passing through the point (2, 3) with a gradient of 3? Give your answer in the form y = mx + c

Use the general equation of a straight line: y - y1 = m(x - x1) and then re-arrange.

y = 3x − 3

### Question 75. If f(x) = (x − 2)2 find f '(x)?

Expand the brackets first

f '(x) = 2x − 4

### Question 76. If g(x) = √ x what is g '(x)?

Write g(x) in index form.

g ' (x) = 1 / (2√x)

### Question 77. Is the sign of cos 1.2 negative or positive? (The angle is in radians)

Check it out on your calculator.

Positive

### Question 78. Factorise 6x2 − 19x + 10

Factors of 6x2 and +10 will be needed.

(3x − 2)(2x − 5)

### Question 79. For which values of x is the function y = tan x UNDEFINED between x = 0° and x = 360°

Whereabouts on the x-axis are the asymptotes?

x = 90° and x = 270°

### Question 80. Is the gradient of the curve y = 5 − x + x3 negative or positive when x = 2?

Differentiate and then substitute x = 2 into the derivative.

Positive!

### Question 81. Solve the equation tan x = 1 for 0° < x < 720°

There are 4 solutions!

45°, 225°, 405°, 585°

### Question 82. The graph of y = − 3sin 2x + 4 passes through the point (25°, y)Find y.

Substitute 25° into the function.

y = 1.7 (to 2 s. f.)

### Question 83. How many two digit numbers can be made using the digits 4, 5 and 6?

How many ways can you fill the first digit and how many ways can you fill the second digit?
Remember both digits can be the same.

There are 9 ways from 44 through to 66

### Question 84. Simplify x2y(3xy2)2

Expand the brackets first.

9x4y5

### Question 85. How long is the line joining the points (−3, −6) to (2, 6)?

Use a diagram or a formula or Pythagoras' Theorem or all three!

13 units

### Question 86. Solve the equation cos 2x = 1 for 0 ≤ x° ≤ 360

You should have three solutions.

x = 0°, 180° and 360°

### Question 87. Solve the equation (x − 2)(x + 3)(2x - 3) = 0

Make each bracket equal to zero.

x = 2, −3, 3/2

### Question 88. What is 210° in radians, in terms of Π?

Multiply by Π / 180

7Π / 6

### Question 89. What type of graph is produced by the function y = 4x?

It is not a straight line or a circle!

An exponential function or growth curve.

### Question 90. Solve the equation 2(x + 4) = 16x - 8

Expand the brackets first then get x terms to one side and number terms to other.

x= 8 / 7

### Question 91. What is the equation of the vertical asymptote of the rectangular hyperbola (x − 2)(y + 3) = 9?

Solve the equation y + 3 = 0 or
Solve the equation x − 2 = 0
Which one?

The vertical asymptote is x = 2

### Question 92. What is the interquartile range of the values {3, 7, 14, 3, 10, 20}?

The interquartile range is the difference between the lower quartile and the upper quartile.

The interquartile range is 14 − 3 = 11

### Question 93. If y = 5(x − 4)2 What is y'?

Expand the brackets first.

y ' = 10x - 40

### Question 94. Find the coordinates of the turning point of the function y = x2 - 4x

Find y ' and then put y ' = 0

(2, −4)

### Question 95. Solve the equation 2p3 = 128

Divide both sides by 2 then take the cube root.

p = 4

### Question 96. What is the name of a survey where ALL of the members of a population are surveyed?

They are held every 5 years in Australia and New Zealand.

Census

Begins with "a"!

Acceleration

### Question 98. Simplify log 3 + log 2 − log 6

Subtract logs - divide

log 1 = 0

### Question 99. Find ∫ 8x3 − 6x2 − 5

Don't forget the "c" when integrating.

2x4 − 2x3 − 5x + c

### Question 100. What is the period of the function y = -2cos 2x + 2?

The period of a trigonometric function is the interval along the x-axis before the pattern of the graph is repeated.

360 / 2 = 180° or Π