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## Purple questions are about TRIGONOMETRY

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### Question 1.  Find the mean, median and mode for the following data:2, 3, 31, 5, 6, 19, 5

The mean is the common average. The mode is the most common.The median is the middle value.

Mean = 10.1 (to 3 sig. fig.) Mode = 5  Median = 5

### Question 2. Write f(x) = x2 - 4x + 11 in the form f(x) = (x − a)2 + b

Complete the square

f(x) = (x − 2)2 + 7

### Question 3. For the graph of the equation x2/3 + y2/9 = 1 What are the x intercepts?

Put y = 0 into the equation

(+√3, 0) and (−√3, 0)

### Question 4. Solve the equation sin2x = 0.5 for 0° ≤ x ≤ 360°

There are 4 solutions. (use the graph of y = sin2x)

15°, 75°, 195°, and 255°

### Question 5. A school contains 500 students. What is the probability that a student chosen at random is born on the first day of a month?

How many first days of a month are there?

12/365

### Question 6. What is the minimum value of x2 − 4x + 11?

Complete the square

x2 − 4x + 11 = (x − 2)2 + 7
So the minimum value is 7

### Question 7. Solve |x + 4| < 1

|x + 4| < 1 means
−1 < x + 4 < 1

−5 < x < −3

### Question 8. Find f '(x) if f(x) = 3x2 - 2/x

Write 2/x as 2x-1 then differentiate

f' '(x) = 6x + 2x−2
= 6x + 2/x2

### Question 9. If f(x) = x3 - 2x2 + 3x + 1find f(-1)

Substitute x = -1 into f(x)

f(-1) = (-1)3 - 2 x (-1)2 + 3(-1) + 1
= -1 - 2 - 3 +1
= -5

### Question 10. For the graph of x2/3 + y2/9 = 1 what are the y-intercepts?

Put x = 0

(0, 3) and (0, -3)

### Question 11. Expand 2(3 - x)2

Expand first and then multiply by 2

y = 18 - 12x + 2x2

### Question 12. Find the median and the interquartile range of 2, 3, 5, 19, 5, 31, 6

Median is the middle value and the interquartile range is the difference between the upper quartile and the lower quartile

Median is 5 and the interquartile range is
19 - 3 = 16

### Question 13. Simplify √(x5) . √(x3)

Write roots as indices and then ADD them.

x5/2 . x3/2 = x4

### Question 14. Solve sinA = 0.188 for 0° ≤ A ≤ 180° (Give answer in degrees to 1 decimal place)

Use sin-1 on calculator. There are 2 solutions.

10.8° and 169.2°

### Question 15. R and Q are two independent events with P(R) = 0.4 and P(Q) = 0.3 What is P(R U Q)?

P(A U B) = P(A) + P(B) - P(A n B)

P(R U Q) = 0.4 + 0.3 - 0.4 x 0.3
= 0.7 - 0.12 = 0.58

### Question 16. Find the median for: 36, 42, 41, 39, 32, 49, 28, 56

Arrange in order first.

The median is 40

### Question 17. Find f '(x) if f(x) = tan(x3)

Use the differentiation formula

f ' (x) = 3x2.sec2(x3)

### Question 18. Simplify (a2 x an) / an + 2

The Laws of Indices must be used.

1

### Question 19. If g(x) = 2x3 + 3x2 - 12x + 3, find g''(x)

g'' means differentiate TWICE

g''(x) = 12x + 6

### Question 20. Solve log5x = 1

use the Properties of Logarithms

x = 5

### Question 21. Evaluate 7C4

Look up the COMBINATION formula or use your calculator

7C4 = 35

### Question 22. What type of graph is given by the equation y = 4√x?

It is one of the conic sections.

It is the equation of part of a parabola.

### Question 23. Evaluate log55

Use the properties of logarithms

log55 = 1

### Question 24. Find ʃ3 + e3x dx

Differentiate term by term

f'(x) = 6x2 − 6x + 4

### Question 25. What are the coordinates of the points of intersection of the graphs of y = x2 and y = 1?

Substitute y = 1 into y = x2

(1, 1) and (-1, 1)

### Question 26. What are the parametric equations of the conic x2/49 - y2/100 = 1

Use x = a secθ and y = b tanθ

x = 7secθ
y = 10tanθ

### Question 27.Find the sample standard deivation for: 36, 42, 41, 39, 32, 49, 28, 56

Use the σn - 1 button on your calculator.

The sample standard deviation is 8.99

### Question 28. Find f '(x) if f(x) = 3(x - 2)5

Use the function of a function (composite function) rule.

f' (x) = 15(x - 2)4

### Question 29. When a2/√ a is written in the form an. What is n?

For the division of terms, SUBTRACT the exponents.

n = 3/2

### Question 30. Find the midpoint of the line joining the points (1, -3) and (-1, 5)

Use the midpoint of a line theorem.

(0, 1)

### Solve tanC = 2.85(Give answer in radians to 3 significant figures.)

Use "shift" and "tan" on your calculator.

C = 1.23

### Question 32. Simplify (x2 - x - 6)/(x2 - 9)

Factorise the numerator

(x + 2)/(x + 3)

### Question 33. What type of graph is given by the equation x2/3 + y2/9 = 1 ?

It is one of the conic sections.

It is an ellipse

### Question 34. What are the coordinates of the centre of the circle given by the equation x2 + y2 - 2x - 4y - 9 = 0?

Complete the square for x2 - 2x and y2 - 4y

The centre of the circle is (1, 2)

### Question 35. The random variable X has the following probability distribution. Find P(X = 7)

The sum of the probabilities is ONE.

P(X = 7) = 0.1

### Question 36. Differentiate 4x3 + 2e2x

Differentiate term by term.

f '(x) = 12x2 + 4e2x

Use calculator

66.69

### Question 38. Evaluate cos2x if sinx = 1/16

Use a double angle formula

cos2x = 127/128

### Question 39. For the equation y = (2x - 3)/(x + 2) What is the equation of the horizontal asymptote?

Try dividing out the rational fraction.
(sometimes known as the cover-up rule)

y = 2

### Question 40. What is the integral of 3/x ?

The integral of 1/x is ln x

3ln x

### Question 41.Find the standard deviation of 2, 3, 31, 5, 6, 19, 5

Use the statistics function of your calculator.

Standard deviation = 10.01

### Question 42. Solve 3cosC = 0.625 for 0 ≤ C ≤ 180°

Divide both sides by 3

C = 78.0°

### Question 43. What is the minimum value of the function f(x) = x2 + 4x + 7?

Complete the square on x2 + 4x

The minimum value of the function is 3

### Question 44. What is the upper quartile of 36, 42, 41, 39, 32, 49, 28, 56?

Arrange the numbers in order first.

The upper quartile is 45.5

### Question 45.Solve x2 - 4x = 8 Leave answer in surd form ( with square roots)

Complete the square

x = 2 ± √12

### Question 46. Differentiate x3(2x + 3)

Expand the brackets first - easiest (unless you use the product rule)

f '(x) = 8x3 + 9x2

### Question 47. W Simplify 12b12 ÷ (2b3)2

Laws of Indices! Remember to square the 2 in the brackets.

3b6

### Question 48. Find the points on the line joining (1, -3) and (-1, 5) which divides it internally in the ratio 2:1

Look up the formula! A diagram may help to check your answer.

(-1/3, 7/3) or (1/3, -1/3)

### Question 49. Make y the subject of the formula x = (3y + 2)/(y - 1)

The first step is to multiply both sides by (y - 1) Then get all of the x terms to one side and all of the y terms to the other.

y = (2 + x)/(x - 3)

### Question 50. Find ʃ(3x2 + 2/x) dx

Integrate term by term and don't forget the "c"!

x3 + 2ln |x| + c

### Question 51.If an experiment has events R and S with P(R) = 0.3 Find P(R')

P(R') is the probability that event R does NOT happen.

P(R') = 1 - 0.3 = 0.7

### Question 52. if e3x = 4.8 find x to 2 significant figures.

Take natural logarithms of both sides.

x = 0.52 (to 2 s.f.)

### Question 53. Evaluate

Integrate cos x first and then substitute the values.

1

### Question 54. Write sin5x - sinx as a product.

Use the sum to product formula

2cos3x sin2x

### Question 55. Suppose that Q and R are events with P(Q) = 0.25, P(R) = 0.4 and P(QnR) = 0.12. Are Q and R independent events? Give a reason for your answer.

Is P(Q) x P(R) = P(QnR)?

P(Q) x P(R) = 0.25 x 0.4 = 0.1
which is NOT equal to 0.12. Therefore the events are NOT independent.

### Question 56. Is the region given by the inequality 3y < 10 - 2x above the line or below it?

Sketch the line 3y = 10 - 2x and then "test" the inequality with the point (0, 0)

The required region is BELOW the line

### Question 57. Find the coordinates of the point of intersection of the simultaneous equations 2y - x = 20, x + y = 100 You should be able to do this one in your head!

The elimination method is probably the easiest method to find the point. (Just add the two equations)

The point of intersection is (60, 40)

### Question 58. Solve 3.1x = 81 (to 3 significant figures)

Take logarithms to base e of both sides.

x = 3.88

### Question 59. If X and Y are independent random variables with E(X) = 7, E(Y) = 9, VAR(X) = 3 and VAR(Y) = 2. Find the mean value of the random variable 4X - Y

E(aX + bY) = aE(X) + bE(Y)

E(4X - Y) = 4 x 7 - 9 = 19

### Question 60. What is the equation of the horizontal asymptote of the hyperbola given by the equation f(x) = 3/(x - 2)?

What happens to the value of the function, as x gets larger?

y = 0 (the x-axis) is the horizontal asymptote.

### Question 61.Find x given that logex2 + logex = 9 (Give answer to 1 decimal place.)

Use the properties of logarithms

x = 20.1

### Question 62. If f(x) = 1/x3 find f '(x)

Write 1/x3 in index form

f '(x) = -3/x4

### Question 63. Simplify (x2 - 2x - 3)/(x + 1)

Factorise the numerator first and then cancel

x - 3

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

Expand the brackets first.

x3 − 3.5x2 − 6x + c

### Question 65. Write log381 = 4 as an exponential statement.

It will be in the form ab = c

34 = 81

### Question 66.If X and Y are independent random variables with E(X) = 7, E(Y) = 9, VAR(X) = 3 and VAR(Y) = 2. Find the variance of 5X + 2Y

VAR(aX + bY) = a2VAR(X) + b2VAR(Y)

VAR(5X + 2Y) = 52 x 3 + 22 x 2 = 83

### Question 67. Find x, in radians, to 3 significant figures, such that sinx = -0.3 for 0 ≤ x ≤ 2π

Use the calculator to find one of the solutions and then use the graph of y = sin x and symmetry to find the ones in the domain given.

x = 3.45 and 5.98

### Question 68. Find x given that logbx3 = 3

Use properties of logarithms.

x = b

### Question 69. Find x in radians such that cos x = -0.3 for 0 ≤ x ≤ 2π Give answers to 3 significant figures.

Use calculator to find one value and then use the graph of y = cos x and its symmetry to find the required values.

x = 1.88 and 4.41

### Question 70. Solve 5(x - 2) = 5x - 10

How many possible solutions are there?

There are an infinite number of solutions so x € R

### Question 71. Find

Factorise then simplify then substitute.

5

### Question 72. Use the statistical mode of your calculator to find the mean of 3.4, 5.6, 8.9 and 4.9?

Most calculators show SD on the display when in statistical mode. Make sure you clear any existing statistical data.

mean = 5.7

### Question 73. What is the maximum value of the function y = -2x2 + 1?

Think of the shape of the graph. What effect does the +1 have on the function?

The maximum value is 1

### Question 74. Use the statistical mode of your calculator to find the standard deviation of 3.4, 5.6, 8.9 and 4.9

Most calculators show the statistical mode with SD on the display. Be sure to clear any stored data from the calculator before starting..

SD = 2.01 (to 3 s.f.)

### Question 75. If g(x) = 2x2 - 3x, calculate the rate of change of g(x) when x = 1

Differentiate to find the rate of change and then substitute x = 1

g '(x) = 4x - 3
When x = 1
g' (1) = 1

### Question 77. Find the 25th term of the sequence 3, 7, 11, ...

Use the formula
tn = a + (n - 1)d

t25 = 99

### Question 78. A circle has equation x2 + y2 - 2x - 4y - 9 = 0 What is its radius?

Complete the square on x2 - 2x and y2 - 4y

### Question 79.State whether the sequence 3, 4, 16/3, ... is arithmetic, geometric or neither

Test to see if it has a common ratio or a common difference.

It is a geometric sequence with a common ratio of 4/3

### Question 80. Differentiate 2x3 + e2x

Differentiate term by term

6x2 + 2e2x

### Question 81. A die is tossed. For the two events A and B where A is getting an odd number and B is getting a number more than 3, what is the probability of A OR B?

The word OR indicates the need to ADD the probabilities but don't count the overlap (the 5) twice

P(A OR B) = 5/6

### Question 82. Find

Substitute n = 1 then n = 2, then n = 3 and finally n = 4 and ADD the resultsof each together to find the SUM.

Sum = 20

### Question 83. Solve the equation sin2x = -0.5 for 0 < x < 2π

Use calculator and the graph of y = sin2x

7π/12, 11π/12, 19π/12, 23π/12

### Question 84. For the graph of x2/3 + y2/9 = 1, what are the y-intercepts?

Put x = 0

The y-intercepts are (0, ± 3)

### Question 85. Find the mean of the following frequency distribution.

Mean = Σ(f(x). x) /Σ f(x)

Mean = 308/47 = 6.6 (to 2 s.f.)

### Question 86. Evaluate 9P3

Use tables, calculator of formula. Just find it!

504

Put y = 0

a2 = 16

### Question 88.How many ways are there of selecting a captain and a vice-captain from a team of 6 players?

(How many ways of picking the captian?) x (How many ways of picking the vice captain?)

There are 6 x 5 ways. i.e 30 ways

### Question 89. The random variable X has the following probability distribution. What is the variance of X?

VAR(X) = E(X2) - [E(X)]2

VAR(X) = 2.44

### Question 90. Find the distance between the points (1, -3) and (-1, 5)

Use the formula or draw a diagram and use Pythagoras' theorem.

Distance = √68

### 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. Solve cos(x + π/6) = √3/2 for 0 ≤ x ≤ 2π

Use "special" triangles or calculator. There are 3 answers in the domain given.

0, 5π/3, 2π

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

Expand the brackets first.

y ' = 10x - 40

### Question 94. If y = 5(x - 4)2 What is y' ?

Expand the brackets first or use the composite function rule.

y ' = 10x - 40

### Question 95.What is the equation of the vertical asymptote of the function f(x) = 3/(x - 2)?

To find the vertical asymptote of a hyperbola find the value of x that makes the denominator equal to 0

x = 2 is the equation of the vertical asymptote.

### Question 96. What is the y-intercept of the graph of the graph of y = sin(x - π/6) ?

Put x = 0 in the equation

y = -0.5

### Question 97. Find ʃ 3x(x - 2)3 dx

Use the substitution u = x - 2

### Question 98. The graph of y = (3x - 4)/(2x + 1) has a vertical asymptote. What is its equation?

The vertical asymptote occurs when the denominator would be 0!

The vertical asymptote is x = -0.5

### Question 99. If y = e6x, find y

Remember: If y = ex then y'= ex

y' = 6e6x

### Question 100. What are the coordinates of the y-intercept of the function f(x) = 2x3 + 3x2 + 4x - 5?

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

(0, -5)