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*If you need help on a particular question, click on the HINT button

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

f(

Complete the square

f(*x*) = (*x* − 2)^{2} + 7

What are the

Put *y* = 0 into the equation

(+√3, 0) and (−√3, 0)

for 0° ≤

There are 4 solutions. (use the graph of *y* = sin2*x*)

15°, 75°, 195°, and 255°

How many first days of a month are there?

12/365

Complete the square

*x*^{2} − 4*x* + 11 = (*x* − 2)^{2} + 7

So the minimum value is 7

|*x* + 4| < 1 means

−1 < *x* + 4 < 1

−5 < *x* < −3

Write 2/*x* as 2*x*^{-1} then differentiate

f' '(*x*) = 6*x* + 2*x*^{−2}

= 6*x* + 2/*x*^{2}

find f(-1)

Substitute *x* = -1 into f(*x*)

f(-1) = (-1)^{3} - 2 x (-1)^{2} + 3(-1) + 1

= -1 - 2 - 3 +1

= -5

Put *x* = 0

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

Expand first and then multiply by 2

*y* = 18 - 12*x* + 2*x*^{2}

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

Write roots as indices and then ADD them.

*x*^{5/2} . *x*^{3/2} = *x*^{4}

(Give answer in degrees to 1 decimal place)

Use sin^{-1} on calculator. There are 2 solutions.

10.8° and 169.2°

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

36, 42, 41, 39, 32, 49, 28, 56

Arrange in order first.

The median is 40

Use the differentiation formula

f ' (x) = 3*x*^{2}.sec^{2}(*x*^{3})

The Laws of Indices must be used.

1

g'' means differentiate TWICE

g''(*x*) = 12*x* + 6

use the Properties of Logarithms

*x* = 5

Look up the COMBINATION formula or use your calculator

^{7}C_{4} = 35

It is one of the conic sections.

It is the equation of part of a parabola.

Use the properties of logarithms

log_{5}5 = 1

Differentiate term by term

*f'( x) = 6x^{2} − 6x + 4*

Substitute *y* = 1 into *y* = *x*^{2}

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

Use *x* = a secθ and *y* = b tanθ

*x* = 7secθ

*y* = 10tanθ

36, 42, 41, 39, 32, 49, 28, 56

Use the σ_{n - 1} button on your calculator.

The sample standard deviation is 8.99

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

f' (*x*) = 15(*x* - 2)^{4}

What is

For the division of terms, SUBTRACT the exponents.

*n* = 3/2

Use the midpoint of a line theorem.

(0, 1)

(Give answer in radians to 3 significant figures.)

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

C = 1.23

Factorise the numerator

(*x* + 2)/(*x* + 3)

It is one of the conic sections.

It is an ellipse

Complete the square for *x*^{2} - 2*x* and *y*^{2} - 4*y*

The centre of the circle is (1, 2)

The sum of the probabilities is ONE.

P(*X* = 7) = 0.1

Differentiate term by term.

f '(*x*) = 12*x*^{2} + 4e^{2x}

Use calculator

66.69

Use a double angle formula

cos2*x* = 127/128

What is the equation of the horizontal asymptote?

Try dividing out the rational fraction.

(sometimes known as the cover-up rule)

*y* = 2

The integral of 1/*x* is ln *x*

3ln *x*

Use the statistics function of your calculator.

Standard deviation = 10.01

Divide both sides by 3

C = 78.0°

f(

Complete the square on *x*^{2} + 4*x*

The minimum value of the function is 3

Arrange the numbers in order first.

The upper quartile is 45.5

Leave answer in surd form ( with square roots)

Complete the square

*x* = 2 ± √12

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

f '(*x*) = 8*x*^{3} + 9*x*^{2}

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

3*b*^{6}

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

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

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)

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

*x*^{3} + 2ln |*x*| + c

Find P(R')

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

P(R') = 1 - 0.3 = 0.7

Take natural logarithms of both sides.

*x* = 0.52 (to 2 s.f.)

Integrate cos *x* first and then substitute the values.

1

Use the sum to product formula

2cos3*x* sin2*x*

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.

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

The required region is BELOW the line

2

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)

Take logarithms to base e of both sides.

*x* = 3.88

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

f(

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

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

Use the properties of logarithms

*x* = 20.1

Write 1/*x*^{3} in index form

f '(*x*) = -3/*x*^{4}

Factorise the numerator first and then cancel

*x* - 3

Expand the brackets first.

*x*^{3} − 3.5*x*^{2} − 6*x* + c

It will be in the form a^{b} = c

3^{4} = 81

Find the variance of 5X + 2Y

VAR(aX + bY) = a^{2}VAR(X) + b^{2}VAR(Y)

VAR(5X + 2Y) = 5^{2} x 3 + 2^{2} x 2 = 83

for 0 ≤

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

Use properties of logarithms.

*x* = *b*

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

How many possible solutions are there?

There are an infinite number of solutions so *x* € R

Factorise then simplify then substitute.

5

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

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

The maximum value is 1

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.)

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

g '(*x*) = 4*x* - 3

When *x* = 1

g' (1) = 1

cos 5

Use the formula ^{t}*n* = *a* + (*n* - 1)d

*t*_{25 = 99}

Complete the square on *x*^{2} - 2*x* and *y*^{2} - 4*y*

The radius is 2

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

Differentiate term by term

6*x*^{2} + 2e^{2x}

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

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

Use calculator and the graph of *y* = sin2*x*

7π/12, 11π/12, 19π/12, 23π/12

Put *x* = 0

The y-intercepts are (0, ± 3)

Mean = Σ(f(*x*). *x*) /Σ f(*x*)

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

Use tables, calculator of formula. Just find it!

504

Put *y* = 0

*a*^{2} = 16

(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

VAR(X) = E(X^{2}) - [E(X)]^{2}

VAR(X) = 2.44

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

Distance = √68

Solve the equation *y* + 3 = 0
or

Solve the equation *x* − 2 = 0

Which one?

The vertical asymptote is *x* = 2

for 0 ≤

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

0, 5π/3, 2π

Expand the brackets first.

*y* ' = 10*x* - 40

Expand the brackets first or use the composite function rule.

*y* ' = 10*x* - 40

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.

Put *x* = 0 in the equation

*y* = -0.5

Use the substitution *u* = *x* - 2

Don't forget to add "+c"

The vertical asymptote occurs when the denominator would be 0!

The vertical asymptote is *x* = -0.5

Remember: If *y* = e^{x} then *y*'= e^{x}

y' = 6e^{6x}

f(

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

(0, -5)