Key Points

Question 1

What must a graph sketch include?

Answer 1

Important coordinates stated

Question 2

What does orthogonal mean?

Answer 2

Perpendicular

Question 3

What does commutative mean?

Answer 3

Order doesn’t matter (+ or x)

Question 4

What does associativity mean?

Answer 4

How signs are grouped (applies to + or x) e.g. a+b+c=(a+b)+c

Question 5

How can you tell if 2 gradients are perpendicular using a calculation?

Answer 5

Multiply the gradients together and if they equal -1 then they’re perpendicular

Question 6

What is a parabola?

Answer 6

A quadratic graph

Question 7

What is an even function?

Answer 7

A graph where f^-1(x)=f(x)

Question 8

What’s another name for stationary points?

Answer 8

Vertices/ turning point

Question 9

What’s the opposite of prime?

Answer 9

Composite (composed of 2 or more numbers)

Question 10

What should you remember about square roots?

Answer 10

They can be have plus or minus answers

Question 11

What’s an exponent?

Answer 11

A power/indice

Question 12

What’s the discriminant?

Answer 12

b^2-4ac (the bit under the radical (square root) sign in the quadratic formula)

Question 13

What 3 cases are there to consider with the discriminant?

Answer 13

1) b^2-4ac=0 (1 real repeated root)
2) b^2-4ac>0 (2 distinct real roots)
3) b^2-4ac<0 (no real roots)

Question 14

What’s the relationship between a logarithmic graph and their accompanying exponential graph?

Answer 14

The graphs are reflections of one another in the line y=x

Question 15

What’s an example of an irrational number (can’t be written as fractions)?

Answer 15

A surd

Question 16

What is a many-to-one function?

Answer 16

Multiple inputs mapping to the same output (one-to-one is one input mapping to one output)

Question 17

What happens to an inequality sign if you divide by a positive or a negative number?

Answer 17

Positive = no change
Negative = the sign changes e.g. from < to >
(This does not apply to multiplying)

Question 18

What is the symbol for a natural log?

Answer 18

ln

Question 19

Is ln(0.98) a positive or negative number?

Answer 19

Negative (so be very careful when dividing across inequalities as this causes the sign to change)

Question 20

Write x+3 / x+1 in the form A + (B / x+1)

Answer 20

x+3 can be written as x+1+2 then you can cross cancel to get 1 + (2 / x+1)

Question 21

What symbol can be used as you move from one line of workings to another?

Answer 21

The implies sign ( ⇒ )

Question 22

What do you call the top and base numbers of fractions? What do you call the answer when dividing?

Answer 22

The top = dividend
The base = divisor
The answer = quotient

Question 23

How do you solve: 4x^5-14x^4-30x^3 ?

Answer 23

Factorise twice: 2x^3(2x^2-7x-15) then 2x^3(2x+3)(x-5)

Question 24

State the degree / order of the following polynomials:
1) 6x^4-9x^2+2
2) 4x^2-3x^5+1
3) 8+4x^3-x^6

Answer 24

1) 4
2) 5
3) 6

Question 25

What does the degree / order of a polynomial mean?

Answer 25

**The highest power any of its terms is raised to** e.g. for: x^4+x^7 7 is the highest power so the degree is 7

Question 26

What happens to the gradient of a log graph as the log’s base increases?

Answer 26

It becomes shallower

Question 27

Are the graphs of y=-e^x and -y=e^x the same?

Answer 27

Yes

Question 28

What does the graph of ln(x) look like compared to e^x?

Answer 28

It’s a reflection in the line y=x

Question 29

How do you work out the y-intercept of (x-3)(x+3)(x-2)?

Answer 29

Multiply the constants together (-3x3x-2) =18

Question 30

What must you state when using the factor theorem to show e.g. (x+2) is a factor of f(x) ?!!

Answer 30

At the end of the proof state: **therefore** (x+2) is a factor of f(x) very simple but you get few marks if you don’t specify this!

Question 31

How do you convert from m/s to km/h?

Answer 31

Multiply by 3600/1000 (3.6)

Question 32

For the equation: T=20+5e^-0.3t, as ‘t’ tends towards infinity what happens to T and why?

Answer 32

T tends towards the constant 20 since the negative power means the last part is a fraction so this term ‘vanishes’ as the denominator gets increasingly larger. This is an example of **exponential decay**

Question 33

For the equation: T=74+36e^4.7t, as ‘t’ tends towards infinity what happens to T and why?

Answer 33

T tends towards infinity too because there’s no negative power so the number simply gets larger and larger in **’exponential growth’**

Question 34

What is the transformation mapping the graph of: y=f(x) onto y=f(x-a)+b

Answer 34

A **translation** by the vector [**a**,b] (a over b)

Question 35

What is the transformation mapping: y=f(x) onto y=f((1/k)x) ?

Answer 35

A stretch parallel to the x-axis by scale factor k

Question 36

What is the transformation mapping: y=f(x) onto y=kf(x) ?

Answer 36

A stretch parallel to the y-axis, scale factor k

Question 37

What is the transformation mapping y=f(x) onto y=-f(x) ?

Answer 37

A reflection in the x-axis

Question 38

What is the transformation mapping y=f(x) onto y=f(-x) ?

Answer 38

Reflection in the y-axis

Question 39

What is the transformation mapping y=f(x) onto x=f(y) ?

Answer 39

Reflection in the line y=x

Question 40

What is the name for what happens when the graph of y=cos(x) is reflected in the line x=0 ?

Answer 40

It’s an even function because the graph is unchanged when mapped from y=cos(x) onto y=cos^-1(x)

Question 41

What is periodicity?

Answer 41

The quality of a function to repeat its values at regular intervals e.g. the sine function

Question 42

What must you remember when writing vectors?

Answer 42

Square brackets

Question 43

The cubic polynomial f(x) is given by f(x) = 2x^3 + ax^2 + bx +15, where a and b are constants. It is given that (x+3) is a factor of f(x) and that, when f(x) is divided by (x-2), the remainder is 35. Find the values of a and b.

Answer 43

1) f(-3) = 0 since (x+3) is a factor, so substitute in -3 for x into the cubic, this gives you 3a - b = 13 2) f(2) = 35 so substitute in 2 for x into the cubic and make it equal to 35. This gives you 2a + b = 2 3) use simultaneous equations to solve. a=3, b=-4

Question 44

Complete the square of y^2 - 4y

Answer 44

(y-2)^2-4

Question 45

Complete the square of 2x^2+3y^2+8x-18y+2

Answer 45

1) 2[x^2+4x] = 2[(x+2)^2-4] = 2(x+2)^2-8 2) 3[y^2-6y]+2 (you can leave the 2 alone) 3[(y-3)^2-9]+2 = 3(y-3)^2-25 3) (2(x+2)^2-8) + (3(y-3)^2-25) = **2(x+2)^2 + 3(y-3)^2 - 33**

Question 46

Work out whether these 2 graphs intersect: (x-2)^2 + (y+2)^2 =20 and 3x + 7 =y

Answer 46

1) substitution: (x-2)^2 + (3x+7+2)^2 =20 2) expand the brackets and collect like terms: 10x^2+50x+65 =0 3) **simplify (divide by 5)**: 2x^2+10x+ 13 =0 4) discriminant: b^2-4ac = -4<0 so no points of intersection

Question 47

What will the discriminant equal when finding the points of intersection between a circle and a tangent?

Answer 47

b^2-4ac=0 because a tangent only meets a circle at **one** point

Question 48

What will the discriminant equal when finding the points of intersection between a circle and a secant line?

Answer 48

b^2-4ac>0 because there’re **two** points of intersection between the lines

Question 49

What is another term for the radius in relation to a tangent?

Answer 49

Normal (orthogonal line)

Question 50

When writing log10(y) = x + 1/2 in exponential form (ax10^bx) why do you make 10 the base e.g. **10**^(log10(y))

Answer 50

**Because it has the same base as the logarithm** so they cancel out and you’re left with just ‘y’ on the left hand side, this makes simplifying and getting it into the correct form easier.

Question 51

Find the values of k for which kx^2+8x+5=0 has a repeated root

Answer 51

b^2-4ac=0 for repeated roots so 8^2-(4xkx5)=0 k=3.2

Question 52

Find the centre and radius of the circle with the equation: x^2+y^2-2x+8y-8=0

Answer 52

x^2-2x+y^2+8y-8=0 (x-1)^2-1+(y+4)^2-16-8=0 (x-1)^2+(y+4)^2=25 c=(1,-4), r=5

Question 53

If f(x) has roots (2,0) & (12,0) and turning point at (3,7) what will the graph of y=2f(x) look like?

Answer 53

The roots will be the same but the maximum point will be multiplied by 2 to become (3,14) because y=2f(x) represents a stretch parallel to the y-axis factor 2

Question 54

If f(x) has roots (2,0) & (12,0) and turning point at (3,7) what will the graph of y=f(-x) look like?

Answer 54

The roots will be (-2,0) & (-12,0), the turning point will be (-3,7) because it represents a stretch parallel to the x-axis factor -1

Question 55

Simplify log h(x/y) + log h(y/z) - log h(z/y)

Answer 55

log h(xy/yz)/(z/y) = log h (x/y)/(z/y) = **log h(xy/z^2)**

Question 56

Put y=kx^n in linear form using logs

Answer 56

log10(y) = log10(kx^n) log10(y) = log10(k) + log10(x^n) log10(y) = log10(k) + n(log10(x))

Question 57

Simplify 3xln(1.5) = ln(7.5) **(where ‘x’ is a letter not a multiplier)**

Answer 57

ln(7.5) / **ln**(1.5) = 3x x = 1.656 (You’re dividing the logs not subtracting so you have ln on the bottom and top of the fraction!!)

Question 58

Solve 2xb^(4logb(x)) = 486

Answer 58

2xb^(logb(x^4)) = 486 2x x x^4 = 486 2x^5 = 486 x^5 = 243 x=3

Question 59

Find using algebra, all real solutions to the following equation giving your answers in exact form. 3^2x + (7 x 3^x) -18 =0

Answer 59

**Use dummy variables** 3^2x = (3^x)^2 Let y = 3^x y^2+7y-18=0 (y+9)(y-2)=0 you can’t have -9 as a solution as the argument of a log can’t be negative so the solution is y=2, substitute into y=3^x 2=3^x, **x=log3(2)**

Question 60

**For this question:** It is given that f(x) = x^2+bx+c and g(x) = x^3+dx^2+e have a common factor of (x-4). Show that 4(b-4d-12)=e-c. Fully justify your answer. **What must you put to get full marks?**

Answer 60

f(4)=0, g(4)=0 so f(4)=g(4) you **must** make these statements otherwise you’ll lose marks because it said to fully justify!!

Question 61

In a SUVAT question how should you round your answer?

Answer 61

To the greatest number of s.f. a number is given as in the question or **if there’s a value given for acceleration due to gravity you must use the number of s.f. in this instead** (it overrides the others)

Question 62

Should you round to decimal places?

Answer 62

No never always to significant figures

Question 63

Explain why the model: Temperature=Ax10^-kt is unlikely to accurately give the value of temperature after 45 minutes

Answer 63

After 45 minutes the outside temperature might have changed.

Question 64

For a cubic graph what are the names of the turning points?

Answer 64

The **local** minimum and the **local** maximum

Question 65

Factorise: 8e^2x -30e^x -27=0 and solve

Answer 65

(2e^x -9)(4e^x +3)=0 e^x=9/2 or e^x=-3/4 e^x=-3/4 not valid so answer is x=In(9/2)

Question 66

The equation for the perpendicular bisector of PQ is y=-9/2x + 30.5. Give this equation in the form ax+by=c where a, b and c are integers

Answer 66

Rearrange to get into the form ax+by=c gives you 9/2x+y=30.5 To make a, b and c **integers multiply by 2 to get whole numbers** gives : 9x+2y=61

Question 67

The temperature of a cup of tea can be modelled by T=Ae^-kt. What does ‘A’ represent in this context? What will ‘A’ represent in the equation: T=Ae^kt?

Answer 67

1) The initial temperature (when t=0) for exponential decay. 2) The room temperature (when ‘t’ tends to infinity) for exponential growth

Question 68

Expand and simplify: 6-2(7-3x) > 8-(3x+7)

Answer 68

1) on the LHS **multiply the bracket by -2**: 6-14+6x 2) on the RHS **multiply the bracket by -1**: 8-3x-7 3) simplify: 9x>9 so x>1

Question 69

Solve the inequality: x^2-x-12

Answer 69

1) factorise: (x-4)(x+3) 2) roots are at (4,0) and (-3,0) 3) sketch a diagram to get the solution: -3

Question 70

What does a sketch of y=sin(x) look like? What does a sketch of y=tan(x) look like?

Answer 70

Sin(x) starts at the origin, crosses the x-axis at 180 and again at 360 (the sine curve repeats every 360° Tan(x) starts at the origin and has asymptotes at 90 & 270° (every 180°)

Question 71

In this question use g=9.8ms^-2 A pen is dropped from rest and 1.5 seconds later it lands on the floor. By modelling the pen as a particle state the speed at which it hits the ground.

Answer 71

v=u+at v=0+(9.8x1.5) v=14.7 so **v=15ms^-1** because you must give your answer to the same number of significant figures as ‘g’ !!