Question 1 How do you find the equation of a perpendicular bisector of a straight line ? 1.1

Answer to Question 1 (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)

How do you find the midpoint of a line joining two points ? Question 2 How do you find the midpoint of a line joining two points ? 1.1

( ) Add the coordinates and divide by two x1+ x2 , y1+ y2 Answer to Question 2 Add the coordinates and divide by two x1+ x2 , y1+ y2 2 2 x y (x2,y2) (x1,y1) ( )

How do you find the altitude AN of ΔABC ? Question 3 How do you find the altitude AN of ΔABC ? 1.1

(i) find the gradient of BC Answer to Question 3 (i) find the gradient of BC (ii) find the gradient of AN, perpendicular to BC (iii) use y-b=m(x-a), using A as (a,b) A N B C

How do you show that x-1 is a factor of the function f(x)=x3-3x+2 ? Question 4 How do you show that x-1 is a factor of the function f(x)=x3-3x+2 ? 2.1

(i) rewrite the function as f(x)=x3+0x2-3x+2 Answer to Question 4 (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

For what values is this function undefined ? f(x) = x Question 5 For what values is this function undefined ? f(x) = x (x+2)(x-3) 1.2

Answer to Question 5 -2 and 3

How do you draw the graph of 2f(x) given the graph of f(x) ? Question 6 How do you draw the graph of 2f(x) given the graph of f(x) ? 1.2

Answer to Question 6 Double the y-coordinates

How do you find the exact value of sin (α-β), given that sinα =4/5 Question 7 How do you find the exact value of sin (α-β), given that sinα =4/5 and cosβ = 12/13 ? 2.3

Answer to Question 7 (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

What is the turning point of y=2(x-a)2+b ? Max or min ? Question 8 What is the turning point of y=2(x-a)2+b ? Max or min ? 2.1

Answer to Question 8 (i) (a,b) minimum (a,b)

How do you draw the graph of f(-x) given the graph of f(x) ? Question 9 How do you draw the graph of f(-x) given the graph of f(x) ? 1.2

Reflect the graph in the y-axis Answer to Question 9 Reflect the graph in the y-axis

How do you draw a graph of the form y = cos(x+a) or y = sin(x+a) ? Question 10 How do you draw a graph of the form y = cos(x+a) or y = sin(x+a) ? 1.2

Move the graph of y=cosx or y=sinx a units to the LEFT Answer to Question 10 Move the graph of y=cosx or y=sinx a units to the LEFT

Name the steps you take in order to differentiate functions like Question 11 Name the steps you take in order to differentiate functions like f(x) = x2+ 3x + 1 √x 1.3

(i) Change roots to powers (ii) split up into 3 fractions Answer to Question 11 (i) Change roots to powers (ii) split up into 3 fractions (iii) simplify each term (iv) differentiate

Question 12 If f(t) is the distance travelled in a certain time t seconds, then what does f’(t) represent ? 1.3

Answer to Question 12 Speed (velocity)

Given f’(x) and a point on the curve, how do you find Question 13 Given f’(x) and a point on the curve, how do you find f(x) ? 2.2

(ii) substitute in a given point to work out value Answer to Question 13 (i) integrate (ii) substitute in a given point to work out value of C

What do you know about the gradients of two parallel lines? Question 14 What do you know about the gradients of two parallel lines? 1.1

Answer to Question 14 They are the same

Question 15 How do you find the equation of a tangent to a curve at the point when x = a ? 1.1

(ii) fit a into f’(x) to get the gradient (m) Answer to Question 15 (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)

Question 16 How do you find the rate of change of a function at a particular point ? 1.3

(ii) fit in given x value Answer to Question 16 (i) differentiate (ii) fit in given x value

If y is the equation of a curve, what is represented by dy/dx ? Question 17 If y is the equation of a curve, what is represented by dy/dx ? 1.3

Answer to Question 17 The gradient

How do you find where a curve is increasing ? Question 18 How do you find where a curve is increasing ? 1.3

Answer to Question 18 (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

Question 19 How would you find the maximum or minimum value of a function given its equation? 1.3

(iii) solve to find the stationary points (iv) draw the nature table Answer to Question 19 (i) differentiate (ii) let f’(x) = 0 (iii) solve to find the stationary points (iv) draw the nature table (v) read off max or min

Question 20 Given a rec. relation in the form un+1 = aun + b and 3 consecutive terms, how do you find the values of a and b? 1.4

(i) fit 1st term into un and 2nd term into un+1 Answer to Question 20 (i) fit 1st term into un and 2nd term into un+1 (ii) fit 2nd term into un and the 3rd term into un+1 (iii) solve simultaneous equations

Question 21 How do you find the value of a in the polynomial x3+ax2+4x+3 given either a factor of the polynomial, or the remainder when the polynomial is divided by a number ? 2.1

(i) do synthetic division (ii) let the expression = 0 or the remainder Answer to Question 21 (i) do synthetic division (ii) let the expression = 0 or the remainder (iii) solve the equation

Question 22 How do you find ∫ (ax + b)n dx ? 3.2

Answer to Question 22 (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

How do you solve equations of the form sin2xo = 0.5 ? (0≤x≤360) Question 23 How do you solve equations of the form sin2xo = 0.5 ? (0≤x≤360) 2.3

(iv) divide each by 2 Answer to Question 23 (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 equations like cos2xo-5sinxo = 0 ? (0≤x≤360) Question 24 How do you solve equations like cos2xo-5sinxo = 0 ? (0≤x≤360) 2.3

(ii) factorise (iii) solve equation (i) fit in 1-2sin2xo for cos2xo Answer to Question 24 (i) fit in 1-2sin2xo for cos2xo (ii) factorise (iii) solve equation

How do you find a unit vector parallel to a given vector ? Question 25 How do you find a unit vector parallel to a given vector ? 3.1

(i) find the length of the given vector Answer to Question 25 (i) find the length of the given vector (ii) divide all the components by this length

How do you prove that a line is a tangent to a circle ? Question 26 How do you prove that a line is a tangent to a circle ? 2.4

Substitute line into circle Answer to Question 26 Rearrange line to make y = or x = Substitute line into circle Prove it has equal roots using b2-4ac = 0 or repeated roots

How do you find the angle between two vectors ? Question 27 How do you find the angle between two vectors ? 3.1

Answer to Question 27 a.b a b cosq = a b q

Question 28 What is a unit vector ? 3.1

Answer to Question 28 A vector of length 1 unit

Question 29 What vector is equal to AB + CD + BC ? 3.1

Answer to Question 29 AD

then what is u in component form ? Question 30 If u = ai+bj+ck then what is u in component form ? 3.1

Answer to Question 30 a b c U =

How do you integrate sin ax ? Question 31 How do you integrate sin ax ? 3.2

Answer to Question 31 -1/a cos ax + C

How would you differentiate a function like Question 32 How would you differentiate a function like y = sin3 x ? 3.2

Answer to Question 32 (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. 3 sin2x cosx

Question 33 Given experimental data, how do you find an equation in the form y=abx or y=axb ? 3.3

(ii) rearrange to get a straight line equation (iii) determine type Answer to Question 33 (i) take logs of both sides (ii) rearrange to get a straight line equation (iii) determine type (iv) Equate and solve for a and b

How do you differentiate an expression like Question 34 How do you differentiate an expression like without multiplying it out ? 3.2

(i) multiply by the power (ii) decrease power by 1 Answer to Question 34 (i) multiply by the power (ii) decrease power by 1 (iii) multiply by derivative of bracket

Question 35 Given an equation like m = moe-3k and an amount by which it has been decayed, how do you find k ? 3.3

(ii) rearrange to get e-3k = (iii) take logs to get -3k = Answer to Question 35 (i) fit in m and mo (ii) rearrange to get e-3k = (iii) take logs to get -3k = (iv) solve for k

Question 36 What is loga x + loga y equal to ? 3.3

Answer to Question 36 Loga xy

How do you solve equations of the form Question 37 How do you solve equations of the form 3x = 0.155 ? 3.3

(i) take logs of both sides (ii) bring x down to front Answer to Question 37 (i) take logs of both sides (ii) bring x down to front (iii) solve the equation

Question 38 What is loga xn equal to ? 3.3

Answer to Question 38 nloga x

If y = How should you rewrite y so it is ready to differentiate? Question 39 If y = How should you rewrite y so it is ready to differentiate? 1.3

Answer to Question 39

How do you find the maximum or minimum values of Question 40 How do you find the maximum or minimum values of acosx + bsinx + c ? 3.4

(i) change acosx+bsinx into Rcos(x-a) Answer to Question 40 (i) change acosx+bsinx into Rcos(x-a) (ii) max is R+c