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

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**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)

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**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

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**( ) 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) ( )

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**How do you find the altitude AN of ΔABC ?**

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

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**(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

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**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

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**(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

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**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

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Answer to Question 5 -2 and 3

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**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

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Answer to Question 6 Double the y-coordinates

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**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

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**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

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**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

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

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**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

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**Reflect the graph in the y-axis**

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

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**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

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**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

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**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

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**(i) Change roots to powers (ii) split up into 3 fractions **

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

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

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Answer to Question 12 Speed (velocity)

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**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

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**(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

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**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

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Answer to Question 14 They are the same

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

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**(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)

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

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**(ii) fit in given x value**

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

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**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

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Answer to Question 17 The gradient

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**How do you find where a curve is increasing ?**

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

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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

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

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**(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

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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

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**(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

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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

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**(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

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

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**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

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**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

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**(iv) divide each by 2 Answer to Question 23**

(i) decide on the quadrants (sin is +ve) (ii) press INV sin to get angle (iii) work out your 2 angles (iv) divide each by 2

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**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

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**(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

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**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

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**(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

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**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

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**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

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**How do you find the angle between two vectors ?**

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

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

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Question 28 What is a unit vector ? 3.1

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Answer to Question 28 A vector of length 1 unit

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

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Answer to Question 29 AD

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**then what is u in component form ?**

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

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Answer to Question 30 a b c U =

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**How do you integrate sin ax ?**

Question 31 How do you integrate sin ax ? 3.2

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

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**How would you differentiate a function like**

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

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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

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

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**(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

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**How do you differentiate an expression like**

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

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**(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

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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

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**(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

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

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Answer to Question 36 Loga xy

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**How do you solve equations of the form**

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

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**(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

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Question 38 What is loga xn equal to ? 3.3

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Answer to Question 38 nloga x

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**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

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Answer to Question 39

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**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

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**(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

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