How do you prove that three 3-D points, A, B and C, are collinear ? Unit 3 Question 1 How do you prove that three 3-D points, A, B and C, are collinear ? 3.1

Answer to Unit 3 Question 1 Prove that vector AB is a multiple of vector BC AB = k BC And state that B is common to both vectors

How do you add or subtract vectors ? Unit 3 Question 2 How do you add or subtract vectors ? 3.1

Answer to Unit 3 Question 2 add or subtract matching components

State the three rules of logs ? Unit 3 Question 3 State the three rules of logs ? 3.3

Answer to Unit 3 Question 3 (i) logaxy = logax + logay (ii) loga = logax – logay (iii) logaxn = nlogax x y

What does ksin(x-a) expand out to? Unit 3 Question 4 What does ksin(x-a) expand out to? 3.4

Answer to Unit 3 Question 4 ksinxcosa-kcosxsina

Unit 3 Question 5 If u = then what is u ? a b c 3.1

Answer to Unit 3 Question 5 work out length √(a2+b2+c2)

Unit 3 Question 6 What does a.a equal ? 3.1

Answer to Unit 3 Question 6

Unit 3 Question 7 Express the equation y=kxn in the form of the equation of a straight line, Y=nX+c. 3.3

Answer to Unit 3 Question 7 logy = nlogx + logk

How do you show that two vectors are perpendicular ? Unit 3 Question 8 How do you show that two vectors are perpendicular ? 3.1

Answer to Unit 3 Question 8 Show that a.b=0 a b

What do you get when you differentiate cosx ? Unit 3 Question 9 What do you get when you differentiate cosx ? 3.2

Answer to Unit 3 Question 9 -sinx

Unit 3 Question 10 What is logax – logay equal to ? 3.3

Answer to Unit 3 Question 10 x loga y

How do you name the angle between a line and a plane ? Unit 3 Question 11 How do you name the angle between a line and a plane ? 3.1

Answer to Unit 3 Question 11 (i) start at end of line (A) (ii) go to where line meets the plane (B) (iii) go to the point on the plane directly under the start of the line (C) (iv) Answer is ABC A B C

Unit 3 Question 12 What is a position vector ? 3.1

Answer to Unit 3 Question 12 A vector which starts at the origin

What do you get when you differentiate sin x ? Unit 3 Question 13 What do you get when you differentiate sin x ? 3.2

Answer to Unit 3 Question 13 cos x

How do you integrate cos ax ? Unit 3 Question 14 How do you integrate cos ax ? 3.2

Answer to Unit 3 Question 14 1/a sin ax + C

What do you get when you differentiate Unit 3 Question 15 What do you get when you differentiate cosax ? 3.2

Answer to Unit 3 Question 15 -asinax

How would you differentiate a function like Unit 3 Question 16 How would you differentiate a function like y = sin ax ? 3.2

Answer to Unit 3 Question 16 dy/dx = acos ax

Unit 3 Question 17 What is logaa equal to ? 3.3 and 1.2

Answer to Unit 3 Question 17

Unit 3 Question 18 What is loga1 equal to ? 3.3 and 1.2

Answer to Unit 3 Question 18

How do you express acosx+bsinx+c in the form kcos(x- α)? Unit 3 Question 19 How do you express acosx+bsinx+c in the form kcos(x- α)? 3.4

Answer to Unit 3 Question 19 (i) expand brackets and equate like terms (ii) find k =√(a2+b2) (iii) identify quadrant α is in (iv) find α using tanα = sinα cosα S A T C

How do you solve an equation of the form acosx + bsinx + c=0 ? Unit 3 Question 20 How do you solve an equation of the form acosx + bsinx + c=0 ? 3.4

Answer to Unit 3 Question 20 Change acosx+bsinx into Rcos(x- a) rearrange and solve