C4

Question 1

sin2A = ?

Answer 1

2sinAcosA

Question 2

cos2A = ?

Answer 2

cos²A - sin²A
2cos²A - 1
1 - 2sin²A

Question 3

tan2A = ?

Answer 3

2tanA/(1 - tan²A)

Question 4

sin4x = ?

Answer 4

2sin2xcos2x

Question 5

d/dx sinax = ?

Answer 5

acosax

Question 6

d/dx cosax = ?

Answer 6

-asinax

Question 7

d/dx sinx = ?

Answer 7

cosx

Question 8

d/dx cosx = ?

Answer 8

-sinx

Question 9

d/dx sin(ax + b) = ?

Answer 9

acos(ax+b)

Question 10

d/dx cos(ax + b) = ?

Answer 10

-asin(ax +b)

Question 11

How would you differentiate sin²x?

Answer 11

Chain rule of (sinx)²
to obtain 2⋅(d/dx sinx)⋅(sinx)²⁻¹
= 2⋅cosx⋅sinx
= sin2x

Question 12

How would you differentiate cos²x?

Answer 12

Chain rule of (cosx)²
to obtain 2⋅(d/dx cosx)⋅(cosx)²⁻¹
= 2⋅-sinx⋅cosx
= -2sinxcosx
= - sin2x

Question 13

How do you make parametric equations cartesian?

Answer 13

Eliminate the parameter

Question 14

dy/dx of a parametric equation?

Answer 14

dy/dx = dy/dt ⋅ dt/dx = g’(t)/f’(t)

Question 15

Implicit: d/dx y = ?

Answer 15

dy/dx

Question 16

Implicit: d/dx y² = ?

Answer 16

2y⋅dy/dx

Question 17

Implicit: d/dx xy = ?

Answer 17

u = x, v = y
u' = 1, v' =dy/dx

∴ d/dx xy = vu’ + uv’
∴ d/dx xy = x⋅dy.dx + y

Question 18

∫ sinax dx = ?

Answer 18

-1/a cosax + c

Question 19

∫ cosax dx = ?

Answer 19

1/a sinax + c

Question 20

∫ sec²ax dx = ?

Answer 20

1/a tanax + c

Question 21

∫ kf’(x)/f(x) dx = ?

Answer 21

kln|f(x)| + c

Question 22

Formula for Integration by parts?

Answer 22

∫ uv’ dx = uv - ∫ vu’ dx

where u is a value that will differentiate to a constant, or is lnx

Question 23

Steps for Integration by substitution?

Answer 23

|> u = f(x)
|> Make obvious substitution (change f(x) for u)
|> Differentiate u = f(x) and rearrange for dx
|> Rearrange u = f(x) for x=
|> Integrate with respect to u
|> Change limits
|> Replace u with f(x)

Question 24

a/b ⋅ c/d = ?

Answer 24

ac/bd

Question 25

a/b ÷ c/d = ?

Answer 25

ad/bc

Question 26

Binomial Expansion formula?

Answer 26

(1 + x)ⁿ = 1 + nx + n(n-1)/2! ⋅ x² + n(n-1)(n-2)/3! ⋅ x³

Question 27

What form must binomial expansions be in?

Answer 27

MUST be in the form (1 + x)ⁿ

Question 28

When do binomial expansions converge?

Answer 28

Converge as |x| < 1

Question 29

How would you write (3x² - 1)/(x-1)(2x-1)² as partial fractions?

Answer 29

= A/x-1 + B/(2x-1) + C(2x-1)²

let x = 1 [cancels (x-1)], 
A = 2

let x = 1/2 [cancels (2x-1)], 
-1/2⋅C = -1/4, 
C = 1/2

let x = 0 [takes note of repeated factor], 
-1 = A + B - C. 
-1 = 2 + B - 1/2
B = -5/2

Question 30

Magnitude of unit vectors?

Answer 30

1

Question 31

Length of unit vectors?

Answer 31

1

Question 32

magnitude/length of unit vectors?

Answer 32

1

Question 33

Where do position vectors start from?

Answer 33

The origin

Question 34

-AB-> = ?

Answer 34

-OB-> − -OA->

Question 35

for x = ai + bj + ck, |x| = ?

Answer 35

√(a² + b² + c²)

magnitude/length

Question 36

Features of a vector line?

Answer 36

have a position vector, a direction vector and a parameter

in the form: position vector + parameter ⋅ direction vector

Question 37

When are vector lines parallel?

Answer 37

Direction vectors are multiples of one another

Question 38

When are vector lines skew?

Answer 38

If you can prove they are not parallel, and do not intersect

Question 39

where do the vector lines
r = (ai + tpi) + (bj + tqj) + (ck + trk)
&
r = (di + smi) + (ej + snj) + (ek + sok)
intersect?

Answer 39

a + tp = d + sm
b + tq = e + sn
c + tr = f + so

Question 40

Equation for scalar product?

Answer 40

(-a->) ⋅ (-b->) = |a||b|cosθ

where θ is the angle between the vectors

Question 41

If lines are perpendicular, (-a->) ⋅ (-b->) = ?

Answer 41

0

because cos90 = 0

Question 42

How to calculate the shortest distance between a vector and a point?

Answer 42

(-a->) ⋅ (-b->) = 0