Pure Year 2

Question 1

What is an improper algebraic fraction?

Answer 1

One whose numerator has a degree equal to or larger than the denominator

Question 2

What is a mapping? (Functions)

Answer 2

A function if every input has a distinct output.

Question 3

What counts as a function and what doesn’t?

Answer 3

One-to-one functions
Many-to-one functions
But One-to-many is not a function!

Question 4

What is the relationship between the graph of f(x) and f⁻¹(x)?

Answer 4

f⁻¹(x) is a reflection of f(x) in the line y=x

Question 5

Describe the transformation f(x+a)

Answer 5

Horizontal translation of -a

Question 6

Describe the transformation f(x) + a

Answer 6

Vertical translation of a

Question 7

Describe the transformation f(ax)

Answer 7

Horizontal stretch of scale factor 1/a

Question 8

Describe the transformation af(x)

Answer 8

Vertical stretch of scale factor a

Question 9

Describe the transformation -f(x)

Answer 9

Reflects f(x) in the x-axis

Question 10

Describe the transformation f(-x)

Answer 10

Reflects f(x) in the y-axis

Question 11

What is the formula for the nth term of an arithmetic sequence?

Answer 11

uₙ = a + (n-1) d

Question 12

What is the formula for the nth term of an geometric sequence?

Answer 12

uₙ = arⁿ⁻¹

Question 13

What is the sum to infinity formula for a geometric series? What is the condition?

Answer 13

Series must be converging, |r|<1

S∞ = a /(1-r)

Question 14

What is the sum of series formula for arithmetic series?

Answer 14

Sₙ = 0.5n(2a + (n-1)d)
or
Sₙ = 0.5n (a + l)

Question 15

What is the sum of series formula for geometric series?

Answer 15

Sₙ = a(1-r ⁿ)/(1-r)

Question 16

When is a sequence increasing?

Answer 16

If uₙ₊₁ > uₙ

Question 17

When is a sequence decreasing?

Answer 17

If uₙ₊₁ < uₙ

Question 18

When is a sequence periodic? What is the order of a periodic sequence?

Answer 18

If the terms repeat in a cycle. For a periodic sequence there is an integer k such that uₙ₊ₖ = uₙ for all n ε ℕ. The value k is the order of the sequence

Question 19

When is the binomial expansion f (1+bx)ⁿ valid, when n is negative or a fraction?

Answer 19

|bx| < 1

or |x| < 1/ |b|

Question 20

When is the binomial expansion f (a+bx)ⁿ valid, when n is negative or a fraction?

Answer 20

|ba/x| < 1

or |x| < |a/b|

Question 21

What is a sector? A segment? (of a circle)

Answer 21

Sector is like a pizza slice

A segment is the area of the circle when the circle is cut by a chord

Question 22

What is the formula for the area of a segment?

Answer 22

A = 0.5 r² (θ - sinθ)

Question 23

What are the small angle approximations?

Answer 23

sinθ ≈ θ
tanθ ≈ θ
cosθ ≈ 1 - θ²/2

Question 24

What does the graph of y = sec x look like? What is the domain and range? Period?

Answer 24

Symmetrical about y-axis
u and n shapes. Asymptotes at π/2, 3π/2 etc.
Domain: all real values of x. Except x ≠ π/2, 3π/2, … or any odd multiple of π/2
Range y ≤ -1, y ≥ 1
Period: 2π

Question 25

What does the graph of y = cosec x look like? What is the domain and range? Period?

Answer 25

u and n shapes. Asymptotes at π, 2π etc. Domain: all real values of x. Except x ≠ π, 2π, ... or any integer multiple of π Range y ≤ -1, y ≥ 1 Period: 2π

Question 26

What does the graph of y = cot x look like? What is the domain and range? Period?

Answer 26

Period of π rads. Vertical asymptotes at 0, π, 2π etc. Domain: All real values of x. Except x ≠ π, 2π, ... or any integer multiple of π Range: All real values of y

Question 27

State the identity including tan and 1

Answer 27

1 + tan²x ≡ sec²x

Question 28

State the identity including cot and 1

Answer 28

1 + cot²x ≡ cosec²x

Question 29

What is the domain and range of y = arcsin x?

Answer 29

Domain: -1 ≤ x ≤ 1 Range: -π/2 ≤ arcsin x ≤ π/2

Question 30

What is the domain and range of y = arccos x?

Answer 30

Domain: -1 ≤ x ≤ 1 Range: 0 ≤ arccos x ≤ π

Question 31

What is the domain and range of y = arctan x?

Answer 31

Domain: all real values of x Range: -π/2 ≤ arctan x ≤ π/2

Question 32

What is the double angle formula for cos(2A)?

Answer 32

Cos(2A) ≡ cos²A - sin²A ≡ 2cos²A -1 ≡ 1 - 2sin²A

Question 33

What is the double angle formula for sin(2A)?

Answer 33

sin(2A) = 2sinAcosA

Question 34

What is the double angle formula for tan(2A)?

Answer 34

2tanA/(1-tan²A)

Question 35

What is the harmonic form of asinx ± bcosx?

Answer 35

``` R sin (x ± α) R > 0 0 < α < π/2 R cos α = a R sin α = b R = √(a² + b²) ```

Question 36

What is the harmonic form of acosx ± bsinx?

Answer 36

R cos (x ∓ α)

Question 37

What is the chain rule?

Answer 37

dy/dx = dy/du * du/dx

Question 38

What is the product rule?

Answer 38

y=uv then | dy/dx = u dv/dx + v du/dx

Question 39

What is the quotient rule?

Answer 39

``` y = u/v then dy/dx = (v du/dx - u dv/dx ) / v² ```

Question 40

When is a function concave?

Answer 40

When f''(x) ≤ 0

Question 41

When is a function convex?

Answer 41

When f''(x) ≥ 0

Question 42

Where is a point of inflection?

Answer 42

A point at which f''(x) changes sign

Question 43

Describe the conclusion when you have shown that there is a root in the interval [a, b]

Answer 43

If the function is CONTINUOUS on the interval [a, b] and f(a) and f(b) have opposite signs, then f(x) has at least one root x, which satisfies a < x < b

Question 44

What is the Newton-Raphson formula?

Answer 44

xₙ₊₁ = xₙ - f(xₙ)/f'(xₙ)

Question 45

What is the trapezium rule?

Answer 45

∫ᵇₐ y dx ≈ ½ h (y₀ + 2( y₁ + y₂ + ... + yₙ₋₁) + yₙ) ``` h = (b-a)/n yᵢ = f(a + hᵢ) ```

Question 46

How can you rewrite dy/dx = f(x)g(y) ?

Answer 46

∫ 1/g(y) dy = ∫ f(x) dx

Question 47

If 𝗮 = x𝗶 + y𝗷 + z𝗸 makes and angle θₓ with the positive x-axis, how do you calculate θₓ?

Answer 47

cos θₓ = x / |a|

Question 48

If 𝗮 = x𝗶 + y𝗷 + z𝗸 makes and angle θᵧ with the positive y-axis, how do you calculate θᵧ?

Answer 48

cos θᵧ = y / |a|

Question 49

If 𝗮 = x𝗶 + y𝗷 + z𝗸 makes and angle θ𝓏 with the positive z-axis, how do you calculate θ𝓏?

Answer 49

cos θ𝓏 = z / |a|