PPQ Corrections

Question 1

How do you find the gradient of a curve?

Answer 1

Find f’(x) and sub in point.
Can sub in point to x and y at any point.

Question 2

How do you find the inverse of a matrix (A-1)?

Answer 2

1/ad-bc . (d -b)
(-c a)

Question 3

When is a matrix singular?

Answer 3

When detA=0

Question 4

Proof by induction steps:

Answer 4

1. Goal
2. Let n=1 (LHS=RHS)
3. Assume true for n=k
4. Prove true for n=k+1 (use step 3)
5. Statement

Question 5

Modulus

Answer 5

r=√x²+y²

Question 6

When gcd=1

Answer 6

a & b are co. prime.

Question 7

Euclidean

Answer 7

1365=4299+169
299=1169+130
…

Question 8

How to find a & b when:
100a+50b=30

Answer 8

Start with the 2nd last row.
Work back eliminating the remainders.

Question 9

How to find gcd:

Answer 9

When remainder=0.

Question 10

ln1=?

Answer 10

0

Question 11

Vertical asymtote

Answer 11

Set denominator=0 and solve for x.

Question 12

Horizontal asymtote

Answer 12

exists when the degree of numerator= degree of denominator.

Question 13

Slant asymtote

Answer 13

Degree of numerator is 1 more than denominator.
Use synthetic division.

Question 14

S∞

Answer 14

a/1-r

Question 15

Clockwise matrix rotation.

Answer 15

( cosθ sinθ)
( -sinθ cosθ)

Question 16

Anticlockwise matrix rotation.

Answer 16

(cosθ -sinθ)
( sinθ cosθ)

Question 17

Proof by induction statement:

Answer 17

Since statement is true for n=1, assumed true for n=k, and by assumption true for n=k+1 the statement is true for all n≥1.

Question 18

Odd & even functions

Answer 18

Even: Symmetrical about y-axis
Odd: Not symmetrical about y-axis
Has half turn rotational symmetry

Question 19

Proof by contradiction:

Answer 19

Prove the opposite is false.

Question 20

Rational & Irrational Numbers

Answer 20

Rational: Can be written as fraction in its simplest form.
Irrational: Cannot be written as fraction in its simplest form.

Question 21

How do you find the point of inflection?

Answer 21

When f’‘(x)=0

Question 22

nth term

Answer 22

Un=ar^(n-1)

Question 23

Σ1=?

Answer 23

n

Question 24

Quadratic formula

Answer 24

-b+or-√b²-4ac
x= ——————–
2a

Question 25

Argument

Answer 25

θ=tan-1(y/x)

Question 26

Cartesian form

Answer 26

z=x+iy

Question 27

Polar form

Answer 27

z=r(cosθ+isinθ)

Question 28

Argand diagram:

Answer 28

Y-axis = Imaginary X-axis = Real Point : z=x+iy

Question 29

Redundancy in 2x2 matrices

Answer 29

When 0=0

Question 30

Inconsistency in 2x2 matrices

Answer 30

When 0=number

Question 31

Modulus of a graph

Answer 31

Make everything positive.

Question 32

ERO's

Answer 32

Put matrix in form: ( 1 0 l x ) ( 0 1 l y )

Question 33

Redundancy in 3x3 matrices

Answer 33

( 1 1 1 l 1 ) ( 1 1 1 l 1 ) ( 0 0 0 l 0 )

Question 34

Inconsistency in 3x3 matrices

Answer 34

( 1 1 1 l 1 ) ( 1 1 1 l 1 ) ( 0 0 0 l 1 )

Question 35

I in matrices

Answer 35

I ( 1 0 ) ( 0 1 ) A-1.A=I

Question 36

Trig values

Answer 36

1/cosx=secx 1/sinx=cosecx 1/tanx=cotx

Question 37

f''(x) (parametric eqns)

Answer 37

y''(t)x'(t)-x''(t)y'(t) = ------------------------- ( x'(t) )^3

Question 38

∫tanx .dx

Answer 38

ln(secx) + c

Question 39

Special common forms

Answer 39

∫f'(x)/f(x) = ln(f(x))+c ∫f'(x).f(x) = 1/2 . (f(x))² + c

Question 40

Area between curve and y axis

Answer 40

A= ∫f(y).dy

Question 41

Volume of revolution

Answer 41

Around x-axis: ∫πy².dx Around y-axis: ∫πx².dy

Question 42

Adding bases

Answer 42

e.g Base6: 4+3=7 6.1+1=11

Question 43

Subtract bases:

Answer 43

e.g 11 - 2 base10: 11=10.1+1=11 base7: 11=7.1+1=8 therefore 8-2 in base7

Question 44

Multiplying bases

Answer 44

e.g 5.4=20 base9: 20=9.2+2=22

Question 45

Dividing bases

Answer 45

e.g 13/5 base10: 13=1.10+3 base7: 1.7+3=10 therefore 10/5 in base7

Question 46

First order differential equations:

Answer 46

Move Y's to left and X's to right and integrate both sides.

Question 47

Special common forms

Answer 47

top derivative of bottom = ln If(x)I + c f(ax+b)* = 1/* . (ax+b) . 1/a +c f'(x)f(x)=1/2 (f(x))^2+ c

Question 48

trig values

Answer 48

1-sinx = cosx

Question 49

Sum to infinity exists when...

Answer 49

r<1