Maths Equations

Question 1

how to prove that there are two distinct real roots

Answer 1

b^2−4ac > 0

Question 2

How to prove that there are equal roots

Answer 2

b^2−4ac = 0

Question 3

How to prove that there are no real roots

Answer 3

b^2−4ac < 0

Question 4

Factor theorem

Answer 4

If f(a) = 0, then (x - a) is a factor of f(x)
If f(b/a) = 0 then (ax - b) is a factor of f(x)

Question 5

Gradient between 2 points

Answer 5

(y2-y1)/(x2-x1)

Question 6

Midpoint

Answer 6

((x1+x2/2) , (y1+y2/2))

Question 7

distance between two points

Answer 7

square root of (x2 - x1)^2 + (y2 - y1)^2

Question 8

equation of a circle

Answer 8

(x - a)^2 + (y - b)^2 = r^2
where (a,b) is the centre and r is the radius

Question 9

how to prove a function is increasing

Answer 9

dy/dx > 0

Question 10

how to prove a function is decreasing

Answer 10

dy/dx < 0

Question 11

prove there is a stationary point / turning point / maximums / minimums

Answer 11

dy/dx = 0

Question 12

prove a point is a maximum

Answer 12

d^2y/dx^2 < 0

Question 13

prove a point is a minimum

Answer 13

d^2y/dx^2 > 0

Question 14

prove a graph is concave

Answer 14

d^2y/dx^2 < 0

Question 15

prove a graph is convex

Answer 15

d^2y/dx^2 > 0

Question 16

prove there is a point of inflection

Answer 16

d^2y/dx^2 = 0
(d^2y/dx^2 = 0 doesn’t guaruntee a point of inflection)

Question 17

chain rule

Answer 17

dy/dx = dy/du x du/dx

Question 18

the product rule

Answer 18

dy/dx = f’(x)g(x) + f(x)g’(x)

Question 19

cosine rule

Answer 19

a^2 = b^2 + c^2 - 2bccosA
cosA = (b^2 + c^2 - a^2) / (2bc)

Question 20

sine rule

Answer 20

a/sinA = b/sinB = c/sinC

Question 21

Area of a triangle (trig)

Answer 21

(1/2)absinC

Question 22

Arc length

Answer 22

rθ
(if θ is in radians)

Question 23

Sector area

Answer 23

(1/2)(r^2)θ
(if θ is in radians)

Question 24

converting radians to degrees

Answer 24

π radians = 180 degrees

Question 25

tanx =

Answer 25

sinx/cosx

Question 26

secx =

Answer 26

1/cosx

Question 27

cosecx =

Answer 27

1/sinx

Question 28

cotx =

Answer 28

1/tanx
cosx/sinx

Question 29

sin2x =

Answer 29

2sinxcosx

Question 30

cos2x =

Answer 30

cos^2x - sin^2x
2cos^2x - 1
1- 2sin^2x

Question 31

tan2x =

Answer 31

2tanx / (1- tan^2x)

Question 32

sin^2x =

Answer 32

1 - cos^2x

Question 33

cos^2x =

Answer 33

1 - sin^2x

Question 34

sec^2x =

Answer 34

tan^2x + 1

Question 35

tan^2x =

Answer 35

sec^2x - 1

Question 36

cosec^2x =

Answer 36

1 + cot^2x

Question 37

cot^2x =

Answer 37

cosec^2x - 1

Question 38

magnitude of vector a (when a = xi + yj +zk)

Answer 38

square root of x^2 + y^2 + z^2

Question 39

unit vector in direction of a (when a = xi + yj +zk)

Answer 39

(1 / (square root of x^2 + y^2 + z^2)) x (xi + yj +zk)

Question 40

loga(b) = c

Answer 40

a^c = b

Question 41

loga(xy)

Answer 41

= loga(x) + loga(y)

Question 42

loga(x/y) =

Answer 42

loga(x) - loga(y)

Question 43

loga(x)^y

Answer 43

yloga(x)