A LEVEL FURTHER MATHS EXAM WITH
QUESTIONS AND VERIFIED ANSWERS||
ALREADY GRADED A+|| GUARANTEED PASS||
LATEST VERSION 2026
what is the midordinate rulε? - ANSWER-area = strip width * (the sum of the
mid points of the rectangular strips)
what is simpson's rule? - ANSWER-h/3 (sum of the end f(x)s + 4(odd f(x)s) +
2(even F(x)s)
with relation to simpson's rule, what do we mean by end, odd and even f(x)s? -
ANSWER-you label all of your f(x)s from left to right starting with 0
end f(x)s are the 0th and last term
odd f((x)s are the 1st, 3rd ect term
even f(x)s are the 2nd, 4th ect term
What is De Moivre's Theorem? - ANSWER-r(cosθ+isinθ)^n =
(r^n)*(cosnθ+isinθ)
what is (cos(π/12)+isin(π/12)^18 =? - ANSWER-cos(3π/2)+isin(3π/2)
what is the general equation for the roots of the equation z^n = 1 ? - ANSWER-
cos(2kπ/n)+isin(2kπ/n)
n = order of the equation
,k = 0,1,2,3......n-1
what is the sum of the nth roots of unity ? - ANSWER-0
what are the roots of unity? - ANSWER-the answers to equation z^n = 1
what shape will n roots of unity make on an argand diagram? - ANSWER-a
regular n sided polygon inscribed in the unit circle
how do we find the roots of z^n = a ? - ANSWER-
r^1/n(cos(Φ+2kπ/2)+ισιν(Φ+2kπ/2))
where Φ = the argument of a
where n.= 0,1,2,3.....n-1
how woould we express cos5θ in terms of cosθ - ANSWER-using de moivre's
theorem we expand (cosθ+isinθ)^5
we then equate real and imaginary parts
express cos5θ in terms of cosθ - ANSWER-15cos^5(θ) - 20cos^3(θ) + 5cosθ
z^n+z^-n = ? - ANSWER-2cos(nθ)
z^n-z^-n /2i - ANSWER-sin(nθ)
express cos^5(θ) in terms of multiple angles - ANSWER-(2cosθ)^5 = (z+z^-
1)^5
expand both sides and equate
remember to double your right hand side as z^n+z^-n = 2cosnθ
final answer:
,(cos5θ+5cos3θ+10cosθ)/16
write r(cosθ+isinθ) in exponential form - ANSWER-re^iθ
write cosnθ+isinθ in exponential form - ANSWER-e^inθ
write r(cosθ-isinθ) in exponential form - ANSWER-re^-iθ
cos(-θ) =?
sin(-θ) = ? - ANSWER-cosθ
-sinθ
A sofa manufacturer makes two different models, x and y. Sofa x makes £20
profit, and y makes £40. If they make 200 of each model, how much profit will
they get? - ANSWER-£12000
A _____ region in a linear programming graph shows all of the possible
solutions that lie within the set constraints. - ANSWER-feasible
The optimum answer to a linear programming problem will always be at (or
very near to) a _____ of the shape called the feasible region. - ANSWER-vertex
A sofa manufacturer makes two different models, x and y. Sofa x makes £20
profit, and y makes £40. The vertices of the feasible region are (100,100),
(100,300) and (300,100). What is the maximum profit that they can make? -
ANSWER-£14000
In a linear programming you have three variables: x, y and z. To be able to draw
a graph you must first put z in terms of x or y. To do this you may need to
_____ an equation to find an expression for z in terms of x or y (or both). -
ANSWER-rearrange
, The objective function can be drawn on the graph as an _____ _____. -
ANSWER-objective line
In the _____ sort, compare the first and second numbers, and swap them if
necessary. Then compare second and third, third and fourth, and so on. At the
end of each pass, the last number compared on that pass will be in the correct
place. - ANSWER-bubble
You are using the bubble sort algorithm to list 2,7,5,3,1,4 in descending order.
What does the list look like after the first two comparisons? - ANSWER-
7,5,2,3,1,4
In the _____ sort, compare the first and second numbers and swap if necessary.
Then compare the second and third numbers, and if you swap them compare the
first and second numbers again. Then continue... - ANSWER-shuttle
What does the list 8,4,1,7,3,6 look like after the second pass of a shuttle sort,
putting them into ascending order? - ANSWER-1,4,8,7,3,6
In the _____ sort (named after its inventor) the list is divided into INT(n/2)
sublists with each pass, and these sublists are then shuttle sorted. - ANSWER-
Shell
The list 2,5,3,9,6,1,7 is Shell sorted. How many sublists is this list divided into
in the first pass? - ANSWER-3
In the _____ algorithm, one number is chosen as a "pivot", and the rest of the
numbers divided into two lists: those smaller than the pivot or those not smaller
than the pivot. Then the process repeats for each list until no sublist has more
than one number in it. - ANSWER-quicksort
QUESTIONS AND VERIFIED ANSWERS||
ALREADY GRADED A+|| GUARANTEED PASS||
LATEST VERSION 2026
what is the midordinate rulε? - ANSWER-area = strip width * (the sum of the
mid points of the rectangular strips)
what is simpson's rule? - ANSWER-h/3 (sum of the end f(x)s + 4(odd f(x)s) +
2(even F(x)s)
with relation to simpson's rule, what do we mean by end, odd and even f(x)s? -
ANSWER-you label all of your f(x)s from left to right starting with 0
end f(x)s are the 0th and last term
odd f((x)s are the 1st, 3rd ect term
even f(x)s are the 2nd, 4th ect term
What is De Moivre's Theorem? - ANSWER-r(cosθ+isinθ)^n =
(r^n)*(cosnθ+isinθ)
what is (cos(π/12)+isin(π/12)^18 =? - ANSWER-cos(3π/2)+isin(3π/2)
what is the general equation for the roots of the equation z^n = 1 ? - ANSWER-
cos(2kπ/n)+isin(2kπ/n)
n = order of the equation
,k = 0,1,2,3......n-1
what is the sum of the nth roots of unity ? - ANSWER-0
what are the roots of unity? - ANSWER-the answers to equation z^n = 1
what shape will n roots of unity make on an argand diagram? - ANSWER-a
regular n sided polygon inscribed in the unit circle
how do we find the roots of z^n = a ? - ANSWER-
r^1/n(cos(Φ+2kπ/2)+ισιν(Φ+2kπ/2))
where Φ = the argument of a
where n.= 0,1,2,3.....n-1
how woould we express cos5θ in terms of cosθ - ANSWER-using de moivre's
theorem we expand (cosθ+isinθ)^5
we then equate real and imaginary parts
express cos5θ in terms of cosθ - ANSWER-15cos^5(θ) - 20cos^3(θ) + 5cosθ
z^n+z^-n = ? - ANSWER-2cos(nθ)
z^n-z^-n /2i - ANSWER-sin(nθ)
express cos^5(θ) in terms of multiple angles - ANSWER-(2cosθ)^5 = (z+z^-
1)^5
expand both sides and equate
remember to double your right hand side as z^n+z^-n = 2cosnθ
final answer:
,(cos5θ+5cos3θ+10cosθ)/16
write r(cosθ+isinθ) in exponential form - ANSWER-re^iθ
write cosnθ+isinθ in exponential form - ANSWER-e^inθ
write r(cosθ-isinθ) in exponential form - ANSWER-re^-iθ
cos(-θ) =?
sin(-θ) = ? - ANSWER-cosθ
-sinθ
A sofa manufacturer makes two different models, x and y. Sofa x makes £20
profit, and y makes £40. If they make 200 of each model, how much profit will
they get? - ANSWER-£12000
A _____ region in a linear programming graph shows all of the possible
solutions that lie within the set constraints. - ANSWER-feasible
The optimum answer to a linear programming problem will always be at (or
very near to) a _____ of the shape called the feasible region. - ANSWER-vertex
A sofa manufacturer makes two different models, x and y. Sofa x makes £20
profit, and y makes £40. The vertices of the feasible region are (100,100),
(100,300) and (300,100). What is the maximum profit that they can make? -
ANSWER-£14000
In a linear programming you have three variables: x, y and z. To be able to draw
a graph you must first put z in terms of x or y. To do this you may need to
_____ an equation to find an expression for z in terms of x or y (or both). -
ANSWER-rearrange
, The objective function can be drawn on the graph as an _____ _____. -
ANSWER-objective line
In the _____ sort, compare the first and second numbers, and swap them if
necessary. Then compare second and third, third and fourth, and so on. At the
end of each pass, the last number compared on that pass will be in the correct
place. - ANSWER-bubble
You are using the bubble sort algorithm to list 2,7,5,3,1,4 in descending order.
What does the list look like after the first two comparisons? - ANSWER-
7,5,2,3,1,4
In the _____ sort, compare the first and second numbers and swap if necessary.
Then compare the second and third numbers, and if you swap them compare the
first and second numbers again. Then continue... - ANSWER-shuttle
What does the list 8,4,1,7,3,6 look like after the second pass of a shuttle sort,
putting them into ascending order? - ANSWER-1,4,8,7,3,6
In the _____ sort (named after its inventor) the list is divided into INT(n/2)
sublists with each pass, and these sublists are then shuttle sorted. - ANSWER-
Shell
The list 2,5,3,9,6,1,7 is Shell sorted. How many sublists is this list divided into
in the first pass? - ANSWER-3
In the _____ algorithm, one number is chosen as a "pivot", and the rest of the
numbers divided into two lists: those smaller than the pivot or those not smaller
than the pivot. Then the process repeats for each list until no sublist has more
than one number in it. - ANSWER-quicksort
