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Remainder theorem
Question Paper 1
Level
Subject
Exam Board
Module
Topic
Sub Topic
Booklet
A Level
Mathematics (Pure)
AQA
Core 4
Algebra
Remainder theorem
Question Paper 1
Time Allowed:
88 minutes
Score:
/74
Percentage:
/100
Grade Boundaries:
A*
>85%
A
777.5%
B
C
D
E
U
70%
62.5%
57.5%
45%
<45%
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Q1.The polynomial f(x) is defined by f(x) = 2x3 + x2 − 8x − 7.
(a)
Use the Remainder Theorem to find the remainder when f(x) is divided by (2x
+ 1).
(2)
(b)
The polynomial g(x) is defined by g(x) = f(x) + d, where d is a constant.
(i)
Given that (2x + 1) is a factor of g(x), show that g(x) = 2x3 + x2 − 8x − 4.
(1)
(ii)
Given that g(x) can be written as g(x) = (2x + 1)(x2 + a), where a is an
integer, express g(x) as a product of three linear factors.
(1)
(iii)
Hence, or otherwise, show that
are integers.
, where p and q
(3)
(Total 7 marks)
Q2.
(a)
(i)
The polynomial f (x) is defined by f (x) = 9x3 + 18x2 – x – 2.
Use the Factor Theorem to show that 3x + 1 is a factor of f (x).
(2)
(ii)
Express f (x) as a product of three linear factors.
(3)
(iii)
Simplify
.
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(3)
(b)
When the polynomial 9x3 + px2 – x – 2 is divided by 3x – 2, the remainder is –
4.
Find the value of the constant p.
(2)
(Total 10 marks)
The polynomial f(x) is defined by f(x) = 4x3 – 13x + 6.
Q3.
(a)
Find f(–2).
(1)
(b)
Use the Factor Theorem to show that 2x – 3 is a factor of f(x).
(2)
(c)
Simplify
(4)
(Total 7 marks)
Q4.
A botanist is investigating the rate of growth of a certain species of toadstool.
She observes that a particular toadstool of this type has a height of 57 millimetres at
a time 12 hours after it begins to grow.
She proposes the model
h millimetres of
, where A is a constant, for the height
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the toadstool, t hours after it begins to grow.
(a)
Use this model to:
(i)
find the height of the toadstool when t = 0;
(1)
(ii)
show that A = 60, correct to two significant figures.
(2)
(b)
Use the model
(i)
to:
show that the time T hours for the toadstool to grow to a height of 48
millimetres is given by
T = a ln b
where a and b are integers;
(3)
(ii)
show that
;
(3)
(iii)
find the height of the toadstool when it is growing at a rate of 13
millimetres per hour.
(1)
(Total 10 marks)
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Q5.
(a)
The polynomial f(x) is defined by f(x) = 8x3 + 6x2 – 14x – 1.
Find the remainder when f(x) is divided by (4x – 1).
(2)
(b)
The polynomial g(x) is defined by g(x) = 8x3 + 6x2 – 14x + d.
(i)
Given that (4x – 1) is a factor of g(x), find the value of the constant d.
(2)
(ii)
Given that g(x) = (4x – 1)(ax2 + bx + c), find the values of the integers a,
b and c.
(3)
(Total 7 marks)
Q6.
(a)
(i)
The polynomial f(x) is defined by f(x) = 4x3 – 7x – 3.
Find f(–1).
(1)
(ii)
Use the Factor Theorem to show that 2x + 1 is a factor of f(x).
(2)
(iii)
Simplify the algebraic fraction
.
(3)
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(b)
The polynomial g(x) is defined by g(x) = 4x3 – 7x + d. When g(x) is divided by
2x + 1, the remainder is 2. Find the value of d.
(2)
(Total 8 marks)
(a) Use the Remainder Theorem to find the remainder when 3x3 + 8x2 – 3x – 5
is divided by 3x – 1.
Q7.
(2)
(b)
in the form ax2 + bx +
Express
are integers.
, where a, b and c
(3)
(Total 5 marks)
The polynomial f(x) is defined by f(x) = 27x3 – 9x + 2.
Q8.
(a)
Find the remainder when f(x) is divided by 3x + 1.
(2)
(b)
(i)
Show that
.
(1)
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(ii)
Express f(x) as a product of three linear factors.
(4)
(iii)
Simplify
(2)
(Total 9 marks)
The polynomial f(x) is defined by f(x) = 2x3 – 7x2 + 13.
Q9.
(a)
Use the Remainder Theorem to find the remainder when f (x) is divided by
(2x – 3).
(2)
(b)
The polynomial g(x) is defined by g(x) = 2x3 – 7x2 + 13 + d, where d is a
constant.
Given that (2x – 3) is a factor of g(x), show that d = −4.
(2)
(c)
Express g(x) in the form (2x – 3)(x2 + ax + b).
(2)
(Total 6 marks)
Q10.
(a)
Find the remainder when 2x2 + x – 3 is divided by 2x + 1.
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(2)
(b)
Simplify the algebraic fraction
(3)
(Total 5 marks)