Further Algebra and Roots - Worksheets, Questions and Revision

13 original exam-style questions - 3 pages of questions with a full mark scheme - free printable PDF.

Download PDFJump to mark scheme (page 4)Read the revision guide
« Previous: MatricesNext: Further Calculus »
Revision Library
revisionlibrary.co.uk
A-Level · Further Pure Mathematics - Core Pure

FP.CP3 Further Algebra and Roots

EDEXCEL 9FM0 · Calculator allowed · about 140 minutes
Total Marks
Name: _______________________________    Date: ____ / ____ / ______
Answer ALL questions. Show all your working.
1
The quadratic equation x2 - 6x + 4 = 0 has roots α and β.
(a)Write down the value of α + β.(1)
(b)Write down the value of α * β.(1)
(c)Find the value of α2 + β2.(3)
(d)Which of the following is the quadratic equation, with integer coefficients, whose roots are α + 2 and β + 2?
A) y2 - 10y + 20 = 0
B) y2 + 10y + 20 = 0
C) y2 - 10y - 20 = 0
D) y2 - 2y + 4 = 0(2)
  • A) y2 - 10y + 20 = 0
  • B) y2 + 10y + 20 = 0
  • C) y2 - 10y - 20 = 0
  • D) y2 - 2y + 4 = 0
(Total for Question 1 is 7 marks)
2
The quadratic equation 3x2 - 7x + 2 = 0 has roots α and β.
(a)State the value of α + β and the value of α*β.(2)
(b)Find the value of α2 + β2.(3)
(c)Find, in the form ay2 + by + c = 0 with integer coefficients, the equation whose roots are 2*α + 1 and 2*β + 1.(4)
(Total for Question 2 is 9 marks)
3
The cubic equation x3 - 6x2 + 11x - 6 = 0 has roots α, β and γ.
(a)State the value of α+β+γ, the value of α*β + β*γ + γ*α, and the value of α*β*γ.(3)
(b)Find the value of α2 + β2 + γ2.(3)
(c)Find the value of 1/α + 1/β + 1/γ, and the value of 1/(α*β) + 1/(β*γ) + 1/(γ*α).(4)
(Total for Question 3 is 10 marks)
4
The cubic equation x3 + 4x2 - 3x + 7 = 0 has roots α, β and γ.
(a)State the value of α+β+γ, the value of α*β+β*γ+γ*α, and the value of α*β*γ.(3)
(b)Find, in the form y3 + by2 + cy + d = 0, the equation whose roots are -α, -β and -γ.(4)
(c)Explain how the equation with roots -α, -β, -γ can be written down directly from x3+4x2-3x+7=0, without carrying out a full substitution.(2)
(Total for Question 4 is 9 marks)
5
The cubic equation 2x3 + 3x2 - 5x + 1 = 0 has roots α, β and γ.
(a)State the value of α+β+γ, the value of α*β+β*γ+γ*α, and the value of α*β*γ.(3)
(b)Find, in the form y3 + by2 + cy + d = 0 with integer coefficients, the equation with roots 1/α, 1/β and 1/γ.(5)
(c)Show that 1/α + 1/β + 1/γ = 5.(2)
(Total for Question 5 is 10 marks)
6
The quartic equation x4 - 2x3 - 7x2 + 8x + 12 = 0 has roots α, β, γ and delta.
(a)State the value of each of: α+β+γ+delta; the sum of the products of the roots taken two at a time; the sum of the products of the roots taken three at a time; α*β*γ*delta.(4)
(b)Find the value of α2+β2+γ2+delta2.(4)
(c)Find the value of 1/α + 1/β + 1/γ + 1/delta.(3)
(Total for Question 6 is 11 marks)
7
The quartic equation x4 - 8x3 + 18x2 - 10x + 3 = 0 has roots that are to be examined using the substitution x = y + 2.
(a)Explain why the substitution x = y + 2, rather than some other shift x = y + k, removes the term in y3 from the resulting quartic in y.(2)
(b)Show that the substitution x = y + 2 transforms x4 - 8x3 + 18x2 - 10x + 3 = 0 into y4 - 6y2 - 2y + 7 = 0.(6)
(Total for Question 7 is 8 marks)
8
The cubic equation x3 - 12x2 + kx - 28 = 0, where k is a constant, has three roots that are in arithmetic progression.
(a)By writing the three roots as (a-d), a and (a+d), explain why a = 4 and hence why x = 4 must satisfy the equation.(2)
(b)Find the value of k.(3)
(c)Hence find the other two roots of the equation.(4)
(Total for Question 8 is 9 marks)
9
Given that 2+i is a root of x3 - 7x2 + 17x - 15 = 0, and that the equation has real coefficients.
(a)State, with a reason, another root of the equation.(2)
(b)Find the third root of the equation.(4)
(c)Hence express x3 - 7x2 + 17x - 15 as the product of a linear factor and a quadratic factor, both with real coefficients.(3)
(Total for Question 9 is 9 marks)
10
The quadratic equation 2x2 + (k-3)x + (k+1) = 0, where k is a constant, has roots α and β such that α = 2*β.
(a)Write down expressions, in terms of k, for α + β and for α*β.(2)
(b)Using α = 2*β, show that k2 - 15k = 0.(5)
(c)Hence find the possible value(s) of k.(2)
(Total for Question 10 is 9 marks)
11
The cubic equation x3 - 3x2 + 5x - 4 = 0 has roots α, β and γ.
(a)State the value of α+β+γ, the value of α*β+β*γ+γ*α, and the value of α*β*γ.(2)
(b)Show that the cubic equation with roots α2, β2 and γ2 is y3 + y2 + y - 16 = 0.(6)
(Total for Question 11 is 8 marks)
12
A cubic equation has roots α, β and γ, with α+β+γ = p, α*β+β*γ+γ*α = q and α*β*γ = r.
(a)Prove that α3 + β3 + γ3 = p3 - 3pq + 3r.(5)
(b)Hence find the value of α3 + β3 + γ3 for the equation x3 - 5x2 + 6x - 2 = 0.(3)
(Total for Question 12 is 8 marks)
13
The quartic equation x4 - 15x3 + 70x2 - 120x + k = 0, where k is a constant, has four roots that form a geometric progression a, ar, ar2, ar3 (with r > 1).
(a)State, in terms of a and r, expressions for m = a*(ar3) (the product of the outer pair of roots), S1 = a + ar3 (the sum of the outer pair), and S2 = ar + ar2 (the sum of the inner pair). Explain why the product of the inner pair, (ar)(ar2), is also equal to m.(2)
(b)By writing the quartic as (x2 - S1*x + m)(x2 - S2*x + m), find the value of k and determine the four roots of the equation.(8)
(Total for Question 13 is 10 marks)
Mark scheme · FP.CP3 Further Algebra and Roots

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Question 11

Question 12

Question 13