CAIE A-Level Mathematics 3.1 Algebra Question Bank

CAIE A-Level Mathematics 3.1 Algebra Question Bank
Cambridge International AS & A Level Mathematics 9709 syllabus for exams in 2028, 2029 and 20302028–2030

Practise Pure 3 algebra skills with modulus graphs, polynomial division, theorem use, partial fractions and binomial expansions.

Exam points

  • solve modulus inequalities using intersections, sign cases or squared forms
  • divide polynomials to find quotients, remainders and values of constants
  • expand binomial expressions and state the valid range using strict inequalities

Covered syllabus points

3.1.1—Modulus equations3.1.2—Polynomial division3.1.3—Factor and remainder theorems3.1.4—Partial fractions3.1.5—Binomial expansion

Question 2

Question 2(a)

(a)

Expand (6−x)(1−2x)−32(6-x)(1-2 x)^{-\frac{3}{2}} in ascending powers of x, up to and including the term in x2x^{2}, simplifying the coefficients.

[ 4 ]

Question 2(b)

(b)

State the set of values of x for which the expansion is valid.

[ 1 ]

Question 3(a)

[Maximum number: 4]

Solve the inequality ∣3x−4∣⩽∣2x+5∣|3 x-4| \leqslant|2 x+5|.

Question 4(a)

[Maximum number: 3]

The polynomial p(x) is defined by

p(x)=x4−10x3+20x2−30x+40.\mathrm{p}(x)=x^{4}-10 x^{3}+20 x^{2}-30 x+40 .

Find the quotient when p(x) is divided by (x2+3)\left(x^{2}+3\right) and show that the remainder is -11.

Question 7

[Maximum number: 6]

The polynomial p(x) is defined by

p(x)=2x4+kx3+kx2+17x+18,\mathrm{p}(x)=2 x^{4}+k x^{3}+k x^{2}+17 x+18,

where k is a constant. It is given that (x+2) is a factor of p(x).

Question 7(a)

(a)

Find the value of k.

It is given that the equation p(x)=0 has exactly two real roots, denoted by α\alpha and β\beta, where α\alpha is an integer and β\beta is not an integer.

[ 2 ]

Question 7(b)

(b)

State the value of α\alpha and show that β\beta satisfies the equation x=−2x−4.53x=\sqrt[3]{-2 x-4.5}.

[ 4 ]

Question 10

[Maximum number: 11]

Let f(x)=x3+2x−11(3+x)(2+x2)\mathrm{f}(x)=\frac{x^{3}+2 x-11}{(3+x)\left(2+x^{2}\right)}.

Question 10(a)

(a)

Express f(x) in partial fractions.

[ 6 ]

Question 10(b)

(b)

Hence obtain the expansion of f(x) in ascending powers of x, up to and including the term in x2x^{2}. [5]

[ 5 ]