CAIE A-Level Mathematics 2.1 Algebra Question Bank

Practise solving modulus equations and inequalities and applying polynomial division, factor and remainder theorems to determine coefficients, factors and real roots.

Exam points

split a modulus relation at its critical points and combine only intervals satisfying all conditions
divide a polynomial by a linear or quadratic divisor and retain the stated remainder
apply p(a) = 0 or p(a) = r to form coefficient equations and complete the factorisation

Covered syllabus points

2.1.1—Modulus equations 2.1.2—Polynomial division 2.1.3—Factor and remainder theorems

[Maximum number: 7]

(a)

Sketch, on the same diagram, the graphs of y=|3 x-5| and y=2 x+7.

[ 2 ]

(b)

Solve the equation |3 x-5|=2 x+7.

[ 3 ]

Solve 3 x-5=2 x+7 to obtain x=12

B1

Attempt solution of linear equation where signs of 3 x and 2 x are
different

M1

3 x-5=-2 x-7 OE.

Obtain $x=-\frac{2}{5}$

A1

Alternative solution for question 4(b)

State or imply non-modulus equation $(3 x-5)^{2}=(2 x+7)^{2}$

B1

Must be working with $(3 x-5)^{2}=(2 x+7)^{2}$ .

Attempt solution of 3-term quadratic equation

M1

Obtain $-\frac{2}{5}$ and 12

A1

(c)

Hence solve the equation $\left|3^{y+1}-5\right|=2 \times 3^{y}+7$ , giving your answer correct to 3 significant figures.

[ 2 ]

Apply logarithms and use power law for $3^{y}=k$ where k>0 or
correct equivalent

M1

Using their positive answer from part (b)
or greater accuracy; and no other values.

Obtain 2.26

A1

[Maximum number: 5]

(a)

Find the quotient when $x^{4}-32 x+55$ is divided by $(x-2)^{2}$ and show that the remainder is 7 .

[ 3 ]

Carry out division at least as far as $x^{2}+k x$ or equivalent ...
M1
OE, e.g. comparing coefficients with coefficient of $x^{2}$ equal to 1
and attempt at a second coefficient.
Obtain quotient $x^{2}+4 x+12$
A1
Confirm remainder is 7
A1
AG

(b)

Factorise $x^{4}-32 x+48$ .

[ 2 ]

Include $(x-2)^{2}$ as a factor
M1
Must be a product of factors only
SC B1 for $\left(x^{2}-4 x+4\right)\left(x^{2}+4 x+12\right)$
Conclude $(x-2)^{2}\left(x^{2}+4 x+12\right)$
A1
isw any attempt to factorise the quotient.

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Covered syllabus points

2.1 Algebra question 1

Question (a)

Question (b)

Question (c)

2.1 Algebra question 2

Question (a)

Question (b)