Practise

Question bank

Original exam-style questions, mapped to your syllabus. Answers are checked for mathematical equivalence and every attempt updates your mastery.

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2730 questions · page 84 of 137 · choose a topic above to narrow them down

SA-9709-P3-AL2-0013 marksL2 StandardCalculatorSimple

The solution of

|3x+2|\ge4

x\le a

x\ge b

. Find

a

and

b

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-AL3-0014 marksL2 StandardCalculatorDivision

Find the quotient and the remainder when

2x^3+3x^2-5x+6

is divided by

(x+3)

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-AL4-0013 marksL2 StandardCalculatorLinear factors

Express

\dfrac{7x-1}{(x+1)(2x-1)}

in the form

\dfrac{A}{x+1}+\dfrac{B}{2x-1}

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-AL4-0024 marksL2 StandardCalculatorLinear factors

Express

\dfrac{2x^2-x+3}{x(x-1)(x+1)}

in the form

\dfrac{A}{x}+\dfrac{B}{x-1}+\dfrac{C}{x+1}

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-AL4-0032 marksL2 StandardCalculatorRepeated factors

Express

\dfrac{2x+1}{(x-1)^2}

in the form

\dfrac{A}{x-1}+\dfrac{B}{(x-1)^2}

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-AL5-0013 marksL2 StandardCalculatorExpanding

Expand

(1+3x)^{-2}

in ascending powers of

x

, up to and including the term in

x^2

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-AL5-0031 markL2 StandardCalculatorValidity

The expansion of

(2-3x)^{-1}

in ascending powers of

x

is valid for

|x|<k

. State

k

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-CX-0023 marksL2 StandardCalculatorModulus–argument

Find the modulus and the exact argument of

z=-1+\sqrt3\,i

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-CX-0033 marksL2 StandardCalculatorModulus–argument

Find the modulus of

z=-3-4i

, and its argument in radians correct to 3 significant figures, with

-\pi<\arg z\le\pi

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-CX-0072 marksL2 StandardCalculatorProducts and quotients

Given

z=2e^{i\pi/3}

and

w=3e^{i\pi/6}

, find

zw

in the form

x+iy

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-CX-0082 marksL2 StandardCalculatorProducts and quotients

With

z=2e^{i\pi/3}

and

w=3e^{i\pi/6}

, find the modulus and argument of

\dfrac zw

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-CX-0093 marksL2 StandardCalculatorRegions

The complex number

z

satisfies

|z-3-4i|=2

. Find the greatest and least possible values of

|z|

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-CX1-0033 marksL2 StandardCalculatorArithmetic

Express

\dfrac{5+i}{2-3i}

in the form

x+iy

, giving

x

and

y

as fractions.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-CX2-0032 marksL2 StandardCalculatorModulus–argument

Find the exact argument of

-3-3i

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-CX3-0013 marksL2 StandardCalculatorPolar form

Express

-2+2\sqrt3\,i

in the form

re^{i\theta}

, where

r>0

and

-\pi<\theta\le\pi

. State

r

and

\theta

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-CX4-0033 marksL2 StandardCalculatorPolynomial roots

2-i

is a root of

z^2+bz+c=0

, where

b

and

c

are real. Find

b

and

c

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-CX5-0032 marksL2 StandardCalculatorLoci

The locus

\arg(z-2)=\dfrac\pi3

is part of a straight line. Find the gradient of this line, giving an exact value.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-CX5-0042 marksL2 StandardCalculatorRegions

For points satisfying

|z-2i|\le1

, find the greatest and least values of

|z|

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-D32-0012 marksL2 StandardCalculatorProduct rule

Differentiate

x^3\ln x

with respect to

x

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-D33-0022 marksL2 StandardCalculatorImplicit

Find the gradient of the curve

x^2+2y^2=9

at the point

(1,2)

Salam Academy Original Practice · Proposed mark scheme