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 32 of 137 · choose a topic above to narrow them down

SA-9709-P3-CX2-0022 marksL1 FoundationCalculatorModulus–argument

Find the exact argument of

-2+2i

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SA-9709-P3-CX3-0022 marksL1 FoundationCalculatorPolar form

Express

z=4\left(\cos\dfrac\pi4+i\sin\dfrac\pi4\right)

in the form

x+iy

, giving exact values.

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SA-9709-P3-CX4-0012 marksL1 FoundationCalculatorPolynomial roots

Solve

z^2+6z+13=0

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SA-9709-P3-CX4-0022 marksL1 FoundationCalculatorPolynomial roots

Solve

z^2-2z+5=0

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SA-9709-P3-CX5-0012 marksL1 FoundationCalculatorLoci

The locus

|z-1+2i|=3

is a circle. State the coordinates of its centre and its radius.

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SA-9709-P3-CX5-0022 marksL1 FoundationCalculatorLoci

The locus

|z+3|=|z-1|

is a straight line. Its equation is

x=c

. Find

c

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SA-9709-P3-CX-G014 marksL1 FoundationCalculatorArithmetic

Given

z=3+2i

and

w=1-i

, find

zw

and

\dfrac zw

, each in the form

x+iy

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SA-9709-P3-D31-0011 markL1 FoundationCalculatorWith the chain rule

Differentiate

e^{5x-2}

with respect to

x

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SA-9709-P3-D31-0021 markL1 FoundationCalculatorWith the chain rule

Differentiate

\ln(3x+1)

with respect to

x

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SA-9709-P3-D32-0022 marksL1 FoundationCalculatorProduct rule

Differentiate

x\sin x

with respect to

x

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SA-9709-P3-D33-G012 marksL1 FoundationCalculatorImplicit

Find the gradient of the circle

x^2+y^2=25

at the point

(3,4)

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SA-9709-P3-D34-G012 marksL1 FoundationCalculatorParametric

A curve has parametric equations

x=2t

y=t^2

. Find the gradient of the curve at the point where

t=3

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SA-9709-P3-D35-G013 marksL1 FoundationCalculatorTangents and normals

Find the equation of the tangent to

y=\ln x

at the point where

x=1

, in the form

y=mx+c

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SA-9709-P3-DE-0041 markL1 FoundationCalculatorForming

The temperature

\theta

of a cooling liquid decreases at a rate proportional to the amount by which it exceeds the room temperature of

20^\circ

C. With

k>0

, which equation models this?

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SA-9709-P3-DE-0101 markL1 FoundationCalculatorForming

The volume

V

of a balloon increases at a rate inversely proportional to

\sqrt V

. Which equation models this?

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SA-9709-P3-DE1-0011 markL1 FoundationCalculatorForming

The mass

m

of a melting block of ice decreases at a rate proportional to the square of

m

. Which equation models this, with

k>0

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SA-9709-P3-DE1-0022 marksL1 FoundationCalculatorRates and constants

A population

N

satisfies

\dfrac{dN}{dt}=kN

. When

N=200

, the population is increasing at

10

per year. Find

k

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SA-9709-P3-DE1-G011 markL1 FoundationCalculatorForming

The area

A

of a patch of mould increases at a rate proportional to

A

. Which equation models this, with

k>0

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SA-9709-P3-DE2-0033 marksL1 FoundationCalculatorGeneral solutions

Solve

\dfrac{dy}{dx}=6x^2y

, given that

y=4

when

x=0

. Give

y

in terms of

x

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SA-9709-P3-DE3-G014 marksL1 FoundationCalculatorParticular solutions

Solve

\dfrac{dy}{dx}=\dfrac{2y}{x}

for

x>0

, given that

y=3

when

x=1

. Give

y

in terms of

x

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