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

SA-9709-P3-D34-0012 marksL2 StandardCalculatorParametric

A curve has parametric equations

x=\cos t

y=\sin t

. Find the exact gradient at the point where

t=\dfrac{\pi}{3}

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SA-9709-P3-D35-0013 marksL2 StandardCalculatorStationary points

Find the

x

-coordinate of the stationary point of

y=xe^{2x}

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SA-9709-P3-D35-0033 marksL2 StandardCalculatorTangents and normals

Find the equation of the normal to

y=e^x

at the point

(0,1)

, in the form

y=mx+c

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SA-9709-P3-DE-0014 marksL2 StandardCalculatorSeparating variables

Solve

\dfrac{dy}{dx}=\dfrac xy

, given that

y=3

when

x=4

and

y>0

. Give

y

in terms of

x

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SA-9709-P3-DE-0034 marksL2 StandardCalculatorSeparating variables

Solve

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

, given that

y=2

when

x=0

. Give

y

in terms of

x

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SA-9709-P3-DE-0074 marksL2 StandardCalculatorIn context

A population

N

grows according to

\dfrac{dN}{dt}=0.02N

, with

N=500

when

t=0

. Find the time taken for the population to double, correct to 3 significant figures.

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SA-9709-P3-DE-0113 marksL2 StandardCalculatorParticular solutions

Given

\dfrac{dy}{dx}=3y

and

y=2

when

x=0

, find the exact value of

y

when

x=1

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SA-9709-P3-DE2-0014 marksL2 StandardCalculatorSeparating variables

Solve

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

, given that

y=1

when

x=0

. Give

y

in terms of

x

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SA-9709-P3-DE3-0034 marksL2 StandardCalculatorHarder integrals

Solve

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

for

-\tfrac\pi2<x<\tfrac\pi2

, given that

y=2

when

x=0

. Give

y

in terms of

x

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SA-9709-P3-DE4-0013 marksL2 StandardCalculatorIn context

A radioactive substance decays according to

\dfrac{dm}{dt}=-km

. Its half-life is

8

days. Find the exact value of

k

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SA-9709-P3-DF-0013 marksL2 StandardCalculatorStandard derivatives

Find

\dfrac{dy}{dx}

when

y=\sin3x+\cos^2x

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SA-9709-P3-DF-0033 marksL2 StandardCalculatorProduct rule

Find

\dfrac{dy}{dx}

when

y=x^2e^{3x}

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SA-9709-P3-DF-0084 marksL2 StandardCalculatorImplicit

The curve

x^2+y^2-4x+6y=12

passes through

(5,1)

. Find the gradient of the curve at this point.

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SA-9709-P3-DF-0144 marksL2 StandardCalculatorStationary points

Find the exact coordinates of the stationary point of

y=xe^{-x}

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SA-9709-P3-I31-0022 marksL2 StandardCalculatorStandard integrals

Find

\displaystyle\int\frac{4}{2x-1}\,dx

, for

x>\tfrac12

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SA-9709-P3-I32-0013 marksL2 StandardCalculatorSquares

Find

\displaystyle\int\cos^2x\,dx

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SA-9709-P3-I32-0022 marksL2 StandardCalculatorDouble angles

Find

\displaystyle\int2\sin3x\cos3x\,dx

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SA-9709-P3-I33-0023 marksL2 StandardCalculatorExact integrals

Find the exact value of

\displaystyle\int_0^{\ln2}e^{2x}\,dx

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SA-9709-P3-I34-0013 marksL2 StandardCalculatorTrapezium rule

Use the trapezium rule with

4

strips to estimate

\displaystyle\int_1^2\frac1x\,dx

, giving your answer to 3 significant figures.

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SA-9709-P3-I34-0023 marksL2 StandardCalculatorTrapezium rule

Use the trapezium rule with

2

strips to estimate

\displaystyle\int_0^1\sqrt{1+x}\,dx

, giving your answer to 2 decimal places.

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