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

SA-FM-CS-0033 marksL3 ExamNon-calculatorSolids

The region under

y=\sqrt x

0\le x\le4

is rotated about the

x

-axis. Find

\bar x

for the solid.

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SA-FM-FK-0033 marksL3 ExamCalculatora(v), a(x)

A particle has

a=-4x

and

v=6

x=0

. Find

v

x=1

(3 s.f.).

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SA-FM-RB-0023 marksL3 ExamCalculatorLadders

A uniform ladder rests against a smooth vertical wall at

60^\circ

to rough horizontal ground and is about to slip. Find

\mu

(3 s.f.).

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SA-FM-RB-0033 marksL3 ExamCalculatorLadders

A uniform ladder on rough ground (

\mu=0.5

) rests against a smooth wall. Find the smallest angle (degrees) with the ground for equilibrium.

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SA-FM-VF-0013 marksL3 ExamCalculatorResistance

2

kg particle moves with resistance

4v

N only, starting at

10

m s

^{-1}

. Find its speed after

0.5

s (3 s.f.).

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SA-FM-VF-0023 marksL3 ExamCalculatorv dv/dx

1

kg particle is repelled from

O

by a force

\frac8{x^2}

N. It starts at rest at

x=1

. Find its speed at

x=4

(3 s.f.).

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SA-FM-VF-0033 marksL3 ExamNon-calculatorResistance

2

kg particle moves with resistance

4v

N only, starting at

10

m s

^{-1}

. Find the total distance it travels.

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SA-FP-CE-0043 marksL3 ExamNon-calculatorTangents

Find the

x

-intercept of the tangent to

\frac{x^2}{25}+\frac{y^2}9=1

\left(3,\frac{12}5\right)

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SA-FP-CT-0033 marksL3 ExamNon-calculatorLoci

The locus

|z-3|=2|z|

is a circle. Find its radius.

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SA-FP-CT-0043 marksL3 ExamNon-calculatorTransformations

Under

w=\frac1z

, the line

\operatorname{Re}z=1

maps to a circle. Find its radius.

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SA-FP-DE1-0024 marksL3 ExamNon-calculatorSolving

Solve

\frac{dy}{dx}+\frac yx=3x

with

y=2

x=1

. Give

y

in terms of

x

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SA-FP-DE1-0034 marksL3 ExamNon-calculatorSolving

\frac{dy}{dx}-2y=4

y(0)=1

. Find the exact value of

y

when

x=\ln2

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SA-FP-DE2-0043 marksL3 ExamNon-calculatorParticular integral

Find a particular integral of

y''+y=3x

, in terms of

x

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SA-FP-DM-0033 marksL3 ExamNon-calculatorRoots

Solve

z^3=8i

. Give the arguments of the roots in degrees, in the range

-180<\theta\le180

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SA-FP-EIG-0033 marksL3 ExamNon-calculatorDiagonalisation

\mathbf M=\mathbf{PDP}^{-1}

with

\mathbf D=\begin{pmatrix}2&0\\0&-1\end{pmatrix}

. Find the trace of

\mathbf M^5

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SA-FP-EIG-0043 marksL3 ExamNon-calculatorEigenvalues

Find the eigenvalues of

\begin{pmatrix}2&0&0\\1&3&0\\4&5&-1\end{pmatrix}

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SA-FP-HYP-0033 marksL3 ExamCalculatorInverses and equations

Solve

\cosh x=\frac54

, giving both solutions to 4 s.f.

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SA-FP-IC-0043 marksL3 ExamCalculatorIntegrating

Evaluate

\displaystyle\int_0^{3/2}\frac{dx}{\sqrt{9-x^2}}

exactly (enter to 4 s.f.).

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SA-FP-INEQ-0023 marksL3 ExamNon-calculatorRational inequalities

Find the smallest value of

x

satisfying

\dfrac{x}{x+1}\ge2

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SA-FP-MAC-0033 marksL3 ExamNon-calculatorStandard series

Find the coefficient of

x^3

in the series for

e^x\sin x

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