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

SA-9709-P3-LE3-G012 marksL1 FoundationCalculatoraˣ = b

Solve

2^x=10

, giving

x

correct to 3 significant figures.

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SA-9709-P3-LE4-0011 markL1 FoundationCalculatorLog equations

Solve

\log_3x=4

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SA-9709-P3-LE4-G011 markL1 FoundationCalculatorLog equations

Solve

\ln x=3

, giving the exact value of

x

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SA-9709-P3-LE5-G012 marksL1 FoundationCalculatorLinear forms

y=kx^n

. The graph of

\ln y

against

\ln x

is a straight line with gradient

2

and intercept

0.7

. Find

n

, and

k

correct to 3 significant figures.

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SA-9709-P3-NM-0031 markL1 FoundationCalculatorRearranging

Which of these is a rearrangement of

x^3-2x-5=0

into the form

x=F(x)

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SA-9709-P3-NM1-G012 marksL1 FoundationCalculatorCounting roots

By sketching

y=\ln x

and

y=3-x

on the same axes, state the number of roots of the equation

\ln x=3-x

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SA-9709-P3-NM2-G012 marksL1 FoundationCalculatorIteration

The iteration

x_{n+1}=\sqrt{x_n+3}

has

x_1=2

. Find

x_2

and

x_3

, each correct to 4 decimal places.

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SA-9709-P3-NM3-G012 marksL1 FoundationCalculatorRearranging

The equation

x^3+4x-1=0

is rearranged as

x=\dfrac{1-x^3}{4}

. Use

x_{n+1}=\dfrac{1-x_n^3}{4}

with

x_1=0

to find

x_2

and

x_3

, giving

x_3

to 4 decimal places.

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SA-9709-P3-NM4-G013 marksL1 FoundationCalculatorNewton–Raphson

Let

f(x)=x^2-2

. Use the Newton–Raphson method with

x_1=1

to find

x_2

and

x_3

, giving

x_3

to 4 decimal places.

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SA-9709-P3-NM-G012 marksL1 FoundationCalculatorSign change

Let

f(x)=x^3-2x-5

. Find

f(2)

and

f(3)

, and hence explain why

f(x)=0

has a root between

2

and

3

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SA-9709-P3-T31-0012 marksL1 FoundationCalculatorReciprocal functions

Find the exact values of

\cot45^\circ

and

\sec180^\circ

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SA-9709-P3-T31-0021 markL1 FoundationCalculatorIdentities

Simplify

\tan^2\theta-\sec^2\theta

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SA-9709-P3-T31-G012 marksL1 FoundationCalculatorReciprocal functions

Find the exact values of

\sec60^\circ

and

\operatorname{cosec}30^\circ

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SA-9709-P3-T33-0021 markL1 FoundationCalculatorDouble angles

Find the exact value of

2\sin15^\circ\cos15^\circ

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SA-9709-P3-T33-G013 marksL1 FoundationCalculatorDouble angles

Given that

\sin\theta=\tfrac35

and

\theta

is acute, find the exact values of

\sin2\theta

and

\cos2\theta

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SA-9709-P3-T34-G013 marksL1 FoundationCalculatorR-form

Express

\sqrt3\sin\theta+\cos\theta

in the form

R\sin(\theta+\alpha)

, where

R>0

and

0^\circ<\alpha<90^\circ

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SA-9709-P3-TR-0012 marksL1 FoundationCalculatorReciprocal functions

Given that

\tan\theta=\tfrac34

and

\theta

is acute, find the exact values of

\sec\theta

and

\csc\theta

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SA-9709-P3-TR-0152 marksL1 FoundationCalculatorDouble angles

Given that

\tan\theta=\tfrac12

, find the exact value of

\tan2\theta

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SA-9709-P3-TR-G013 marksL1 FoundationCalculatorExact values

Use

\sin(45^\circ+30^\circ)

to find the exact value of

\sin75^\circ

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SA-9709-P3-VE-0012 marksL1 FoundationCalculatorMagnitude

The points

A

and

B

have position vectors

\overrightarrow{OA}=2\mathbf{i}-\mathbf{j}+4\mathbf{k}

and

\overrightarrow{OB}=5\mathbf{i}+3\mathbf{j}+4\mathbf{k}

. Find

|\overrightarrow{AB}|

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