Practise

Question bank

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

TopicSkillDifficulty

12 questions

SA-9709-P3-NM1-0032 marksL1 FoundationCalculatorSign change

Let

f(x)=x^3+x-3

. Find

f(1)

and

f(1.5)

, and hence show that

f(x)=0

has a root between

1

and

1.5

Salam Academy Original Practice · Proposed mark scheme

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

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-NM1-Q011 markL1 FoundationCalculatorSign change

To show that a root is

1.796

correct to 3 decimal places, you should check for a sign change between

Salam Academy Original Practice · Proposed mark scheme

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

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-NM-Q011 markL1 FoundationCalculatorSign-change failures

f(x)=\dfrac{1}{x-2}

has

f(1)<0

and

f(3)>0

. What can you conclude?

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-NM-0012 marksL2 StandardCalculatorSign change

Let

f(x)=e^x+x-3

. Find

f(0.5)

and

f(1)

correct to 3 decimal places, and hence show that

e^x=3-x

has a root between

0.5

and

1

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-NM-0062 marksL2 StandardCalculatorSign change

Let

f(x)=x-2\cos x

(

x

in radians). Find

f(1)

and

f(1.1)

correct to 3 decimal places, and hence show that

x=2\cos x

has a root between

1

and

1.1

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-NM1-0012 marksL2 StandardCalculatorCounting roots

By sketching suitable graphs, state the number of real roots of the equation

x^3=3x-1

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-NM1-0041 markL2 StandardCalculatorSign-change failures

Let

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

. Then

f(1)=1

and

f(2)=1

. Which statement is correct?

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-NM1-0051 markL2 StandardCalculatorSign-change failures

Let

f(x)=\tan x-1

(

x

in radians). Then

f(1)=0.557

and

f(2)=-3.19

. Why does this not show a root of

\tan x=1

between

1

and

2

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-NM1-0022 marksL3 ExamCalculatorCounting roots

By sketching

y=e^{-x}

and

y=x^2

, state the number of real roots of

e^{-x}=x^2

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-NM1-0063 marksL4 ChallengingCalculatorSign-change failures

Let

f(x)=10x^2-31x+24

. Then

f(1)=3

and

f(2)=2

. Find

f(1.55)

, and hence state the number of roots of

f(x)=0

between

1

and

2

Salam Academy Original Practice · Proposed mark scheme