Practise questions · Salam Academy

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.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-NM2-Q011 markL1 FoundationCalculatorAccuracy

An iteration gives

x_5=2.0942

and

x_6=2.0945

. The root correct to 2 decimal places is

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-NM2-Q021 markL1 FoundationCalculatorIteration

The iteration

x_{n+1}=\sqrt[3]{x_n+6}

starts at

x_1=2

. Find

x_2

.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-NM-Q031 markL1 FoundationCalculatorAccuracy

To give a root correct to 2 decimal places from an iteration, you should

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-NM-0023 marksL2 StandardCalculatorIteration

Use the iteration

x_{n+1}=\sqrt[3]{2x_n+5}

with

x_1=2

to find the root of

x^3-2x-5=0

correct to 2 decimal places.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-NM-0053 marksL2 StandardCalculatorIteration

Use the iteration

x_{n+1}=\ln(x_n+4)

with

x_1=2

to find the root of

x=\ln(x+4)

correct to 3 decimal places.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-NM2-0013 marksL2 StandardCalculatorIteration

Use the iteration

x_{n+1}=1+\dfrac1{x_n}

with

x_1=1

to find the positive root of

x^2-x-1=0

correct to 2 decimal places.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-NM2-0033 marksL2 StandardCalculatorIteration

Use the iteration

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

with

x_1=2

to find the positive root of

x^2-x-3=0

correct to 3 decimal places.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-NM2-0042 marksL2 StandardCalculatorAccuracy

An iteration suggests that the root of

x^3-x-4=0

is

1.796

to 3 decimal places. Let

f(x)=x^3-x-4

. Find

f(1.7955)

and

f(1.7965)

, each to 2 significant figures.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-NM-0073 marksL3 ExamCalculatorIteration

Use the iteration

x_{n+1}=\tfrac12\left(x_n+2\cos x_n\right)

with

x_1=1

to find the root of

x=2\cos x

correct to 3 decimal places.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-NM-0103 marksL3 ExamCalculatorIteration

Use the iteration

x_{n+1}=\sqrt[3]{12-x_n}

with

x_1=2

to find the root of

x^3+x-12=0

correct to 3 decimal places.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-NM2-0023 marksL3 ExamCalculatorIteration

Use the iteration

x_{n+1}=e^{-x_n}

with

x_1=0.5

to find the root of

x=e^{-x}

correct to 2 decimal places. Give each iterate to 4 decimal places.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-NM-0095 marksL4 ChallengingCalculatorIteration

It is given that

\displaystyle\int_1^a\left(\frac1x+1\right)dx=3

, where

a>1

. Show that

a=4-\ln a

, and use an iteration based on this, with

a_1=3

, to find

a

correct to 2 decimal places.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-NM2-0055 marksL4 ChallengingCalculatorIteration

A chord of a circle with radius

r

subtends an angle

\theta

radians at the centre. The segment it cuts off has area equal to one quarter of the area of the circle. Show that

\theta=\sin\theta+\tfrac\pi2

, and use the iteration

\theta_{n+1}=\sin\theta_n+\tfrac\pi2

with

\theta_1=2.3

to find

\theta

correct to 2 decimal places.

Salam Academy Original Practice · Proposed mark scheme

Question bank