Practise questions | Salam Academy

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-0042 marksL2 StandardCalculatorRearranging

The sequence defined by

x_{n+1}=\tfrac12\left(x_n+\dfrac7{x_n}\right)

,

x_1=2

, converges to

\alpha

. Find the exact value of

\alpha

.

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-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-NM-0082 marksL2 StandardCalculatorRearranging

The iteration

x_{n+1}=\dfrac{x_n^3+3}{5}

converges to

\alpha

. The number

\alpha

is a root of

x^3+px+q=0

. Find

p

and

q

.

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-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-NM3-0012 marksL2 StandardCalculatorRearranging

The sequence

x_{n+1}=\dfrac{2x_n}{3}+\dfrac{4}{3x_n^2}

, with

x_1=2

, converges to

\alpha

. Find the exact value of

\alpha

.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-NM4-0013 marksL2 StandardCalculatorNewton–Raphson

Use the Newton–Raphson method with

x_1=2

to find

x_3

for the equation

x^3-2x-5=0

. Give your answer to 4 decimal places.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-NM4-0023 marksL2 StandardCalculatorNewton–Raphson

Use one step of the Newton–Raphson method with

x_1=1

to find an approximation

x_2

to the root of

x=\cos x

(

x

in radians). Give

x_2

to 4 decimal places.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-NM4-0031 markL2 StandardCalculatorFailures

Why does the Newton–Raphson method fail for

f(x)=x^3-3x+1

with

x_1=1

?

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-NM4-0083 marksL2 StandardCalculatorNewton–Raphson

Use one step of the Newton–Raphson method with

x_1=1.5

on

f(x)=e^x-3x

to find

x_2

, giving your answer to 4 decimal places.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-T31-0032 marksL2 StandardCalculatorIdentities

Given that

\sec\theta=3

and

\theta

is acute, find the exact value of

\tan\theta

.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-T32-0012 marksL2 StandardCalculatorExact values

Find the exact value of

\cos75^\circ

.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-T32-0023 marksL2 StandardCalculatorCompound angles

Angles

A

and

B

are acute, with

\sin A=\tfrac45

and

\cos B=\tfrac5{13}

. Find the exact value of

\sin(A-B)

.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-T33-0031 markL2 StandardCalculatorDouble angles

Find the exact value of

\cos^222.5^\circ-\sin^222.5^\circ

.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-T34-0013 marksL2 StandardCalculatorR-form

Express

5\cos x+12\sin x

in the form

R\cos(x-\alpha)

, where

R>0

and

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

. Give

\alpha

to 2 decimal places.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-T34-0022 marksL2 StandardCalculatorMax and min

Find the exact greatest value of

2\sin\theta+2\cos\theta

.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-T34-0053 marksL2 StandardCalculatorR-form

Express

\cos\theta+\sin\theta

in the form

R\cos(\theta-\alpha)

, where

R>0

and

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

.

Salam Academy Original Practice · Proposed mark scheme

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