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

SA-9709-P3-I35-0013 marksL2 StandardCalculatorSubstitution

Use the substitution

u=x^2+4

to find

\displaystyle\int x\sqrt{x^2+4}\,dx

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-I35-0024 marksL2 StandardCalculatorChanging limits

Use the substitution

u=x^2+1

to find the exact value of

\displaystyle\int_0^1\frac{x}{(x^2+1)^2}\,dx

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-I35-0032 marksL2 StandardCalculatorRecognising

Find

\displaystyle\int\cos x\,e^{\sin x}\,dx

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SA-9709-P3-I36-0013 marksL2 StandardCalculatorBy parts

Find

\displaystyle\int x\sin x\,dx

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SA-9709-P3-I36-0023 marksL2 StandardCalculatorChoosing u

Find

\displaystyle\int\ln x\,dx

, for

x>0

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SA-9709-P3-IN-0104 marksL2 StandardCalculatorChanging limits

Use the substitution

u=x^2+1

to find

\displaystyle\int_0^1x(x^2+1)^3\,dx

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SA-9709-P3-IN-0133 marksL2 StandardCalculatorBy parts

Find

\displaystyle\int xe^x\,dx

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SA-9709-P3-LE-0023 marksL2 StandardCalculatoraˣ = b

Solve

2^{x+1}=5^x

, giving

x

correct to 3 significant figures.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-LE-0034 marksL2 StandardCalculatorDisguised quadratics

Solve the equation

3^{2x}-4\left(3^x\right)+3=0

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SA-9709-P3-LE-0063 marksL2 StandardCalculatorCombining

Solve

\log_2(x+1)-\log_2x=3

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SA-9709-P3-LE-0114 marksL2 StandardCalculatorLinear forms

Variables

x

and

y

satisfy

y=kx^n

. The graph of

\ln y

against

\ln x

is a straight line through

(1,\,2.5)

and

(3,\,6.5)

. Find

n

, and

k

correct to 3 significant figures.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-LE-0143 marksL2 StandardCalculatorGraphs

The curve

y=e^{2x}-3

crosses the

x

-axis at

P

. Find the exact

x

-coordinate of

P

and the equation of the horizontal asymptote,

y=c

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SA-9709-P3-LE1-0021 markL2 StandardCalculatorDefinition

Find the value of

\log_9 3

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SA-9709-P3-LE2-0022 marksL2 StandardCalculatorGraphs

For

y=2-e^{-x}

, state the equation of the asymptote,

y=c

, and the

y

-intercept.

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SA-9709-P3-LE2-0032 marksL2 StandardCalculatorGraphs

For

y=\ln(x+3)

, state the equation of the vertical asymptote,

x=a

, and the

x

-intercept.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-LE3-0013 marksL2 StandardCalculatoraˣ = b

Solve

5^{2x-1}=30

, giving

x

correct to 3 significant figures.

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SA-9709-P3-LE4-0032 marksL2 StandardCalculatorCombining

Solve

\ln(2x+1)=\ln x+\ln3

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SA-9709-P3-LE5-0012 marksL2 StandardCalculatorLinear forms

y=ka^x

. The graph of

\ln y

against

x

is a straight line with intercept

0.5

and gradient

0.3

. Find

k

and

a

, each correct to 3 significant figures.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-LE5-0023 marksL2 StandardCalculatorFinding constants

y=kx^n

. The graph of

\lg y

against

\lg x

is a straight line through

(1,3)

and

(3,7)

. Find

n

and

k

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