Practise questions | Salam Academy

SA-9709-P2-DF-0114 marksL3 ExamCalculatorImplicit

Find the gradient of the curve

x^3+xy+y^3=11

at the point

(1,2)

.

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SA-9709-P3-CX-0055 marksL3 ExamCalculatorPolynomial roots

Given that

1+2i

is a root of

z^3-z^2+3z+5=0

, find the other two roots.

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SA-9709-P3-DE-0055 marksL3 ExamCalculatorIn context

A liquid cools so that its temperature is

\theta=20+60e^{-kt}

after

t

minutes. After

10

minutes the temperature is

50^\circ

C. Find the exact value of

k

, and the temperature after

20

minutes.

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SA-9709-P3-DE3-0016 marksL3 ExamCalculatorHarder integrals

Solve

y\dfrac{dy}{dx}=xe^x

, given that

y=2

when

x=0

and

y>0

. Give

y

in terms of

x

.

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SA-9709-P3-DE3-0026 marksL3 ExamCalculatorHarder integrals

Solve

\dfrac{dy}{dx}=\dfrac{2y}{x(x+2)}

for

x>0

, given that

y=1

when

x=2

. Give

y

in terms of

x

.

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SA-9709-P3-DE4-0026 marksL3 ExamCalculatorIn context

A liquid at

100^\circ

C is placed in a room at

20^\circ

C and cools according to

\dfrac{d\theta}{dt}=-k(\theta-20)

, with

t

in minutes. After

5

minutes it is at

60^\circ

C. Find the time at which it reaches

30^\circ

C.

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SA-9709-P3-IN-0154 marksL3 ExamCalculatorBy parts

Find

\displaystyle\int x\cos2x\,dx

.

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SA-9709-P3-LE-0124 marksL3 ExamCalculatorLinear forms

Variables

x

and

y

satisfy

y=ka^x

. The graph of

\ln y

against

x

is a straight line through

(0,\,1.2)

and

(4,\,3.0)

. Find

k

and

a

, each correct to 3 significant figures.

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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

.

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SA-9709-P3-TR-0064 marksL3 ExamCalculatorDouble angles

Given that

\cos\theta=-\tfrac13

and

90^\circ<\theta<180^\circ

, find the exact values of

\sin2\theta

and

\cos2\theta

.

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SA-9709-P3-VE-0043 marksL3 ExamCalculatorScalar product

Find the angle between

\mathbf{p}=\mathbf{i}+2\mathbf{j}+2\mathbf{k}

and

\mathbf{q}=2\mathbf{i}-\mathbf{j}+2\mathbf{k}

, giving your answer in degrees correct to 1 decimal place.

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SA-9709-P3-VE-0073 marksL3 ExamCalculatorLine equation

The line

l

has equation

\mathbf{r}=\begin{pmatrix}1\\-2\\3\end{pmatrix}+\lambda\begin{pmatrix}3\\2\\-2\end{pmatrix}

. The point

C(-5,-6,7)

lies on

l

. Find the value of

\lambda

at

C

.

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SA-9709-P3-VE4-0015 marksL3 ExamCalculatorIntersections

Show that the lines

\mathbf{r}=\begin{pmatrix}2\\0\\1\end{pmatrix}+\lambda\begin{pmatrix}1\\-1\\2\end{pmatrix}

and

\mathbf{r}=\begin{pmatrix}5\\0\\6\end{pmatrix}+\mu\begin{pmatrix}1\\2\\1\end{pmatrix}

intersect, and find the coordinates of the point of intersection.

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SA-9709-P3-VE5-0043 marksL3 ExamCalculatorFoot of perpendicular

The foot of the perpendicular from

A(5,1,0)

to the line

l

is

F(3,3,1)

. Find the coordinates of the reflection of

A

in

l

.

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SA-9709-P3-VE5-0054 marksL3 ExamCalculatorFoot of perpendicular

Find the coordinates of the foot of the perpendicular from

A(3,3,1)

to the line

\mathbf{r}=\begin{pmatrix}0\\1\\2\end{pmatrix}+\lambda\begin{pmatrix}1\\0\\-1\end{pmatrix}

.

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SA-9709-P4-EW-0055 marksL3 ExamCalculatorWork–energy

A sledge of mass

40

kg starts from rest and slides

30

m down a slope, descending

12

m vertically. A constant friction force of

60

N acts. Find its speed at the bottom, correct to 3 significant figures.

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SA-9709-P4-FE3-0045 marksL3 ExamCalculatorAngled forces

A block of mass

5

kg rests on a rough horizontal floor with

\mu=0.4

. It is pulled by a force

P

N at

30^\circ

above the horizontal and is on the point of moving. Find

P

, correct to 3 significant figures.

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SA-9709-P4-MO-0043 marksL3 ExamCalculatorDirections

A (

3

kg,

4\ \text{m s}^{-1}

) collides directly with B (

2

kg) moving at

2\ \text{m s}^{-1}

in the opposite direction. After the collision A moves back at

1\ \text{m s}^{-1}

. Find B's speed after the collision.

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SA-9709-P5-ND-0084 marksL3 ExamCalculatorApproximation

X\sim B(100,0.4)

. Use a suitable approximation to find

P(X\le45)

, correct to 3 significant figures.

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SA-9709-P5-ND-0104 marksL3 ExamCalculatorApproximation

X\sim B(60,0.25)

. Use a normal approximation to find

P(X>20)

, correct to 3 significant figures.

Salam Academy Original Practice · Proposed mark scheme

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