Questions - Edexcel - A-Level Maths

Browse by module

All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 S1 S2 S3 S4 Unit 1 Unit 2 Unit 3 Unit 4

Browse by board

AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 PURE S1 S2 S3 S4 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4

Sort by: Default | Easiest first | Hardest first

Edexcel M3 2001 June Q2

7 marks Challenging +1.2

\includegraphics{figure_1} A smooth solid hemisphere, of radius 0.8 m and centre $O$, is fixed with its plane face on a horizontal table. A particle of mass 0.5 kg is projected horizontally with speed $u$ m s$^{-1}$ from the highest point $A$ of the hemisphere. The particle leaves the hemisphere at the point $B$, which is a vertical distance of 0.2 m below the level of $A$. The speed of the particle at $B$ is $v$ m s$^{-1}$ and the angle between $OA$ and $OB$ is $\theta$, as shown in Fig. 1.

Find the value of $\cos \theta$. [1]
Show that $v^2 = 5.88$. [3]
Find the value of $u$. [3]

Edexcel M3 2001 June Q3

10 marks Standard +0.3

\includegraphics{figure_2} A light horizontal spring, of natural length 0.25 m and modulus of elasticity 52 N, is fastened at one end to a point $A$. The other end of the spring is fastened to a small wooden block $B$ of mass 1.5 kg which is on a horizontal table, as shown in Fig. 2. The block is modelled as a particle. The table is initially assumed to be smooth. The block is released from rest when it is a distance 0.3 m from $A$. By using the principle of the conservation of energy,

find, to 3 significant figures, the speed of $B$ when it is a distance 0.25 m from $A$. [5]

It is now assumed that the table is rough and the coefficient of friction between $B$ and the table is 0.6.

Find, to 3 significant figures, the minimum distance from $A$ at which $B$ can rest in equilibrium. [5]

Edexcel M3 2001 June Q4

10 marks Standard +0.8

A projectile $P$ is fired vertically upwards from a point on the earth's surface. When $P$ is at a distance $x$ from the centre of the earth its speed is $v$. Its acceleration is directed towards the centre of the earth and has magnitude $\frac{k}{x^2}$, where $k$ is a constant. The earth may be assumed to be a sphere of radius $R$.

Show that the motion of $P$ may be modelled by the differential equation $$v \frac{dv}{dx} = -\frac{gR^2}{x^2}.$$ [4]

The initial speed of $P$ is $U$, where $U^2 < 2gR$. The greatest distance of $P$ from the centre of the earth is $X$.

Find $X$ in terms of $U$, $R$ and $g$. [6]

Edexcel M3 2001 June Q5

11 marks Standard +0.3

\includegraphics{figure_3} An ornament $S$ is formed by removing a solid right circular cone, of radius $r$ and height $\frac{1}{2}h$, from a solid uniform cylinder, of radius $r$ and height $h$, as shown in Fig. 3.

Show that the distance of the centre of mass $S$ from its plane face is $\frac{19}{30}h$. [7]

The ornament is suspended from a point on the circular rim of its open end. It hangs in equilibrium with its axis of symmetry inclined at an angle $\alpha$ to the horizontal. Given that $h = 4r$,

find, in degrees to one decimal place, the value of $\alpha$. [4]

Edexcel M3 2001 June Q6

14 marks Standard +0.3

\includegraphics{figure_4} A particle $P$ of mass $m$ is attached to two light inextensible strings. The other ends of the string are attached to fixed points $A$ and $B$. The point $A$ is a distance $h$ vertically above $B$. The system rotates about the line $AB$ with constant angular speed $\omega$. Both strings are taut and inclined at $60°$ to $AB$, as shown in Fig. 4. The particle moves in a circle of radius $r$.

Show that $r = \frac{\sqrt{3}}{2}h$. [2]
Find, in terms of $m$, $g$, $h$ and $\omega$, the tension in $AP$ and the tension in $BP$. [8]

The time taken for $P$ to complete one circle is $T$.

Show that $T < \pi\sqrt{\left(\frac{2h}{g}\right)}$. [4]

Edexcel M3 2001 June Q7

16 marks Challenging +1.2

\includegraphics{figure_5} A small ring $R$ of mass $m$ is free to slide on a smooth straight wire which is fixed at an angle of $30°$ to the horizontal. The ring is attached to one end of a light elastic string of natural length $a$ and modulus of elasticity $\lambda$. The other end of the string is attached to a fixed point $A$ of the wire, as shown in Fig. 5. The ring rests in equilibrium at the point $B$, where $AB = \frac{a}{2}$.

Show that $\lambda = 4mg$. [3]

The ring is pulled down to the point $C$, where $BC = \frac{1}{4}a$, and released from rest. At time $t$ after $R$ is released the extension of the string is $(\frac{1}{4}a + x)$.

Obtain a differential equation for the motion of $R$ while the string remains taut, and show that it represents simple harmonic motion with period $\pi\sqrt{\left(\frac{a}{g}\right)}$. [6]
Find, in terms of $g$, the greatest magnitude of the acceleration of $R$ while the string remains taut. [2]
Find, in terms of $a$ and $g$, the time taken for $R$ to move from the point at which it first reaches maximum speed to the point where the string becomes slack for the first time. [5]

Edexcel M3 2002 June Q1

6 marks Standard +0.3

A particle $P$ moves in a straight line with simple harmonic motion about a fixed centre $O$ with period 2 s. At time $t$ seconds the speed of $P$ is $v$ m s$^{-1}$. When $t = 0$, $v = 0$ and $P$ is at a point $A$ where $OA = 0.25$ m. Find the smallest positive value of $t$ for which $AP = 0.375$ m. [6]

Edexcel M3 2002 June Q2

9 marks Standard +0.3

\includegraphics{figure_1} A metal ball $B$ of mass $m$ is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point $A$. The ball $B$ moves in a horizontal circle with centre $O$ vertically below $A$, as shown in Fig. 1. The string makes a constant angle $\alpha°$ with the downward vertical and $B$ moves with constant angular speed $\sqrt{(2gk)}$, where $k$ is a constant. The tension in the string is $3mg$. By modelling $B$ as a particle, find

the value of $\alpha$, [4]
the length of the string. [5]

Edexcel M3 2002 June Q3

10 marks Standard +0.3

A particle $P$ of mass 2.5 kg moves along the positive $x$-axis. It moves away from a fixed origin $O$, under the action of a force directed away from $O$. When $OP = x$ metres the magnitude of the force is $2e^{-0.1x}$ newtons and the speed of $P$ is $v$ m s$^{-1}$. When $x = 0$, $v = 2$. Find

$v^2$ in terms of $x$, [6]
the value of $x$ when $v = 4$. [3]
Give a reason why the speed of $P$ does not exceed $\sqrt{20}$ m s$^{-1}$. [1]

Edexcel M3 2002 June Q4

10 marks Standard +0.3

A light elastic string $AB$ of natural length 1.5 m has modulus of elasticity 20 N. The end $A$ is fixed to a point on a smooth horizontal table. A small ball $S$ of mass 0.2 kg is attached to the end $B$. Initially $S$ is at rest on the table with $AB = 1.5$ m. The ball $S$ is then projected horizontally directly away from $A$ with a speed of 5 m s$^{-1}$. By modelling $S$ as a particle,

find the speed of $S$ when $AS = 2$ m. [5]

When the speed of $S$ is 1.5 m s$^{-1}$, the string breaks.

Find the tension in the string immediately before the string breaks. [5]

Edexcel M3 2002 June Q5

12 marks Standard +0.3

\includegraphics{figure_2} A model tree is made by joining a uniform solid cylinder to a uniform solid cone made of the same material. The centre $O$ of the base of the cone is also the centre of one end of the cylinder, as shown in Fig. 2. The radius of the cylinder is $r$ and the radius of the base of the cone is $2r$. The height of the cone and the height of the cylinder are each $h$. The centre of mass of the model is at the point $G$.

Show that $OG = \frac{1}{14}h$. [8]

The model stands on a desk top with its plane face in contact with the desk top. The desk top is tilted until it makes an angle $\alpha$ with the horizontal, where $\tan \alpha = \frac{r}{7h}$. The desk top is rough enough to prevent slipping and the model is about to topple.

Find $r$ in terms of $h$. [4]

Edexcel M3 2002 June Q6

14 marks Standard +0.3

A light elastic string, of natural length $4a$ and modulus of elasticity $8mg$, has one end attached to a fixed point $A$. A particle $P$ of mass $m$ is attached to the other end of the string and hangs in equilibrium at the point $O$.

Find the distance $AO$. [2]

The particle is now pulled down to a point $C$ vertically below $O$, where $OC = d$. It is released from rest. In the subsequent motion the string does not become slack.

Show that $P$ moves with simple harmonic motion of period $\pi\sqrt{\frac{2a}{g}}$. [7]

The greatest speed of $P$ during this motion is $\frac{1}{2}\sqrt{(ga)}$.

Find $d$ in terms of $a$. [3]

Instead of being pulled down a distance $d$, the particle is pulled down a distance $a$. Without further calculation,

describe briefly the subsequent motion of $P$. [2]

Edexcel M3 2002 June Q7

14 marks Standard +0.3

\includegraphics{figure_3} A particle of mass $m$ is attached to one end of a light inextensible string of length $l$. The other end of the string is attached to a fixed point $O$. The particle is hanging at the point $A$, which is vertically below $O$. It is projected horizontally with speed $u$. When the particle is at the point $P$, $\angle AOP = \theta$, as shown in Fig. 3. The string oscillates through an angle $\alpha$ on either side of $OA$ where $\cos \alpha = \frac{2}{3}$.

Find $u$ in terms of $g$ and $l$. [4]

When $\angle AOP = \theta$, the tension in the string is $T$.

Show that $T = \frac{mg}{3}(9\cos\theta - 4)$. [6]
Find the range of values of $T$. [4]

an expression for $v^2$ in terms of $x$, [7]
the initial speed of the car. [2]

Edexcel M3 2003 June Q4

11 marks Standard +0.3

\includegraphics{figure_1} A particle $P$ of mass $m$ is attached to the ends of two light inextensible strings $AP$ and $BP$ each of length $l$. The ends $A$ and $B$ are attached to fixed points, with $A$ vertically above $B$ and $AB = \frac{3}{4}l$, as shown in Fig. 1. The particle $P$ moves in a horizontal circle with constant angular speed $\omega$. The centre of the circle is the mid-point of $AB$ and both strings remain taut.

Show that the tension $AP$ is $\frac{1}{6}m(3l\omega^2 + 4g)$. [7]
Find, in terms of $m$, $l$, $\omega$ and $g$, an expression for the tension in $BP$. [2]
Deduce that $\omega^2 \geq \frac{4g}{3l}$. [2]

Edexcel M3 2003 June Q5

13 marks Standard +0.3

A particle $P$ of mass $0.8$ kg is attached to one end $A$ of a light elastic spring $OA$, of natural length $60$ cm and modulus of elasticity $12$ N. The spring is placed on a smooth horizontal table and the end $O$ is fixed. The particle $P$ is pulled away from $O$ to a point $B$, where $OB = 85$ cm, and is released from rest.

Prove that the motion of $P$ is simple harmonic with period $\frac{2\pi}{5}$ s. [5]
Find the greatest magnitude of the acceleration of $P$ during the motion. [2]

Two seconds after being released from rest, $P$ passes through the point $C$.

Find, to 2 significant figures, the speed of $P$ as it passes through $C$. [5]
State the direction in which $P$ is moving 2 s after being released. [1]

Edexcel M3 2003 June Q6

14 marks Challenging +1.2

\includegraphics{figure_2} A particle is at the highest point $A$ on the outer surface of a fixed smooth sphere of radius $a$ and centre $O$. The lowest point $B$ of the sphere is fixed to a horizontal plane. The particle is projected horizontally from $A$ with speed $u$, where $u < \sqrt{ag}$. The particle leaves the sphere at the point $C$, where $OC$ makes an angle $\theta$ with the upward vertical, as shown in Fig. 2.

Find an expression for $\cos \theta$ in terms of $u$, $g$ and $a$. [7]

The particle strikes the plane with speed $\sqrt{\frac{9ag}{2}}$.

Find, to the nearest degree, the value of $\theta$. [7]

Edexcel M3 2003 June Q7

16 marks Standard +0.3

\includegraphics{figure_3} The shaded region $R$ is bounded by part of the curve with equation $y = \frac{1}{4}(x - 2)^2$, the $x$-axis and the $y$-axis, as shown in Fig. 3. The unit of length on both axes is 1 cm. A uniform solid $S$ is made by rotating $R$ through $360°$ about the $x$-axis. Using integration,

calculate the volume of the solid $S$, leaving your answer in terms of $\pi$, [4]
show that the centre of mass of $S$ is $\frac{4}{5}$ cm from its plane face. [7]

\includegraphics{figure_4} A tool is modelled as having two components, a solid uniform cylinder $C$ and the solid $S$. The diameter of $C$ is 4 cm and the length of $C$ is 8 cm. One end of $C$ coincides with the plane face of $S$. The components are made of different materials. The weight of $C$ is $10W$ newtons and the weight of $S$ is $2W$ newtons. The tool lies in equilibrium with its axis of symmetry horizontal on two smooth supports $A$ and $B$, which are at the ends of the cylinder, as shown in Fig. 4.