Questions — CAIE (7646 questions)

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
CAIE Further Paper 3 2023 November Q4
8 marks Challenging +1.2
\includegraphics{figure_4} A light elastic string has natural length \(8a\) and modulus of elasticity \(5mg\). A particle \(P\) of mass \(m\) is attached to the midpoint of the string. The ends of the string are attached to points \(A\) and \(B\) which are a distance \(12a\) apart on a smooth horizontal table. The particle \(P\) is held on the table so that \(AP = BP = L\) (see diagram). The particle \(P\) is released from rest. When \(P\) is at the midpoint of \(AB\) it has speed \(\sqrt{80ag}\).
  1. Find \(L\) in terms of \(a\). [5]
  2. Find the initial acceleration of \(P\) in terms of \(g\). [3]
CAIE Further Paper 3 2023 November Q5
9 marks Challenging +1.2
A particle \(P\) is projected with speed \(u\text{ ms}^{-1}\) at an angle \(\theta\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. During its flight \(P\) passes through the point which is a horizontal distance \(3a\) from \(O\) and a vertical distance \(\frac{3}{8}a\) above the horizontal plane. It is given that \(\tan\theta = \frac{1}{3}\).
  1. Show that \(u^2 = 8ag\). [2]
A particle \(Q\) is projected with speed \(V\text{ ms}^{-1}\) at an angle \(\alpha\) above the horizontal from \(O\) at the instant when \(P\) is at its highest point. Particles \(P\) and \(Q\) both land at the same point on the horizontal plane at the same time.
  1. Find \(V\) in terms of \(a\) and \(g\). [7]
CAIE Further Paper 3 2023 November Q6
11 marks Challenging +1.8
A particle \(P\) of mass \(m\) is attached to one end of a light inextensible rod of length \(3a\). An identical particle \(Q\) is attached to the other end of the rod. The rod is smoothly pivoted at a point \(O\) on the rod, where \(OQ = x\). The system, of rod and particles, rotates about \(O\) in a vertical plane. At an instant when the rod is vertical, with \(P\) above \(Q\), the particle \(P\) is moving horizontally with speed \(u\). When the rod has turned through an angle of \(60°\) from the vertical, the speed of \(P\) is \(2\sqrt{ag}\), and the tensions in the two parts of the rod, \(OP\) and \(OQ\), have equal magnitudes.
  1. Show that the speed of \(Q\) when the rod has turned through an angle of \(60°\) from the vertical is \(\frac{2x}{3a-x}\sqrt{ag}\). [2]
  2. Find \(x\) in terms of \(a\). [5]
  3. Find \(u\) in terms of \(a\) and \(g\). [4]
CAIE Further Paper 3 2023 November Q1
4 marks Standard +0.3
One end of a light inextensible string of length \(a\) is attached to a fixed point \(O\). The other end of the string is attached to a particle of mass \(m\). The string is taut and makes an angle \(\theta\) with the downward vertical through \(O\), where \(\cos \theta = \frac{2}{3}\). The particle moves in a horizontal circle with speed \(v\). Find \(v\) in terms of \(a\) and \(g\). [4]
CAIE Further Paper 3 2023 November Q2
6 marks Challenging +1.2
A particle \(P\) of mass \(0.5\) kg moves in a straight line. At time \(t\) s the velocity of \(P\) is \(v\) m s\(^{-1}\) and its displacement from a fixed point \(O\) on the line is \(x\) m. The only forces acting on \(P\) are a force of magnitude \(\frac{150}{(x+1)^2}\) N in the direction of increasing displacement and a resistive force of magnitude \(\frac{450}{(x+1)^3}\) N. When \(t = 0\), \(x = 0\) and \(v = 20\). Find \(v\) in terms of \(x\), giving your answer in the form \(v = \frac{Ax + B}{(x + 1)}\), where \(A\) and \(B\) are constants to be determined. [6]
CAIE Further Paper 3 2023 November Q3
7 marks Challenging +1.2
\includegraphics{figure_3} A uniform lamina is in the form of an isosceles triangle \(ABC\) in which \(AC = 2a\) and angle \(ABC = 90°\). The point \(D\) on \(AB\) is such that the ratio \(DB : AB = 1 : k\). The point \(E\) on \(CB\) is such that \(DE\) is parallel to \(AC\). The triangle \(DBE\) is removed from the lamina (see diagram).
  1. Find, in terms of \(k\), the distance of the centre of mass of the remaining lamina \(ADEC\) from the midpoint of \(AC\). [4]
When the lamina \(ADEC\) is freely suspended from the vertex \(A\), the edge \(AC\) makes an angle \(\theta\) with the downward vertical, where \(\tan \theta = \frac{2}{15}\).
  1. Find the value of \(k\). [3]
CAIE Further Paper 3 2023 November Q4
7 marks Challenging +1.8
\includegraphics{figure_4} Two smooth vertical walls meet at right angles. The smooth sphere \(A\), with mass \(m\), is at rest on a smooth horizontal surface and is at a distance \(d\) from each wall. An identical smooth sphere \(B\) is moving on the horizontal surface with speed \(u\) at an angle \(\theta\) with the line of centres when the spheres collide (see diagram). After the collision, the spheres take the same time to reach a wall. The coefficient of restitution between the spheres is \(\frac{1}{2}\).
  1. Find the value of \(\tan \theta\). [4]
  2. Find the percentage loss in the total kinetic energy of the spheres as a result of this collision. [3]
CAIE Further Paper 3 2023 November Q5
8 marks Challenging +1.8
\includegraphics{figure_5} A bead of mass \(m\) moves on a smooth circular wire, with centre \(O\) and radius \(a\), in a vertical plane. The bead has speed \(v_A\) when it is at the point \(A\) where \(OA\) makes an angle \(\alpha\) with the downward vertical through \(O\), and \(\cos \alpha = \frac{2}{3}\). Subsequently the bead has speed \(v_B\) at the point \(B\), where \(OB\) makes an angle \(\theta\) with the upward vertical through \(O\). Angle \(AOB\) is a right angle (see diagram). The reaction of the wire on the bead at \(B\) is in the direction \(OB\) and has magnitude equal to \(\frac{1}{6}\) of the magnitude of the reaction when the bead is at \(A\).
  1. Find, in terms of \(m\) and \(g\), the magnitude of the reaction at \(B\). [6]
  2. Given that \(v_A = \sqrt{kag}\), find the value of \(k\). [2]
CAIE Further Paper 3 2023 November Q6
9 marks Standard +0.8
A particle \(P\) is projected with speed \(u\) at an angle \(\alpha\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. The horizontal and vertical displacements of \(P\) from \(O\) at a subsequent time \(t\) are denoted by \(x\) and \(y\) respectively.
  1. Derive the equation of the trajectory of \(P\) in the form $$y = x \tan \alpha - \frac{gx^2}{2u^2} \sec^2 \alpha.$$ [3]
During its flight, \(P\) must clear an obstacle of height \(h\) m that is at a horizontal distance of \(32\) m from the point of projection. When \(u = 40\sqrt{2}\) m s\(^{-1}\), \(P\) just clears the obstacle. When \(u = 40\) m s\(^{-1}\), \(P\) only achieves \(80\%\) of the height required to clear the obstacle.
  1. Find the two possible values of \(h\). [6]
CAIE Further Paper 3 2023 November Q7
9 marks Challenging +1.8
\includegraphics{figure_7} A particle \(P\) of mass \(m\) is attached to one end of a light rod of length \(3a\). The other end of the rod is able to pivot smoothly about the fixed point \(A\). The particle is also attached to one end of a light spring of natural length \(a\) and modulus of elasticity \(kmg\). The other end of the spring is attached to a fixed point \(B\). The points \(A\) and \(B\) are in a horizontal line, a distance \(5a\) apart, and these two points and the rod are in a vertical plane. Initially, \(P\) is held in equilibrium by a vertical force \(F\) with the stretched length of the spring equal to \(4a\) (see diagram). The particle is released from rest in this position and has a speed of \(\frac{6}{5}\sqrt{2ag}\) when the rod becomes horizontal.
  1. Find the value of \(k\). [5]
  2. Find \(F\) in terms of \(m\) and \(g\). [2]
  3. Find, in terms of \(m\) and \(g\), the tension in the rod immediately before it is released. [2]
CAIE Further Paper 3 2024 November Q1
5 marks Standard +0.8
A particle \(P\) is projected with speed \(u\text{ms}^{-1}\) at an angle \(\tan^{-1}2\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. When \(P\) has travelled a distance \(56\text{m}\) horizontally from \(O\), it is at a vertical height \(H\text{m}\) above the plane. When \(P\) has travelled a distance \(84\text{m}\) horizontally from \(O\), it is at a vertical height \(\frac{1}{2}H\text{m}\) above the plane. Find, in either order, the value of \(u\) and the value of \(H\). [5]
CAIE Further Paper 3 2024 November Q2
5 marks Challenging +1.2
A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The particle \(P\) is held at the point \(A\) with the string taut. It is given that \(OA\) makes an angle \(\theta\) with the downward vertical through \(O\), where \(\tan\theta = \frac{3}{4}\). The particle \(P\) is projected perpendicular to \(OA\) in an upwards direction with speed \(\sqrt{5ag}\), and it starts to move along a circular path in a vertical plane. When \(P\) is at the point \(B\), where angle \(AOB\) is a right angle, the tension in the string is \(T\). Find \(T\) in terms of \(m\) and \(g\). [5]
CAIE Further Paper 3 2024 November Q3
6 marks Challenging +1.2
A particle \(P\) of mass \(m\text{kg}\) is attached to one end of a light elastic string of natural length \(2\text{m}\) and modulus of elasticity \(2mg\text{N}\). The other end of the string is attached to a fixed point \(O\). The particle \(P\) hangs in equilibrium vertically below \(O\). The particle \(P\) is pulled down vertically a distance \(d\text{m}\) below its equilibrium position and released from rest.
  1. Given that the particle just reaches \(O\) in the subsequent motion, find the value of \(d\). [6]
CAIE Further Paper 3 2024 November Q3
2 marks Moderate -0.5
  1. Hence find the speed of \(P\) when it is \(2\text{m}\) below \(O\). [2]
CAIE Further Paper 3 2024 November Q4
4 marks Standard +0.3
\includegraphics{figure_4} An object is formed by removing a cylinder of radius \(\frac{2}{3}a\) and height \(kh\) (\(k < 1\)) from a uniform solid cylinder of radius \(a\) and height \(h\). The vertical axes of symmetry of the two cylinders coincide. The upper faces of the two cylinders are in the same plane as each other. The points \(A\) and \(B\) are the opposite ends of a diameter of the upper face of the object (see diagram).
  1. Find, in terms of \(h\) and \(k\), the distance of the centre of mass of the object from \(AB\). [4]
CAIE Further Paper 3 2024 November Q4
3 marks Standard +0.8
When the object is suspended from \(A\), the angle between \(AB\) and the vertical is \(\theta\), where \(\tan\theta = \frac{1}{2}\).
  1. Given that \(h = \frac{8}{3}a\), find the possible values of \(k\). [3]
CAIE Further Paper 3 2024 November Q5
4 marks Challenging +1.2
A particle \(P\) of mass \(2\text{kg}\) moving on a horizontal straight line has displacement \(x\text{m}\) from a fixed point \(O\) on the line and velocity \(v\text{ms}^{-1}\) at time \(t\). The only horizontal force acting on \(P\) is a variable force \(F\text{N}\) which can be expressed as a function of \(t\). It is given that $$\frac{v}{x} = \frac{3-t}{1+t}$$ and when \(t = 0\), \(x = 5\).
  1. Find an expression for \(x\) in terms of \(t\). [4]
CAIE Further Paper 3 2024 November Q5
3 marks Moderate -0.5
  1. Find the magnitude of \(F\) when \(t = 3\). [3]
CAIE Further Paper 3 2024 November Q6
3 marks Standard +0.3
\includegraphics{figure_6} A particle \(P\) of mass \(0.05\text{kg}\) is attached to one end of a light inextensible string of length \(1\text{m}\). The other end of the string is attached to a fixed point \(O\). A particle \(Q\) of mass \(0.04\text{kg}\) is attached to one end of a second light inextensible string. The other end of this string is attached to \(P\). The particle \(P\) moves in a horizontal circle of radius \(0.8\text{m}\) with angular speed \(\omega\text{rads}^{-1}\). The particle \(Q\) moves in a horizontal circle of radius \(1.4\text{m}\) also with angular speed \(\omega\text{rads}^{-1}\). The centres of the circles are vertically below \(O\), and \(O\), \(P\) and \(Q\) are always in the same vertical plane. The strings \(OP\) and \(PQ\) remain at constant angles \(\alpha\) and \(\beta\) respectively to the vertical (see diagram).
  1. Find the tension in the string \(OP\). [3]
CAIE Further Paper 3 2024 November Q6
3 marks Moderate -0.5
  1. Find the value of \(\omega\). [3]
CAIE Further Paper 3 2024 November Q6
2 marks Moderate -1.0
  1. Find the value of \(\beta\). [2]
CAIE Further Paper 3 2024 November Q7
4 marks Challenging +1.8
A particle \(P\) is projected with speed \(u\) at an angle \(\tan^{-1}\left(\frac{4}{3}\right)\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. When \(P\) is moving horizontally, it strikes a smooth inclined plane at the point \(A\). This plane is inclined to the horizontal at an angle \(\alpha\), and the line of greatest slope through \(A\) lies in the vertical plane through \(O\) and \(A\). As a result of the impact, \(P\) moves vertically upwards. The coefficient of restitution between \(P\) and the inclined plane is \(e\).
  1. Show that \(e\tan^2\alpha = 1\). [4]
CAIE Further Paper 3 2024 November Q7
6 marks Challenging +1.2
In its subsequent motion, the greatest height reached by \(P\) above \(A\) is \(\frac{3}{10}\) of the vertical height of \(A\) above the horizontal plane.
  1. Find the value of \(e\). [6]
CAIE Further Paper 3 2024 November Q1
3 marks Standard +0.3
A particle of mass \(2\) kg is attached to one end of a light elastic string of natural length \(0.8\) m and modulus of elasticity \(100\) N. The other end of the string is attached to a fixed point \(O\) on a smooth horizontal surface. The particle is moving in a horizontal circle about \(O\) with the string taut and with constant angular speed \(5\) radians per second. Find the extension of the string. [3]
CAIE Further Paper 3 2024 November Q2
4 marks Standard +0.3
A particle \(P\) of mass \(m\) is attached to one end of a light elastic spring of natural length \(a\) and modulus of elasticity \(5mg\). The other end of the spring is attached to a fixed point \(O\). The spring hangs vertically with \(P\) below \(O\). The particle \(P\) is pulled down vertically and released from rest when the length of the spring is \(\frac{7}{5}a\). Find the distance of \(P\) below \(O\) when \(P\) first comes to instantaneous rest. [4]