Cambridge Past Paper Questions
Browse 23,045questions from 25 years of O-Level & A-Level exams. Click any question to practice.
The sequence of values given by the iterative formula xn+1 = (6+8xn) / (8+xn²) with initial value x₁ = 2 converges to α.
It is given that 3 sin 2θ = cos θ where θ is an angle such that 0° < θ < 90°.
A curve is defined by the parametric equations x = 3t - 2 sin t, y = 5t + 4 cos t, where 0 ≤ t ≤ 2π. At each of the points P and Q on the curve, th...
A curve has equation y = f(x) where f(x) = (4x³ + 8x - 4) / (2x - 1)
Solve the equation 7 cot θ = 3 cosec θ for 0° < θ < 90°.
Given that 2³ˣ⁺² + 8 2³ˣ - 7 to 4 significant figures. = 5, find the value of 2³ˣ and hence, using logarithms, find the value of x correct
The diagram shows the curve with equation y = x-2 x² + 8 and the lines x = 14 and y = 0. [Figure 4.1]
The equation of a curve is 2e²ˣy - y³ + 4 = 0.
A curve has equation y = f(x) where f(x) = x⁴ – 5x³ + 6x² + 5x – 15. As shown in the diagram, the curve crosses the x-axis at the points A and B wi...
Given that ln(2x + 1) – ln(x − 3) = 2,
The sequence of values given by the iterative formula xₙ₊₁ = (6+8xₙ)/(8+xₙ²) with initial value x₁ = 2 converges to α.
A curve is defined by the parametric equations x = 3t – 2 sin t, y = 5t + 4 cos t, where 0 ≤ t ≤ 2π. At each of the points P and Q on the curve, th...
A curve has equation y = f(x) where f(x) = (4x³ + 8x – 4) / (2x – 1).
Solve the inequality 2 – 5x > 2|x-3|.
The parametric equations of a curve are x = 3 - cos 2θ, y = 2θ + sin 2θ, for 0 < θ < ½π. Show that dy/dx = cot θ.
Solve the equation log₁₀(2x + 1) = 2log₁₀(x + 1) – 1. Give your answers correct to 3 decimal places.
By sketching a suitable pair of graphs, show that the equation cosec x = 1 + e¯½ˣ has exactly two roots in the interval 0 < x < π.
Express √6 cos θ + 3 sin θ in the form R cos(θ – α), where R > 0 and 0° < α < 90°. State the exact value of R and give α correct to 2 decimal places.
Verify that –1 + √5i is a root of the equation 2x³ + x² + 6x – 18 = 0.
The coordinates (x, y) of a general point of a curve satisfy the differential equation x dy/dx = (1 - 2x²)y, for x > 0. It is given that y = 1 when...
Let f(x) = (8 + 5x + 12x²)/((1-x)(2 + 3x)²).
The diagram shows the curve y = (2 – x)e¯½ˣ, and its minimum point M. [Figure 10.1]
Two lines have equations r = i + 2j + k + λ(ai + 2j – k) and r = 2i + j – k + μ(2i – j + k), where a is a constant.
Solve the equation ln(1 + e¯3x) = 2. Give the answer correct to 3 decimal places.
The variables x and y satisfy the relation 2ʸ = 3¹⁻²ˣ.
The diagram shows the curve with parametric equations x = tan θ, y = cos² θ, for −π < θ < π. [Figure X.X]
The complex number u is defined by u = 7+ i 1-i
The variables x and t satisfy the differential equation e³ᵗ dx/dt = cos² 2x, for t ≥ 0. It is given that x = 0 when t = 0.
With respect to the origin O, the position vectors of the points A, B, C and D are given by OA = (2, 1, 5), OB = (4, -1, 1), OC = (1, 1, 2) and OD ...
Let f(x) = 7x + 18 (3x + 2)(x² + 4)
The diagram shows the curve y = √x cos x, for 0 < x < ¾π, and its minimum point M, where x = a. The shaded region between the curve and the x-axis ...
The diagram shows the curve y = (2 – x)e⁻½ˣ, and its minimum point M. [Figure 10.1]
Two lines have equations r = i + 2j + k + λ(ai + 2j – k) and r = 2i + j – k + µ(2i – j + k), where a is a constant.
A particle B of mass 5 kg is at rest on a smooth horizontal table. A particle A of mass 2.5 kg moves on the table with a speed of 6 m s⁻¹ and colli...
A car of mass 1400 kg is moving along a straight horizontal road against a resistance of magnitude 350 N.
Coplanar forces of magnitudes 8 N, 12 N, 10 N and P N act at a point in the directions shown in the diagram. The system is in equilibrium. [Figure ...
A particle P moves in a straight line. It starts from rest at a point O on the line and at time t s after leaving O it has acceleration a m s⁻², wh...
Two particles of masses 0.8 kg and 0.2 kg are connected by a light inextensible string that passes over a fixed smooth pulley. The system is releas...
A car of mass 1500 kg is pulling a trailer of mass 750 kg up a straight hill of length 800 m inclined at an angle of sin⁻¹ 0.08 to the horizontal. ...
Three points A, B and C lie on a line of greatest slope of a plane inclined at an angle of 30° to the horizontal, with AB = 1 m and BC = 1m, as sho...
Two particles P and Q, of masses 0.2 kg and 0.5 kg respectively, are at rest on a smooth horizontal plane. P is projected towards Q with speed 2 m ...
A car of mass 1800 kg is travelling along a straight horizontal road. The power of the car's engine is constant. There is a constant resistance to ...
A block of mass m kg is held in equilibrium below a horizontal ceiling by two strings, as shown in the diagram. One of the strings is inclined at 4...
The diagram shows a velocity-time graph which models the motion of a car. The graph consists of four straight line segments. The car accelerates at...