Cambridge Past Paper Questions

The diagram shows the curve with parametric equations x = 3 ln(2t – 3), y = 4t ln t. The curve crosses the y-axis at the point A. At the point B, t...

The curve with equation e^(2x) – 18x + y³ + y = 11 has a stationary point at (p, q).

Find the exact coordinates of the points on the curve y = x² / (1-3x) at which the gradient of the tangent is equal to 8.

The variables x and y are related by the equation y = ab^x, where a and b are constants. The diagram shows the result of plotting In y against x fo...

The complex number u is defined by u = (3 + 2i) / (a - 5i), where a is real.

The parametric equations of a curve are x = √t + 3, y = ln t, for t > 0.

The variables x and θ satisfy the differential equation (x / tan θ) (dx / dθ) = x² + 3. It is given that x = 1 when θ = 0. Solve the differential e...

The diagram shows the curve y = xe^(-x²/4), for x ≥ 0, and its maximum point M. [Figure 9]

Let f(x) = (24x + 13) / ((1 - 2x)(2 + x)²)

In the diagram, OABCDEFG is a cuboid in which OA = 3 units, OC = 2 units and OD = 2 units. Unit vectors i, j and k are parallel to OA, OD and OC re...

1 (a) Sketch the graph of y = |4x − 2|. (b) Solve the inequality 1 + 3x < |4x − 2|.

2 The parametric equations of a curve are x = (ln t)², y = e^(2-t²), for t > 0. Find the gradient of the curve at the point where t = e, simplifyin...

3 The polynomial 2x³ + ax² – 11x + b is denoted by p(x). It is given that p(x) is divisible by (2x – 1) and that when p(x) is divided by (x + 1) th...

4 (a) On a sketch of an Argand diagram, shade the region whose points represent complex numbers z satisfying the inequalities |z − 4 − 3i| ≤ 2 and ...

5 Find the exact value of ∫(from 0 to 6) [x(x + 1) / (x² + 4)] dx.

6 (a) By sketching a suitable pair of graphs, show that the equation cot x = 2 − cos x has one root in the interval 0 < x < ½π. (b) Show by calcula...

7 (a) By expressing 3θ as 2θ + θ, prove the identity cos 3θ = 4 cos³ θ – 3 cos θ. (b) Hence solve the equation cos 3θ + cos θ cos 2θ = cos² θ for 0...

8 It is given that (2 + 3ai) / (a + 2i) = λ(2 – i), where a and λ are real constants. (a) Show that 3a² + 4a – 4 = 0. (b) Hence find the possible v...

9 The diagram shows the curve y = sin x cos 2x, for 0 ≤ x ≤ π, and a maximum point M, where x = a. The shaded region between the curve and the x-ax...

10 The equations of the lines l and m are given by l: r = (3i - 2j + k) + λ(3i + j + 2k) and m: r = (6i - 3j + 6k) + μ(-2i + 4j + ck), where c is a...

11 The variables x and y satisfy the differential equation x² dy/dx + y² + y = 0. It is given that x = 1 when y = 1. (a) Solve the differential equ...

Find the set of values of x satisfying the inequality | 2x+1 – 2| < 0.5, giving your answer to 3 significant figures.

The polynomial 2x³ + ax² + bx + 6, where a and b are constants, is denoted by p(x). When p(x) is divided by (x + 2) the remainder is –38 and when p...

Solve the quadratic equation (3 + i)w² – 2w + 3 − i = 0, giving your answers in the form x + iy, where x and y are real.

Find the exact coordinates of the stationary points of the curve y = e^(3x²-1) / (1-x²).

The equation of a curve is x³ + y² + 3x² + 3y = 4.

The variables x and y satisfy the differential equation e^(4x) dy/dx = cos² 3y. It is given that y = 0 when x = 2. Solve the differential equation,...

Let f(x) = (17x² − 7x + 16) / ((2 + 3x²)(2−x)).

The diagram shows the curve y = x cos 2x, for x ≥ 0. [Figure X.X]

The line l has equation r = i – 2j – 3k + λ(−i + j + 2k). The points A and B have position vectors –2i + 2j – k and 3i – j + k respectively.

A particle of mass 1.6kg is projected with a speed of 20ms¯¹ up a line of greatest slope of a smooth plane inclined at α to the horizontal, where t...

A particle of mass 2.4kg is held in equilibrium by two light inextensible strings, one of which is attached to point A and the other attached to po...

The diagram shows the velocity-time graph for the motion of a bus [Figure 3.1]. The bus starts from rest and accelerates uniformly for 8 seconds un...

Two particles P and Q, of masses 6kg and 2kg respectively, lie at rest 12.5m apart on a rough horizontal plane. The coefficient of friction between...

The diagram shows a particle A, of mass 1.2kg, which lies on a plane inclined at an angle of 40° to the horizontal and a particle B, of mass 1.6kg,...

A car of mass 1300kg is moving on a straight road.

A particle moves in a straight line starting from a point O before coming to instantaneous rest at a point X. At time t s after leaving O, the velo...

A block of mass 15kg slides down a line of greatest slope of an inclined plane. The top of the plane is at a vertical height of 1.6m above the leve...

The diagram shows a smooth ring R, of mass mkg, threaded on a light inextensible string. A horizontal force of magnitude 2N acts on R. The ends of ...

A block of mass 10kg is at rest on a rough plane inclined at an angle of 30° to the horizontal. A force of 120N is applied to the block at an angle...

A particle P of mass 0.2kg lies at rest on a rough horizontal plane. A horizontal force of 1.2N is applied to P.

A particle A of mass 0.5 kg is projected vertically upwards from horizontal ground with speed 25 m s⁻¹.

A railway engine of mass 120000kg is towing a coach of mass 60000kg up a straight track inclined at an angle of α to the horizontal where sin α = 0...

A particle X travels in a straight line. The velocity of X at time ts after leaving a fixed point O is denoted by vms⁻¹, where v = −0.1t³ + 1.8t² –...

A particle is projected vertically upwards from horizontal ground with a speed of ums⁻¹. The particle has height sm above the ground at times 3 sec...