Cambridge & Pearson Edexcel Past Paper Questions

Let f(x) = x² + x + 6 x²(x + 2) (i) Express f(x) in partial fractions. (ii) Hence, showing full working, show that the exact value of S 4 f(x) dx...

(i) By first expanding cos(2x + x), show that cos 3x = 4 cos³ x − 3 cos x. 3 [4] (ii) Hence solve the equation cos 3x + 3 cos x + 1 = 0, for 0 ≤ x...

10 (a) The complex number u is given by u = −3 – (2√10)i. Showing all necessary working and without using a calculator, find the square roots of u....

Solve the equation 5 ln(4 – 3ˣ) = 6. Show all necessary working and give the answer correct to 3 decimal places.

The curve with equation y = e⁻²ˣ / (1-x²) has a stationary point in the interval −1 < x < 1. Find dy/dx and hence find the x-coordinate of this sta...

The polynomial x⁴ + 3x³ + ax + b, where a and b are constants, is denoted by p(x). When p(x) is divided by x² + x − 1 the remainder is 2x + 3. Find...

The context for the parts is: Express (√6) sin x + cos x in the form R sin(x + α), where R > 0 and 0° < α < 90°. Hence solve an equation.

The equation of a curve is 2x²y – xy² = a³, where a is a positive constant. Show that there is only one point on the curve at which the tangent is ...

The variables x and θ satisfy the differential equation sin(θ/2) dx/dθ = (x + 2) cos(θ/2) for 0 < θ < π. It is given that x = 1 when θ = π/3. Solve...

The context for the parts involves complex numbers.

Let f(x) = (2x² + x + 8) / ((2x−1)(x² + 2)).

It is given that ∫(from 0 to a) x cos(x/3) dx = 3, where the constant a is such that 0 < a < (3/2)π.

The line l has equation r = i + 3j – 2k + λ(i – 2j + 3k). The plane p has equation 2x + y − 3z = 5.

Solve the inequality 2|x + 2| > |3x – 1|.

The polynomial 6x³ + ax² + bx − 2, where a and b are constants, is denoted by p(x). It is given that (2x + 1) is a factor of p(x) and that when p(x...

Showing all necessary working, solve the equation (3^(2x) + 3^(-x)) / (3^(2x) - 3^(-x)) = 4. Give your answer correct to 3 decimal places.

By first expanding tan(2x + x), show that the equation tan 3x = 3 cot x can be written in the form tan⁴x - 12 tan²x + 3 = 0.

By sketching a suitable pair of graphs, show that the equation ln(x + 2) = 4e⁻ˣ has exactly one real root.

Throughout this question the use of a calculator is not permitted. The complex number with modulus 1 and argument 1/3π is denoted by w. The complex...

The plane m has equation x + 4y – 8z = 2. The plane n is parallel to m and passes through the point P with coordinates (5, 2, −2).

The diagram shows the graph of y = sec x for 0 ≤ x < 1/2π.

The variables x and t satisfy the differential equation 5 dx/dt = (20−x)(40 – x). It is given that x = 10 when t = 0.

The diagram shows the graph of y = e^(cosx) sin³x for 0 ≤ x ≤ π, and its maximum point M. The shaded region R is bounded by the curve and the x-axis.

A crane is lifting a load of 1250 kg vertically at a constant speed Vm s⁻¹. Given that the power of the crane is a constant 20 kW, find the value o...

The total mass of a cyclist and her bicycle is 75 kg. The cyclist ascends a straight hill of length 0.7 km inclined at 1.5° to the horizontal. Her ...

A block of mass 3 kg is at rest on a rough plane inclined at 60° to the horizontal. A force of magnitude 15 N acting up a line of greatest slope of...

Two blocks A and B of masses 4 kg and 5 kg respectively are joined by a light inextensible string. The blocks rest on a smooth plane inclined at an...

A small ring P is threaded on a fixed smooth horizontal rod AB. Three horizontal forces of magnitudes 4.5 N, 7.5 N and F N act on P [Figure X.X].

A particle of mass 0.4 kg is released from rest at a height of 1.8 m above the surface of the water in a tank. There is no instantaneous change of ...

A particle moves in a straight line, starting from rest at a point O, and comes to instantaneous rest at a point P. The velocity of the particle at...

A particle moves in a straight line. The displacement of the particle at time ts is sm, where s = t³ – 6t² + 4t. Find the velocity of the particle ...

The diagram shows a velocity-time graph which models the motion of a tractor. The graph consists of four straight line segments. The tractor passes...

A particle P of mass 0.3 kg is held in equilibrium above a horizontal plane by a force of magnitude 5 N, acting vertically upwards. The particle is...

A lorry of mass 25 000 kg travels along a straight horizontal road. There is a constant force of 3000 N resisting the motion.

Two particles A and B move in the same vertical line. Particle A is projected vertically upwards from the ground with speed 20m s⁻¹. One second lat...

A block of mass 3 kg is initially at rest on a rough horizontal plane. A force of magnitude 6N is applied to the block at an angle of θ above the h...

Two particles P and Q, of masses 0.3 kg and 0.2 kg respectively, are attached to the ends of a light inextensible string. The string passes over a ...

A crate of mass 500 kg is being pulled along rough horizontal ground by a horizontal rope attached to a winch. The winch produces a constant pullin...

A train of mass 150000 kg ascends a straight slope inclined at α° to the horizontal with a constant driving force of 16000 N. At a point A on the s...

Three coplanar forces of magnitudes 50 N, 60 N and 100 N act at a point. The resultant of the forces has magnitude R N. The directions of these for...

A car travels along a straight road with constant acceleration. It passes through points P, Q, R and S. The times taken for the car to travel from ...

A cyclist is travelling along a straight horizontal road. The total mass of the cyclist and his bicycle is 80 kg. His power output is a constant 24...

Particle P travels in a straight line from A to B. The velocity of P at time t s after leaving A is denoted by v m s⁻¹, where v = 0.04t³ + ct² + kt...

Two particles A and B have masses m kg and km kg respectively, where k > 1. The particles are attached to the ends of a light inextensible string. ...

A uniform solid cone has weight 5 N and base radius 0.1 m. AB is a diameter of the base of the cone. The cone is held in equilibrium, with A in con...

A particle is projected from a point on horizontal ground with speed 15 m s¯¹ at an angle of θ° above the horizontal. The particle strikes the grou...

A smooth horizontal surface has two fixed points O and A which are 0.8 m apart. A particle P of mass 0.25 kg is projected with velocity 3 m s¯¹ hor...

A small ball B is projected with speed 30 m s¯¹ at an angle of 60° above the horizontal from a point O. At time t s after projection the horizontal...

A and B are two fixed points on a vertical axis with A 0.6 m above B. A particle P of mass 0.3 kg is attached to A by a light inextensible string o...

ABC is the cross-section through the centre of mass of a uniform prism which rests with AB on a rough horizontal surface. AB = 0.4 m and C is 0.9 m...