OCR A Level Maths B (MEI) Pure Mathematics and Mechanics June 2019 question paper H640/01

In this question you must show detailed reasoning.

Show that the line which passes through the points (2, - 4) and (- 1, 5) does not intersect the line

3 x + y = 10.

The function f(x) is given by f(x) = (1 - ax)^-3, where a is a non-zero constant. In the binomial expansion of f(x), the coefficients of x and x² are equal.

Find the value of a.

Using this value for a,

state the set of values of x for which the binomial expansion is valid,

write down the quadratic function which approximates f (x) when x is small.

Fig. 4 shows a uniform beam of mass 4 kg and length 2.4 m resting on two supports P and Q. P is at one end of the beam and Q is 0.3 m from the other end.

Determine whether a person of mass 50 kg can tip the beam by standing on it.

Fig. 4

A car of mass 1200 kg travels from rest along a straight horizontal road. The driving force is 4000 N and the total of all resistances to motion is 800 N.

Calculate the velocity of the car after 9 seconds.

The velocity v ms^-1 of a particle at time t s is given by

v = 0.5 t (7 - t).

Determine whether the speed of the particle is increasing or decreasing when t = 8.

An arithmetic series has first term 9300 and 10th term 3900.

Show that the 20th term of the series is negative.

The sum of the first n terms is denoted by S. Find the greatest value of S as n varies.

A cannonball is fired from a point on horizontal ground at 100ms^-1 at an angle of 25° above the horizontal. Ignoring air resistance, calculate

the greatest height the cannonball reaches,

the range of the cannonball.

Express 7 cos x - 2 sin x in the form R cos(x + α) where R > 0 and 0 < α < , giving the

exact value of R and the value of α correct to 3 significant figures.

Give details of a sequence of two transformations which maps the curve y = sec x onto the

Write down the value of k.

Initially the velocity of the particle is (4i + 7j)ms^-1.

Find the velocity of the particle 10 seconds later.

Fig. 12 shows a curve C with parametric equations x = 4t², y = 4t. The point P, with parameter t is a general point on the curve. Q is the point on the line x + 4 = 0 such that PQ is parallel to the x-axis. R is the point (4, 0).

Show algebraically that P is equidistant from Q and R.

Find a cartesian equation of C.

A 15 kg box is suspended in the air by a rope which makes an angle of 30° with the vertical. The box is held in place by a string which is horizontal.

Draw a diagram showing the forces acting on the box.

Calculate the tension in the rope.

Calculate the tension in the string.

Fig. 14 shows a circle with centre O and radius r cm. The chord AB is such that angle AOB = x radians. The area of the shaded segment formed by AB is 5% of the area of the circle.

The Newton-Raphson method is to be used to find x.

Write down the iterative formula to be used for the equation in part (a).

Use three iterations of the Newton-Raphson method with x₀ = 1.2 to find the value of x to a

suitable degree of accuracy.

A model for the motion of a small object falling through a thick fluid can be expressed using the differential equation

where v ms^-1 is the velocity after t s and k is a positive constant.

Given that v = 0 when t = 0, solve the differential equation to find v in terms of t and k.

Sketch the graph of v against t.

Experiments show that for large values of t, the velocity tends to 7ms^-1.

Find the value of k.

Find the value of t for which v = 3.5.

A particle of mass 2 kg slides down a plane inclined at 20° to the horizontal. The particle has an initial velocity of 1.4ms^-1 down the plane. Two models for the particle’s motion are proposed.

In model A the plane is taken to be smooth.

Calculate the time that model A predicts for the particle to slide the first 0.7 m.

Explain why model A is likely to underestimate the time taken.

In model B the plane is taken to be rough, with a constant coefficient of friction between the particle and the plane.

Calculate the acceleration of the particle predicted by model B given that it takes 0.8 s to slide

the first 0.7 m.

Find the coefficient of friction predicted by model B, giving your answer correct to 3 significant

figures.