UNIVERSITIES OF MANCHESTER LIVERPOOL
LEEDS SHEFFIELD AND BIRMINGHAM
JOINT MATRICULATION BOARD
GENERAL CERTIFICATE OF EDUCATION
Friday 4 June 1965 9.30-12.30
Careless and untidy work will be penalized.
Answer seven questions.
83 ADV EE Turn over
1. (a) Solve for x and y the simultaneous equations
where a and b are positive.
if x and y also satisfy the relation
(b) If x is real and
show that k cannot lie between certain limits, and find these limits.
2. (a) Express the following complex numbers in the form a + ib, where a and b are real:
(ii) , where c is real.
(b) If: and , where x, y, a, b are rea1, prove that
By solving the equation
for , or otherwise. express each of the four roots of the equation in the form x+ iy.
3. (a) Find (i) the sum and (ii) the product of the first n terms of the geometric sequence 18, 6, 2,...
(b) If , show that provided
that y is either positive or less than -1.
Use the expansions of and to show that
for y > 0 and for y < -1.
From the first three terms of this series, calculate giving your answer to three decimal places.
4. (a) Prove that, in the triangle ABC.
Show that, if cot A. cot B, cot C are in arithmetic progression, then so are a2, b2, c2.
(b) Find the solutions between -1800 and 1800 of the equation
giving your answers correct to the nearest tenth of a degree.
83 ADV Turn over
5. (a) Show that the equation of the tangent at the origin to the circle is
The circle cuts a circle C at right angles at the origin. If C passes through the
point (2, 0), find its equation. -
(b) Show that the equation of the tangent to the parabola at the’ point is
Find the equation of the common tangent to the parabolas
6. Show that the equation of the normal to the ellipse
at the point is
If J and K are the feet of the perpendiculars from the centre O to the normal and the tangent at P respectively. find the lengths of OJ and OK. and deduce that the area of the rectangle OJPK is
If the normal meets Ox at G and Oy at H. show that the locus of the mid-point of GH as varies is an ellipse.
7. (a) Find the values of x for which the function
has its maximum and minimum values.
Find the coordinates of the maximum and minimum points, and of the point of inflexion on the graph of the function. Sketch the graph.
(b) Find the volume generated by the rotation through one revolution about the x-axis of the region between the x-axjs and that part of the curve for which and y > 0.
8. (a) A tower stands on level ground. A man observes that the top of the tower is at an elevation to the horizontal from a fixed point on the ground. He walks a distance x straight towards the tower, and finds that the elevation is now. Express the height h of the tower in terms of , x and .
Find dx/d in terms of h and and show that, if increases by as x increases by z. where and z are small, then
(b) A curve is described by the equation
Obtain an expression for dy/dx.
If (a, b) is the point of the curve where dy/dx = 0, prove that and hence find a and b.
9. (a) Prove that
(c) By means of the substitution , or otherwise, evaluate