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



prove that             


(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.


83 ADV.


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

    and  .





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



(b)  Evaluate


(c)  By means of the substitution , or otherwise, evaluate