INTERMEDIATE HONOURS.

SYNTHETIC SOLID GEOMETRY.

1. Show how to draw a normal to a plane from a point in the plane.

2. Develop formulae to find the number of faces, edges and corners, in any polyhedron, given the number of faces at a corner, and the number of sides to a face; and apply to the regular dodecahedron.

3. In any tetrahedron prove that

(a) nine times the sum of the squares on the medians is equal to four times the sum of the squares on the edges. (b) R2: p2: r2=9:3:1 where R, p and r are the radii of the circumscribed sphere, tangent sphere to the edges, and inscribed sphere, respectively.

4. Find an expression for the volume of a ppd. in terms of the direction-edges and face-angles at any corner.

5. Show that the prismoidal formula applies to the sphere.

6. A plane figure X, guided by the semi-cubical parabola =pr3, moves from the vertex, perpendicularly to the axis, through a distance h. Find the volume generated.

7. Find the volume of a circular spindle in terms of the chord and area of the revolving segment.

8. Prove one of the theorems of Apollonius by making sections of a cylinder.

9. (a) A cylindrical hole, of radius a, is bored centrically through a sphere of radius r. Show that the volume removed is 4r (1-cos30)/3, where sin @=a/r.

moved.

(b) Find the portion of the surface of the sphere re

10. By using the principle of perspective projection, prove that every line parallel to the axis of a parabola is a dia

meter.

FINAL HONOURS.

QUATERNIONS.

1. Prove that, in general, vector multiplication is not commutative.

2. Show that By/ay is not the same as yẞ/ya, and explain why.

3. Prove that V.aVBy=ySaß - ẞSay; and thence deduce the corresponding equivalent for V. VaßVyd.

4. In a spheric triangle show that sin A sin b sin c= -Saẞy, where a,B,y are vectors from the centre of the unit sphere to the vertices.

5. Find the vector to the point where the line V(p-8)B=0 meets the plane p=y+аT, SаT=0.

6. (a) Show that S(p-y) (y) a2 is tangent to a circle at the point p.

(b) If π is fixed, find the perpendicular upon the line determined by p.

7. In the ellipse with equation Spop=1, find op when p=xi+yj; thence derive the function p, and explain the relation between these two functions.

8. If w,' are asymptotes of a hyperbola, deduce the result that xy=a constant, where p = xw+yw' &c.

9. Obtain the equations of a parabola (a) with the vertex as origin, and (b) with the focus as origin.

10. OA is perpendicular from a fixed point to a fixed plane, and T is a variable point in the plane. On OT, OP is taken so that TP= TA. Find the locus of P, and show that the parallel plane at distance 20A is asymptotic.

FINAL HONOURS.

FINITE DIFFERENCES.

1. Find the value of each of the following, and deduce the analogous expressions in the differential calculus :-A"a*, (x) as a series of factorials, Arun as a series of differences, Exm), "cos(ax+b). 2. Find the value of Ex sin(ax+b), and interpret the meaning of it in the summation of series.

3. Develop the inverted form of Maclaurin's series. Apply it to log(1+x).

4. Find Weddle's formula for areas. What error would result in using it to find the area of y=16−x2 lying above the x-axis?

5. Express Eau, as a series of differences of u. Sum 3'.1+3°.23 +33.33 +34.43 + . . . . . .

terms.

to n

6. (a) Find the sum of the rth powers of the first n natural numbers in terms of the Bernouillian numbers. (b) Indicate two methods for finding the Bernouillian numbers, and find the first three.

1

7. By difference methods sum -4+ 10-13+..

7

correct to 4 places of decimals.