2. Prove that (ƒD).eaxX=eaxƒ(D+a).X where X is a function of x; and apply it to develope Sex log x dx in descending powers of x.

3. If

dy

dx

=

d2 x cos2x, find in terms of tan x, and dy2

establish the principle of operation.

3. Express p in terms of a, b, 0 so that p=cos 0 ty sin may touch xm/am+m/6m1; and explain the relation of the polar in p, 0 to the original curve.

5. Show that if a curve of n dimensions has n nonparallel asymptotes, these intersect the curve in n points which lie on a curve of n 2 dimensions.

6. Prove that at a point of contrary flexure u+ d2u/d02 changes sign.

7. Find the conic which has 3rd-order contact with x-y=x -3x3+2x2.

8. If fμ-o be normal to a curve where μ is the tangent of its direction, show that the eliminant of fμ=0 and dƒ/dμ=0 is the evolute.

Apply this method to find the evolute of the parabola.

2. Investigate the position and character of S-K'S'-o, when So and S'=o denote two spheres. 3. Develope the equation of the Hyperbolic Paraboloid, and show that it is a ruled surface.

4. A variable parabola of the form y=px moves with its axis parallel to the x axis, its plane parallel to the xy plane, and its vertex guided by the line Investigate the character of the surface generated, as shown by sections parallel to the three principal planes.

z = tan a.

5. Determine two planes which give circular sections on the ellipsoid; and show that if these be of opposite inclinations they are sections of the same sphere.

6. The semiaxes of an ellipsoid are 3, 4, 5, and a plane through the centre is equally inclined to the three axes. Find the area of the section made by

the plane.

7. If u=f(xyz)=o be a concoid, and p be the radius of curvature at the point l,m,n, show that

2

2

2

p=√ {ux2+u ̧2 + uz2 } / (Σl2 U xx+2Σmnuμz)

where uduəx, ux=Ə2u/Əxǝy, &c.

8. Define the osculating plane of a curve of double curvature at a given point, and find its equation.

TRIGONOMETRY II.

FINAL HONOURS.

I. In the series cos 0=1+a02 +

4a2 4!

8a3

-04+

examine and explain the function and meaning of a.

2. Develope sec-1x in descending powers of x.

1

3. Find a value for at in the form RV.

4. Find in terms of sines of multiples of a when sin 0=m sin(0+a).

5. Prove that S2n(2n) !=22n−1B2", where B is a Bernoullian number, and S=1+2+3+.... 6. Expand i sin(cos ◊)sin(i sin ) in terms of sines of multiples of 0.

7. Prove in any way that the limit of

sin 0+ sin 30+ sin 50+.... is π.

n

n

8. Sum the series Ex cos xe, and Ex sin xe.