ADVANCED MATHEMATICS. (1.)

Time allowed, 3 hours.

1. The diameter of the fore-wheel of a bicycle is a feet, and of the hinder-wheel b feet; what is the distance travelled when the hinder-wheel has made m revolutions more than the fore-wheel?

2. Prove the Binomial Theorem for exponents which are positive integers. If the difference between a and b be small compared to either of them, show that 5/a- 5/b

3/a-3/b

is nearly equal to

3 62

3. Eliminate a, b, c from the equations

5 a

4. In an equation of the form ax + by =c, show how to find the solutions in positive integers, and their number.

5. Prove that sin (A + B)= sin A cos B + cos A sin B. Being given y sin C+z sin2 B=z sin2 A+ sin 2 C x sin2 B+ y sin2 A, and A+B+C=180°;

prove that

Ꮳ

y =

sin 2A sin 2B sin 20

6. Prove that cos 0-2 cos 0 cos a cos (a+0) + cos2(a + 0)

is independent of 0.

7. Prove that 2 cos no

Indicate briefly the development of crystallographic theory which follows from this series.

8. Prove that in a spherical triangle

Deduce the relation which connects the small variation of a side and the opposite angle.

9. If from the vertices A, B, C of a spherical triangle arcs of great circles are drawn intersecting in P and meeting the opposite sides in D, E, F respectively, prove that

Point out the use made by Professor Miller of this relation in crystallography.

10. Prove the equations employed in the solution of rightangled triangles when two of its elements are given.

State Napier's rules which summarise these results.

11. A line is drawn bisecting the vertical angle C of a triangle ABC and meeting the opposite side in D ; show that DA:DB:: AC: BC.

If from the angle A of any parallelogram any line be drawn cutting the diagonal in P, and the sides BC, CD, produced if necessary in Q, R; show that AP2 PQ. PR.

=

ADVANCED MATHEMATICS. (2.)

Time allowed, 3 hours.

1. Show that the equation x cos a + y sin a p = 0 represents a straight line; and state the geometric meaning of the constants.

Obtain expressions which give the angle between

the two lines+1= Q, and
0,

-1=0;

and show when they will be at right angles.

2. Find the equation of the tangent to the parabola, 4ax, in terms of the tangent of its inclination to the axis.

y3

Show that tangents at right angles intersect on the directrix; and find the locus of intersection of

tangents which cut one another at a constant angle.

3. Determine the condition that the general equation ax2+2hxy + by2 + 2yx + 2fy + c = 0

should represent two straight lines; and also the condition that they should be at right angles to one another.

4, Find the polar equation of the ellipse, taking the focus as origin and the major axis as initial line.

Show that any focal chord is a third proportional to the transverse axis and the parallel diameter. 5. Find the equation to the perpendicular through the origin on the plane h+k+12=d, assuming

α

the axes to be rectangular.

Find the condition that the two planes

should be simultaneously at right angles to the first plane.

6. If u be a given function of У and 2, and y: & given functions of x, prove that

dv

du dy

=

+

du dz,

dx dy dx dz dx

Differentiate sin (x sin x) and log (tan-i x).

7. State and prove Taylor's theorem

Expand log (1 + x) in terms of x, and find an expression for the remainder after n terms.

8. Find the limiting value, when x = 0, of

(1) (cos x) cot; (2)

sin x

in x ) a

X

9. Obtain an expression for the radius of curvature of a plane curve; and find the values of the greatest and least radii of curvature of an ellipse.

10. What is meant by an envelope in the theory of curves? Show how to find the envelope of ƒ (x,y,a) = 0.

Find the envelope of a series of circles having their centres in a straight line and such that the radius is proportional to the distance of the centre from a fixed point on the straight line.

11, Integrate the following differentials

12. Show that the area common to two ellipses, which have the same centre and equal axes, but in which the major axes are at right angles to one another

is 4ab tan-1

y2

b

α

Also find the length of an arc of the parabola 4ax, from a given point to the vertex.

ADVANCED MATHEMATICS (3).

Time allowed, 3 hours.

1. Explain what is meant by an integrating factor of a differential equation; and find the equation which gives the integrating factors of Mdx + Ndy

Find an integrating factor of

(x3y3 + 1) ydx + (x3y3 — 1) xdy = 0

and solve the equation.

= 0.

medy

=

3yx-y3,

2. Show how to transform the equation x2

da

into a linear differential equation, and thus, or in other way, find the primitive.

any

3. Being given the equation of the caustic produced by a series of rays, incident in parallel directions on a plane curve, find the differential equation of the reflecting curve; and show that the complete primitive represents a series of parabolæ, having their axes parallel to the incident rays.

4. Prove the parallelogram of forces for commensurable forces.

Two unequal weights are connected by a string and rest on the convex surface of a smooth vertical circle; find the position of equilibrium. Show that in this position the centre of gravity of the two weights lies in the vertical through the centre.

5. Define the term "moment of a force." Show that the sum of the moments of two forces with respect to a point in their plane, is equal to the moment of their resultant with respect to the point.

6. A uniform beam is supported horizontally at its extremities on two vertical props; show that the

bending moment at any point of it is Wy

(21-y)

21

where W is the weight of the beam, y the distance of the point from one extremity, and 27 the length of the beam.

7. Show how to find the coordinates of the centre of mass of a plane curve of varying section and density.

Find the centre of mass of a uniform wire in the shape of a cardioid r = a (1 + cos 0); and also of a uniform plate which has the shape of a loop of the lemniscate r2 = a2 cos 20.

8. Find the equations of equilibrium of a heavy flexible string suspended at its two extremities. And show that the catenary of uniform strength is represented by

9. Enunciate Newton's Laws of Motion; and from Law II. write down the equations of motion of a particle, under the action of given forces.

10. Obtain and solve the equation representing the motion of a stone dropped from a very great height, neglecting the resistance of the atmosphere.

Deduce from the solution the ordinary expression for the velocity acquired in falling through h feet at the earth's surface.

11. Find expressions for the acceleration of a particle in polar coordinates.

Find the differential equation of the path of a particle acted on by a central force varying inversely