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9. Investigate the expression for cos ne in a series of descending powers of cos 0, when n is a positive integer.

IO. Show how to transform an equation into another, the roots of which shall be less than those of the proposed equation by a constant difference.

Transform x-6x2+5=0 into an equation without its second term, and hence find the roots of the original equation.

II. Determine the roots of the equation

2x1 — 5x3+6x2 - 5x+2=0.

12. Describe fully the process known as Des Cartes' method of solving biquadratic equations.

Example. x-3x2 - 42x — 40=0.

VII. PURE MATHEMATICS (3).

[Great importance will be attached to accuracy in results. Full marks may be obtained by doing about four-fifths of this paper.]

1. Explain the terms dependent variable, independent variable, limit of a function.

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2. Find the differential coefficients of x" and sin-1 x.

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5. If f(x) be a function which vanishes when x=a and x=b, and ƒ(x) and ƒ'(x) be both continuous between those values of x, prove that f'(x) will vanish for some value of x between a and b.

Deduce a proof of Taylor's Theorem.

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7. The ordinate of any point P of an ellipse, whose centre is C, meets the axis major in N. Find the positions of P which make the area PCN a maximum.

8. Draw all the asymptotes of the curve

xy ( y − x)=a (3x2 + 2y2).

9. If p be the perpendicular let fall from the pole upon the tangent at any point of the curve r=f(0), prove that

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In the curve 2-a2 sin 20 show that p∞ and that the radius of

curvature ∞

10. Explain what is meant by the evolute of a plane curve.

Show that 473=27ax2 is the equation of the evolute of the parabola x2=4a (y+2a).

II. Find the integrals of the functions

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12. Show how to find by integration the length of a curve whose equation is given in rectangular co-ordinates.

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VIII. STATICS.

[Great importance will be attached to accuracy.]

1. If the parallelogram of forces be true for the direction of the resultant of one force and each of two other forces taken separately, prove that it will be true for the direction of the resultant of the former force and the two others taken together.

2. State and prove the triangle of forces.

The resultant of two forces P and Q acting in the directions AB and CB respectively meets the line AC in the point D, and the resultant of the same forces acting in the directions AB and BC respectively meets the same line AC produced in the point E; prove that AC is divided harmonically at D and E.

Where are D and E respectively situated when AB and CB are proportional to P and Q?

3. Define the moment of a force about a point, and prove that the moments of two non-parallel forces, about any point in the line of their resultant, are equal and opposite.

State and prove the converse of this proposition.

4. Four forces, P, Q, R, and S, no two of which are parallel, act in a plane; the resultant of P and Q meets that of R and S in A, the resultant of P and R meets that of Q and S in B, and the resultant P and S meets that of Q and R in C; prove that A, B, and C are in the same straight line.

5. Three weightless rods, AB, AC, and BC, are fastened together at the angles to form the isosceles triangle ABC, whose vertical angle BAC is 30°. The figure is kept at rest by a force Pat A bisecting BAC, and two forces, each equal to P, at B and C. Prove that the equilibrium will not be affected if the rods be replaced by flexible strings knotted together at the ends, the forces acting outwards from the triangle. Find the tensions of the strings.

6. Show how to determine the centre of gravity of known weights rigidly connected.

Prove that, if weights 1, 2, 3, 4, 5, 6 are situated at the angles of a regular hexagon, the distance of their centre of gravity from the centre of the circumscribing circle is two-sevenths of the radius of that circle.

7. Show how to graduate the common steelyard.

The length of the rod is 2 feet, its weight 2 lbs., the distance of its centre of gravity from the fulcrum 1 inch towards the end of the shorter arm, the distance of the point where the weight is suspended from the fulcrum 2 inches, and the moveable weight 6 ozs. Find the greatest weight which can be weighed.

8. Find the ratio of the power to the weight in the system of pulleys wherein each hangs by a separate string, neglecting the weights of the pulleys themselves.

If the weights of the pulleys be taken into account, and be all equal, find the least value of the weight of each in order that there may be any mechanical disadvantage in supporting a weight W.

9. State the laws of friction.

A weightless rod of length (2a) is loaded with equal weights (W) at its extremities, and supported with its middle point on the highest point A of the circumference of a rough vertical circle.

The coefficient of friction is such that a particle of the same material as the rod would be just upon the point of sliding if placed on the point B of the circle where the arc AB=6. If the weight at one end of the rod be very slowly increased, prove that the greatest weight which can be thus 26 W a-b'

added for equilibrium to be possible is

If half this weight be added, find the position of equilibrium.

10. Explain the principle of Virtual Velocities. Apply it to find the ratio of the power to the weight in the Differential Screw.

IX. DYNAMICS.

[Great importance will be attached to accuracy.]

[N.B. When needed, the measure of the accelerating force of gravity may be taken as 32.]

I. Give some explanation why different measures of force are adopted in Statics and Dynamics. Show how the one measure may be derived from the other.

A body that weighs 2 cwt. is moved along a smooth horizontal table by a constant pressure which, acting for two seconds, generates a velocity of 16 feet per second in the body; find the weight which the pressure would statically support.

2. If a body be projected vertically downwards with a velocity V, find the space described and the velocity acquired at the end of (t) seconds.

A body is projected downwards with a velocity (V), and at the end of (t) seconds has a velocity (v); show that the space described is the same as that described by the body moving uniformly for (t) seconds with a velocity that is the arithmetic mean of (V) and (v).

3. State the second law of motion, and show how it is applicable to the theory of projectiles.

If a projectile be projected from the foot of a given inclined plane, find the distance from the point of projection at which the projectile strikes the inclined plane.

There is a hill whose inclination is 30°. From a point on the hill one projectile is projected up the hill and the other down it with equal velocities; the angle of projection in each case is inclined to the horizon at 45°; show that the range of one projectile is nearly 3 times the range of the other.

4. A weight (W) is placed on a rough horizontal table, and is moved along the table by a vertical weight (P) which hangs over the edge of the table, and is attached to (W) by an inextensible string; show that the P-μW accelerating force on (W) is g, where (u) is the coefficient of P+W friction.

If W=5P, find (u) when the roughness of the plane is such that it takes twice the time from rest to acquire the same velocity in the moving body that it acquires when the plane is smooth.

5. A ball weighing 12 lbs. leaves the mouth of a cannon horizontally, with a velocity of 1000 feet per second; the gun and carriage, together weighing 12 cwt., slide upon a smooth plane whose inclination to the horizon is 30°; find the space through which the gun and carriage will be driven up the plane by the recoil, it being given that the pressure caused by the explosion on the ball and on the end of the bore of the gun at each instant is the same.

6. How is the work accumulated in a moving body estimated? If one inelastic ball overtake another inelastic ball in direct impact, show that work is lost by the impact. If the balls be equal, and the velocity of one be double the velocity of the other, find the work lost.

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