6. Find the equations of the tangents to the parabola y2 = 4x at the points whose a co-ordinates are each equal to 1.

Show that they are at right angles and intersect where the directrix cuts the axis.

7. Defining the ellipse as the locus of a point the sum of whose distances from two fixed points is constant, find its equation, taking the line joining the fixed points as axis of x, and the middle point of this line as origin.

If (x1, Y1)(x2, Y2) are the co-ordinates of two points on the ellipse

is the equation of the line joining them. Deduce the equation of the tangent at any point.

8. Give an example of (1) stable, (2) unstable, (3) neutral equilibrium.

A number of cubical blocks are piled one on the other so as to form a staircase, the breadth of each step being 2", and the side of each block being 1 foot. How many can be piled before the whole begins to topple ?

9. A board in the shape of an equilateral triangle is hung horizontally by three vertical strings attached to its corners. Show where to place a weight so that the tensions in the strings may be as 1:2:3. (Neglect the weight of the board.)

IO. An elevator-cage is suspended by a rope passing over a smooth pulley and having a counterpoise equal to the weight of the cage at its other end. The rope between the weight and the pulley passes through the cage and can be handled by anyone within. A person of weight W steps inside, and by the friction of his hands on the rope reduces the downward acceleration to g. Find the tension on the rope produced by the friction of his hands. Take the weight of the cage to be W, and that of the rope to be negligible.

II. A wedge has one face resting on a smooth table and is free to move. A smooth particle of half its mass is placed on the other face, one foot from the edge. How long will it take to reach the table, the angle of the wedge being 30°?

12. In Atwood's machine where two masses m1 and m1⁄2 (m1>mą) are connected by a light string passing over a smooth pulley, find the tension of the string during the motion.

If the pulley ascends with a uniform acceleration equal to g, find the change in the tension.

MATHEMATICAL EXAMINATION PAPERS

FOR ADMISSION INTO

Royal Military Academy, Woolwich,

NOVEMBER, 1899.

I.

I. PART I. FIRST PAPER.

[Great importance is attached to accuracy.]

From + take (− +), and simplify the fraction

74-7+78-713
20+30-28

2. If one perch of wire weigh 24 lb. 12 oz. and one mile of the same wire cost £27. 10s. od., how much would 8 cwt. 2 qr. 8 lb. of the wire cost?

3. Find the present worth of £504. 15s. od. due 73 days hence at 4 per cent.

4. Extract the square root of 108241.

How many whole numbers between 100 and 100,000 are perfect cubes?

5. A certain length of pathway has to be constructed; it is found that three men can construct one-fifth all but one mile in two days, whilst 18 men can construct one mile more than two-fifths in one day. What is the length of the path?

6. Two numbers differ by 6; the sum will be a square number. find the two numbers.

show that if 9 be added to their product If the square root of this sum be 13,

7. Multiply 2x2 - 19x+35 by 2x2- 13x+15, and divide the result by

10. Having given the first term and the common difference of an Arithmetic Progression, find the expression for the sum of n terms of the series.

Find the sum of 25 terms of the series

1+2+3+4+ ....

and find the value of n when the sum of n terms of the series

I+2+4+8+...

exceeds by 186 the sum of 25 terms of the preceding series.

II.

Write down the expression for the number of permutations of n things taken together, and hence deduce the expression for the number of combinations of n things taken together.

Of 15 men, 10 can row and cannot steer, and five can steer and cannot row; find how many boats crews of eight rowers and a coxswain can be formed out of the 15 men.

12.

Write down the expression for the coefficient of x in the expansion of (1+x)", and find the greatest coefficient in the expansion of (1+x)8. Also find the value of x when the fifth term in the expansion of (1 + x) is equal to the number 70.

II. PART I. SECOND PAPER.

N.B.-In questions on Geometry ordinary abbreviations may be employed, but the method of proof must be geometrical. Proofs other than Euclid's must not violate Euclid's sequence of propositions. In the absence of special directions to Candidates, any of the propositions within the limits prescribed for Examination may be used in the solution of problems and riders.

I.

Define an isosceles triangle, a circle, and parallel straight lines. Construct an isosceles triangle for which each of the equal sides shall be double the base.

2.

Show that any two sides of a triangle are together greater than the third side.

area,

Show that of triangles described on a given base, and having a given that which has the least perimeter it isosceles.

3. Divide a given straight line into two parts so that the rectangle contained by the whole and one part may be equal to the square on the other part.

If the line be divided so that nine times the rectangle contained by the whole and one part is equal to four times the square on the other part, find the point of division.

4.

Prove that the angles contained by a tangent of a circle and a chord of the circle drawn from the point of contact of the tangent are respectively equal to the opposite angles subtended at the circle by the chord.

Describe two circles to touch two given circles, the point of contact with one of these given circles being given.

5. The radii of two intersecting circles are respectively 15 inches and 13 inches, and the common chord of the circles 24 inches long. What length of the line joining their centres lies within both circles?

6. Construct the centre of the circle inscribed in a given triangle. Prove that, if ABC is the triangle, I the centre of the circle, D the middle point of BC, L the point where AI produced meets BC, and P the foot of the perpendicular from A on BC, then Z lies between P and D, the sides AB and AC being unequal.