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III. ALGEBRA.

(Up to and including the Binomial Theorem, the theory and use of Logarithms.)

N.B. Great importance will be attached to accuracy.

I. Find the squares of x+y-2z+1, and of x+y-2z-1. What is

the value of the difference of these squares when z = (x+y)?

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+

(x−y)(x−z) + (y−z)(y - x) + (z−x)(z−y) = x+y+z;

.(I)

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5. A and B have the same birthday. A's age is represented, on his birthday in the year 1895, by the two right-hand digits of the year in which B was born; also the product of A's age and the number represented by the two right-hand digits of the year in which he was born gives the year when B was 9 years old. Find their ages, A being older than B.

6. Find the sum of an infinite geometrical progression whose first term is a and ratio r

To what restriction is subject, and why?

During any year the excess of births over deaths causes an increase of h per cent. of the population, and at the end of the year a fixed number A of people emigrate. Prove that at the end of n years a population P becomes

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7. Investigate the number of different ways in which n men may stand in a row.

If two specified men are, neither of them, to be at either extremity

of the row, show that the number of arrangements is

(n-2)(n-3) x (n-2)!

8. Write down the (s+1)th term in the expansion of

(a+b)n.

Find the sum of all the coefficients; and show, by using a table of logarithms that if a+b = I and n = 10, the first term is greater than the sum of all the remainder if a is greater than (about) '933.

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Employ tables to find the square root of " to 5 places of decimals

10. If

18+5 – 514 – 413 + 6y2 + 3y − 1 = 0,

and

I

show that

y = x +,

x13 - I

= 0.

X-I

IV. PLANE TRIGONOMETRY AND MENSURATION.

(Including the Solution of Triangles.)

[N.B. Great importance will be attached to accuracy.]

1. Explain the measurement of angles in circular measure. If an angle contains A seconds and its circular measure is a, show that, approximately,

A" = 206265 × a.

and

sec

Find from the tables the values of cos

seconds.

2.

58

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Show how to construct the angle whose cotangent is Ts, and find

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3. If x and y are any two numbers, show that the equation

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6. Prove the formulæ

COS 2A = 2 cos2A - 1,

cos 3A = 4 cos3A - 3 cos A.

7. Show that the sines of the angles of a plane triangle are proportional

to the lengths of the opposite sides, and deduce the relation

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8. ABC is a plane triangle, and Pa point in the side AB such that

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If the angles ACP and BCP are a and respectively, show also that

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9. Given b = 14, c = 13, and A = 67° 22' 48" in a triangle, find C by logarithmic calculation.

10. The radii of the circular faces of a frustum of a right cone are 12 and 8 feet, and the area of its curved surface is 207/241 square feet; find the thickness of the frustum.

Show that the vertical angle of the cone, of which this is a frustum, is 29° 51′ 46′′.

I.

V. STATICS AND DYNAMICS.

[Assume that w = 27, and that g = 32.]

If a number of forces, lying in one plane, act at a point, explain how their resultant may be found.

A, B, C, D are the angular points of a square taken in order, and forces represented in direction by the lines AB, BD, DA and AC, and in magnitude by the numbers 1, 2/2, 3 and 2, act at a point; find their resultant graphically or otherwise.

2.

Prove that two forces, whose lines of action intersect, have moments about a point in their plane, that are together equal to the moment of their resultant about the same point.

Pis a fixed point on the circumference of a fixed circle; PM and PN are any two chords of the circle at right angles to one another; Qis any other fixed point whatever in the plane of the circle. Show that if PM and PN represent forces, the algebraic sum of their moments about

is constant.

3. Define the centre of gravity of a body; and show that a body cannot possess more than one centre of gravity.

Prove that the centre of gravity of three particles of equal mass placed at the angular points of a triangular lamina of uniform thickness coincides with the centre of gravity of the lamina.

4. A bent lever consists of two uniform, heavy, straight rods, whose lengths are as 3 to 4; find the weight which must be attached to the end of the shorter rod in order that the fulcrum being at the junction of the two rods-they may make equal angles with the horizon.

5. Find the relation between the Effort or "Power" and the Resistance in the case of the frictionless screw press, when motion is just about to take place.

The step or distance between two threads of a screw is 0.187 of an inch, the length of the arm (reckoned from the centre of the screw) on which the effort acts is 25 inches, and the effort is 11.9 lbs.; find the resistance when the screw is on the point of moving.

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