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GEOMETRY.

1. Draw a straight line perpendicular to a given straight line of unlimited length from a given point without it.

2. Parallelograms on the same base and between the same parallels are equal to one another.

3. If a straight line be bisected, and produced to any point, the rectangle contained by the whole line. thus produced, and the part of it produced, together with the square on half the line bisected, is equal to the square on the straight line which is made up of the half and the part produced.

4. If a straight line drawn through the centre of a circle bisect a straight line in it which does not pass through the centre, it shall cut it at right angles.

A straight line is drawn intersecting two concentric circles; prove that the portions of the straight line, intercepted between the two circles, are equal.

5. If a straight line touch a circle, the straight line drawn from the centre to the point of contact shall be perpendicular to the line touching the circle.

6. Describe a circle of given radius passing through a given point and touching a given straight line.

Also describe a circle of given radius touching a given straight line and a given circle.

In each case find how many circles can be drawn.

7. If from a point without a circle two straight lines be drawn, one of which cuts the circle and the other touches it, the rectangle contained by the whole line which cuts the circle and the part of it without the circle shall be equal to the square on the line which touches it.

If two circles intersect each other, their common chord bisects their common tangent.

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8. Inscribe in a given circle a triangle equiangular to a given triangle.

9. Triangles and parallelograms of the same altitude are to one another as their bases.

10. Find a mean proportional between two given straight lines.

11. Similar triangles are one to another in the duplicate ratio of their homologous sides.

ALGEBRA.

1. If x=4, y = 3, find the numerical value of

(x* — 4x3y + 6x2y2 — 4xy3 + y1),

and show that for these values of x and y, √(x3+x2+1)=y. 2. Reduce to their simplest forms the following expressions

(1)

x2 - xy + y2 _ x2 + xy + y2

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(2) {a + b √( − 1)}2 × {a − b √√(− 1)}”.

3. Investigate a rule for finding what is called the least common multiple of two algebraical expressions. Find the least common multiple of (x - a) and x-2xła‡ + a3. acx2 - (ad+bc) x+bd

Reduce to its lowest terms

4. Solve the following equations:

a2x2-b2.

(1) (x−1)3+(x−2)3+ (x−3)3=3(x−1)(x−2)(x−·3).

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(3) (2x-1) (3x + 1) − (x − 1) (2x + 1) = 26.

(4) x2 + y2+z2=77,

xy+yz - xz = 26,

y + z = 11.

5. A person bought two farms, one of pasture, the other of arable land, the latter exceeding the former by 20 acres; for the pasture land he paid £10 an acre more than he paid for the arable, and the whole cost of the pasture farm was £9,000 and of the arable £8,000. How many acres were there in each farm, and what price per acre was paid in each case?

6. Express algebraically the condition that one quantity shall vary directly as a second and inversely as a third. State any known geometrical example of such a law of variation.

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of which varies as

is equal to the sum of two terms, one

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and the other as x, and that when

x is 1 or 2, y is equal to 7, find the equation between y and x.

7. A merchant transmits to a retailer a cask of spirits containing (a) gallons, the cask is consigned to three carriers in succession, the first when he receives it draws off one-fourth of a gallon from the cask and fills it up with water, the second when he receives it does likewise, and so does the third, find the ratio of the quantity of spirit in the cask to the quantity of water when the retailer receives it, and find the actual quantities of water and spirit remaining if (a) is nine gallons.

8. Without assuming the general expression for the sum of an arithmetic series, find the sum of (n) terms of the series

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There are (x) terms in arithmetical progression between 5 and 25, and the second term is ths of the last term but one, express the terms of the series.

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If a,b,c are in harmonical progression, prove a3+c>263.

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10. Distinguish between permutations and combinations. Deduce from first principles the expression for the number of permutations of (n) things taken (4) at a time, and assuming the general form, show that the number of permutations of (n) things taken altogether when they all differ is double the number of permutations of (n) things taken together when two of them are alike.

11. Assuming the form of the expansion of a binomial, express the (r+ 1)th term of (1 + x)"; show how to find the greatest term of (1+x)" when (x) is a proper fraction and (n) an integer, determine the place of the greatest term in the expansion of (1 + 4)→✨.

1. Prove that

PURE MATHEMATICS.

(y+z−x)2+(z+x−y)2+(x+y−z)2=4(x2+ y2+z2)−(x+y+z)*, and that (a2+b2) (c2 + d2) = (ac ± bd)2 + (ad + bc)2. In how many ways can the expression

(a2 + b2) (c2 + d2) (e2 +ƒ2)

be written in the form of the sum of two squares?

2. Solve the equation

6x289x+323 = 0.

If y and z be the roots of the equation ax2+b+c=0, prove that a (y + z)

- b, and ayz =

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Find, in terms of a, b, c, sions y2+z2, and y3+23.

= C.

the values of the expres

3. Sum the series

and

1+3+5+7+... to n terms,

1+3+9+27 +...to n terms,

1a + 22 + 32 + 42 +... to n terms.

4. Write down the coefficients of x3, and of x", in the expansions of (1-x), and (1-x).

If

prove that

n (n-1) (n-2)... (n − r + 1)

n+1

1 199

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5. Express 209, and √(10) in the form of continued fractions, and find the first four convergents to each of the continued fractions.

6. Define the sine, and the cosine, of an angle; and prove the formulæ

sin Asin (180° - A),

sin (A + B) = sin A cos B+ cos A sin B.

Find the values of sin 15° and sin 75°.

7. Prove that in any triangle ABC, of which a, b, c are the sides,

b2 + c2 - a2 = 2bc cos A.

Find the angles of a triangle, the sides of which are 6 inches, 3 inches, and 3 √(3) inches.

8. If R be the radius of the circumscribing circle of a triangle ABC, prove that

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State the ambiguous case in the solution of triangles, and prove that, in that case, the circumscribing circles of the two triangles are equal.

9. Define the logarithm of a number to a given base, and find the logarithms of 64 to the base 2, and of 128 to the base.

Prove that log(m × n) = log m+logn.

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