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III. ALGEBRA.

(Including Equations, Progressions, Permutations and Combinations, and the Binomial Theorem.)

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

I. Add together x+y, 3x − y − 3%, 2y - 2x+z and multiply the result by x-y-z.

x5 −x+y+x3y3

Find the value of

when x=2y, and prove that

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2.

(x+y) (x2+y2) (x4 +34) = ~

x-y

Divide 15x5 - 17xa − 24x3 +138x2 - 130x+63 by 5x3+6x2-9x+7 and verify your result by multiplication.

3. Prove that any common factor of two expressions is a factor of the sum or difference of any multiples of them.

What use is often made of this fact in the process of finding the highest common divisor of two algebraic expressions?

Express 4x2 - 6yz − (9y2+z2), 912 + 4x2 − (4x2 + 22), z2 – 12xy − (4x2+912) in factors, and hence write down their L.C.M.

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5. Extract the square root of the expression

(a - b)2 {(a - b)2 – 2 (a2 + b2)} + 2 (a1 +b1).

6. Show how to determine when the roots of the equation

are impossible.

ax2 + bx+c=0

If a, ẞ be the roots of the equation x2+px+q=0, prove that the roots

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of the equation qx2 +p (1+9) x + (1+q)2=0 are a +

and ẞ+

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7. Prove that when m and n are positive integers (am)n = amn,

and explain why the symbol aTM” is used to denote

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9. Find two numbers which are in the ratio of to, but which, if respectively increased by 6 and 5, will be in the ratio of § to §.

10. A slow train takes 5 hours longer in journeying between two given termini than an express, and the two trains when started at the same time, one from each terminus, meet 6 hours afterwards. Find how long each takes in travelling the whole journey.

11. Explain the meaning of "fourth proportional," tional," "duplicate ratio."

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mean propor

= 2 and show that if x and y bd'

are unequal, and xy is the duplicate ratio of x-z: y-z, then z is a mean proportional to x and y.

12. If the first term of a Geometric series be (a) and the last term be (/), the number of terms (2) being odd, what is the middle term?

Sum the series:

(1) √√ +} √2 + √+ &c. to infinity.

(2) (2n-1)+(4n+ †) + (6n − 1)+ &c. to n terms.

13. Prove that the number of different arrangements of ʼn things, taken three together is n (n − 1) (n − 2).

In how many ways can a picket of 3 men and an officer be chosen out of a company consisting of 80 men and 3 officers?

14. Expand (1 − x2)-5 to 6 terms by the binomial theorem.

Show that the nth terms of (1 − x)−” and (1+x)2n −2 are equal.

IV. PLANE TRIGONOMETRY.

(Including the Solution of Triangles.)

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

I. Define the tangent and secant of an angle, and obtain a formula connecting them. Deduce the corresponding formula connecting the cotangent and cosecant, and prove that

2.

(tan a+cosec ẞ)2 - (cot ẞ — sec a)2 = 2 tan a cot ẞ (cosec a + sec ß).

Prove by geometrical construction that

sin Asin (180° – A), and

cos (180°+4)=cos (180° - A).

3. Express cos 24 in terms of cos A or sin A, and show that

8 (sin* A - sin2 A) + 1 = cos 4A.

If 44 contains 40°, what four angles less than 180° will have their sines determined by this formula?

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4 COS 2A

(3) cot (4+15°) - tan (A - 15°) = 2 sin 2A + 1

5. Find all the values of which satisfy the equation

4 cos 0-3 sec = 2 tan 0.

6. Prove that the circular measure of an angle less than a right angle is intermediate in value between the sine and the tangent of the angle.

Find approximately the number of minutes denoting the inclination to the horizon of an incline which rises 5 feet in 420 yards (π=22).

7. Show that in any plane triangle

(1) a=b cos C+ccos B.

(2) a sin (B-C)+b sin (C-A)+c sin (A - B)=0.

8. Two adjacent sides of a parallelogram, 5 inches and 8 inches long respectively, include an angle of 60°. Find the length of the two diagonals and the area of the figure.

9. In the triangle ABC, a=13, b=7, C=60°, find A and B. Having given log 3='4771213. Z tan 27°. 27′ =9°7155508.

Diff. 1'=3087.

10.

A man observes that, when he has walked c feet up an inclined plane, the angular depression of an object in the horizontal plane through the foot of the slope is a; and that, when he has walked a further distance of c feet, the angular depression of the same object is B. Show that the inclination of the slope to the horizon is cot-1 {2 cot B-cot a}, and determine the distance of the object observed from the foot of the slope.

II.

Find the radius of the circle inscribed in a triangle in terms of the lengths of the sides.

If d1, da, da be the distances of the centre of this circle from the angular points A, B, C respectively, prove that

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(In answering the questions on Geometry ordinary abbreviations may be employed, but the method of proof must be geometrical.)

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

I. If two straight lines are at right angles to the same plane, prove that they are parallel to one another.

2. Draw a straight line perpendicular to a given plane from a given point without it.

Draw a straight line perpendicular to each of two given straight lines which are not in the same plane.

3. Show that there cannot be more than five different regular polyhedra.

Find the volume of a regular octohedron in terms of one of its edges.

4.

Show how a right circular cone may be cut by a plane so that the curve of section may be a parabola.

Determine the relative positions of two such cutting planes if they produce parabolas of which the one latus rectum is double the other.

5. TP, TP' being two tangents to a parabola whose focus is F, prove that TP, TP' subtend equal angles at F; and that the triangles TFP, TFP' are similar.

6. Any point P on an ellipse being joined with A, A' the extremities of the major axis and PK, PK' being drawn respectively perpendicular to AP, A'P to meet the major axis in Kand K'; prove that KK' is equal to the latus rectum.

7. If SY, HZ be the perpendiculars from the foci S and H upon a tangent to a hyperbola, prove that CY-CZ-CA: and that

SY.HZ=BC2.

8. Find the equations to the straight lines which pass through a given point and make a given angle with a given straight line.

Show that the equation y3 -- x3+3xy ( y − x)=0 represents three straight lines equally inclined to one another.

9. Find the equation to a circle referred to oblique axes, and deduce that for rectangular axes.

Obtain the equations to two circles passing through a point h, k and touching the two straight lines y±x=0.

IO. Find the equation to a tangent at any point of the parabola y2=4ax, and the length of the perpendicular to it drawn from the focus.

II.

If CP, CD are two conjugate semi-diameters of the ellipse

a2y2+b2x2=a2b2,

find the coordinates of D in terms of those of P.

If S be a focus and 0 and the angles SP, SD make with the tangents at P and D, prove that cot 20+ cot 20 is constant.

12. Find the equation of a hyperbola referred to its asymptotes as

axes.

Find the eccentricity and the equations of the asymptotes of the hyperbola 4xy-3x2- 2ay=0, the axes being rectangular.

13. Obtain the equation to the normal at the origin of the curve ax2+ bxy+cy2+dx+ey=0.

Hence show that all chords of any conic section which subtend a right angle at a given point of the curve intersect on the normal at that point.

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