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7. Show geometrically that the harmonic mean between two lines equals (i.) the rectangle under them, divided by half their sum, and also (ii.) the rectangle under the differences divided by half the difference of these differences.
8. The sides of one triangle ABC pass through the vertices a, b, c of another; show that if the lines Aa, Bb, Cc intersect at a point, and the sides of each triangle be produced to meet the opposite sides of the other, every line in the figure is cut harmonically.
9. Given two points in the circumference of a circle and a chord, to find a third point such that the chords joining it to the given points may intercept on the given chords two segments in a given ratio.
10. From a point in the diameter of a circle produced, to draw a chord cutting the circle so that the rectangle under the lines joining the points of section to the extremities of the diameter may be given.
11. A quadrilateral circumscribes a circle, and the points of contact are joined so as to form an inscribed quadrilateral; show that the diagonals of the two quadrilaterals all intersect in the same point and form a harmonic pencil.
12. Describe a circle passing through a given point, and cutting two given right lines, so that the intercepted chords may subtend given angles at the centre.
13. If from any point on one circle tangents be drawn to another, find the envelope of the chord joining the points of their intersection with the first circle.
14. Given all the sides of a polygon, prove that its area is a maximum when it is capable of being inscribed in a circle.
15. Divide a given circle into five equal parts geometrically, by describing concentric circles.
16. If the bisectors of the base angles of a triangle be equal, prove that it is isosceles.
17. Describe a circle which shall touch two given circles, and intersect a third at a given angle.
18. Given of a triangle its inscribed circle, and the difference of its sides, find the locus of its vertex.
1. Clear of radicals the equation
2. Find the values of a which satisfy the equation
V x2 + ax - 1 + x2 + bx – 1=7a+ . 3. Find the values of x and y which satisfy the equations
✓ x+y+wc-y=va, ✓ x2 + y2 + v 32 y2 = 72. 4. Find by any method the sum to infinity of the series
1 + 2x + 3x2 + 433+ &c. 5. State and prove the rule for reducing a mixed circulating decimal to its equivalent ordinary fraction in any scale of notation.
6. A waterman rows a given distance a, and back again in h hours, and finds that he rows 6 miles with the tide for b' against it. Required the rate of the tide, and the rate of rowing; also the times of rowing down and up the stream.
7. If o be the sum, and 02 the sum of the squares of the terms of an infinite geometrical progression, then will
0 + 02
02 + 8. Prove that the coefficient of är in
(1 + 2x + 3x2 + &c. ad inf.)2
is (r+1) (r+2) (r+3). 9. Let ax=by = cm; show that if x, y, z be in harmonic progression, then a, b, and c will be in geometrical.
10. In a bag are 10 balls, of which I draw 4; find the chance that there shall be among them
(i.) 2 marked ones.
11. Solve the equation
zat xi (x+ 3al):-(at - 3al)t =
bt 12. Show that the sum of the series
2+6+14+30 + &c. to n terms is 2n+2 - (2n + 4). 13. Decompose into its partial fractions the expression
32 + 38 – 4
(+2)(x-3) 14. Reduce to its simplest form the quantity
1. Prove the formula for the tangent of the sum of any number of angles in terms of the tangents of the several angles.
2. Given that tan (a + B ✓ - 1) = a +b/-1, required the values of a and b in terms of a and ß, and conversely.
3. Given that x = a cos (Ø + a), y = b cos (0+), =ccos (+ y),
• sin (B – y) +Ž. sin (y-a) +. sin (a – B) =0. 4. If x be any finite angle, and 8 any small change in it positive or negative, prove that approximately
loge sin (x + 3) = loge sin x + cotx.d,
loge cos (x + 0) = loge cos tan x.d. 5. Prove the logarithmic formulæ for the base and base angles of a plane triangle, in terms of the sides and the vertical angle.
6. By means of the formula
a3 sin a = a
1.2.3 calculate the numerical value of sin 1° 54' 35".5 to five places of decimals. 7. Prove geometrically that
cot tan - = 2 cote. 8. Assuming the preceding formula, prove that the sum of the series
8 8 to n terms is
9. If the tangents of the angles of a triangle are in geometric progression, so that
tan A=rtan B=g2 tan C, show that (i.)
trt r21 tan C=
3 and that (ü.)
sin 2C = y sin 2 A.
10. Looking at two distant towers, their summits seem in the same straight line from my eye. On approaching C yards towards them, the angle of elevation of the nearer becomes twice that of the more remote. Show that the distance of the towers from me being, respectively, a and 0 yards, their heights are
(2bc - ab - ac) 71
11. If tan A = }, and tan B= ì, find the value of tan (3A + B). 12. Eliminate 0 between the equations
a cos (@+a)= = x and b cos (O+B) =y. 13. Prove Machin's formula, viz.,
= 4 tan -1
239 14. Apply this formula to the calculation of a to five places of decimals, assuming the expansion for 0 in terms of tan 0. 15. Find the sum of the infinite series,
a3 'I + a sin 0+
sin 20 +
sin 30+ &c.
1.2.3 a being less than unity.
1. If two circles intersect at right angles, the polar of either centre of similitude with respect to either circle is the polar of the other centre of similitude with respect to the other circle ?
2. If a variable circle intersect two fixed circles, its chord of contact with both is cut in a constant anharmonic ratio by their radical axis and by their centre of similitude through which it passes ?
3. If three of the six tangents to a circle from the three vertices of a triangle, meet the three opposite sides of the latter in three collinear points, the remaining three meet them also in three collinear points ?
4. Apply the method of homographic division to the solution of the problem : “to draw a chord of a given angle which shall be cut in two given anharmonical ratios by two other given angles."
5. If three coaxal circles touch the three sides of a triangle and three more of the same coaxal system pass through the three vertices, the six centres are in involution?
6. Describe a circle of a given coaxal system cutting a given circular arc in a given anharmonic ratio.
7. If two intersecting circles be inverted from any point, the inverse circles intersect at the same angles as the originals. Prove this, and show from it that every two intersecting circles and their two circles of inversion invert from any point into two circles and their two circles of inversion.
8. A triangle of given species is inscribed in a fixed triangle and a circle is inscribed in the variable triangle, find the locus of its centre.
9. Tangents to two circles intersect at a given angle; find the locus of the point dividing in a given ratio the line joining the points of contact.
10. Construct an isosceles triangle, being given the perpendiculars on its sides from the centre of its circumscribing circle.
11. Describe a circle coaxal with two given circles, and cutting off a given arc from a third.
12. If a circle touch two fixed circles, it divides every coaxal circle homographically ?
Translate the following passages into English :1. Beginning, θήβαι δε, θήβαι, πόλις αστυγείτων, κ. τ.λ. Ending, άλλ' ήδη περί του της πατρίδος εδάφους.
ÆSCHINES, Contra Ctesiphontem, s. 116. 2. Beginning, Καίτοι πόσα χρήματα τους ηγεμόνας των συμμοριών, Ending, τόν τ' έκ τού προτέρου νόμου και τον κατά τον εμόν. Λέγε.
DEMOSTHENES, De Corona, s. 81. 3. Beginning, Δεϊν δ' ώετο μηδένα αποστερείσθαι του δίκης τυχεϊν, Ending, ίν' εκινδύνευες περί χιλίων, γέλως αν είη δήπουθεν.
DEMOSTHENES, Adv. Androt., s. 26. 4. Beginning, Λαβε δή και τον μεταλλικών νόμον, κ. τ.λ. Ending, ά προεϊντό σοι, μεθ' όπλων ήκειν νομίζεις.
DEMOSTHENES, Adv. Pantæn, ss. 35, 36.
κ. τ. λ.
1. What adjectives are regularly placed before the article and substantive, or after them?