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5. What is the difference between Mill's dictum and Aristotle's dictum ?
6. How are disjunctives reduced? What is the meaning of U?
MR MÓCAY. 1. Similar figures are described on the sides of a triangle whose sides are 5, 7, and 9 feet respectively in length ; being given the area of the figure described on the side 5 equal to 75 square feet, determine the areas of the other figures.
2. Divide a given line into two parts, the difference of the squares of which shall be equal to a given square.
3. Prove that the area of a regular dodecagon inscribed in a circle is equal to į (diameter)2. 4. Find the value of
23 + 2x2 + 2x – 5 when x = I. 5. Solve the equations
22 + 2y = a,
4y2 + x = a. 6. Find the coefficient of 26 in
7. Prove that the bisectors of the external angles at the base of a triangle meet on the bisector of the vertical angle.
8. If two circles touch externally, prove that the square of the common tangent is equal to the rectangle under the diameters.
9. Prove that the product of the area of any triangle by the diameter of the circumscribing circle is equal to half the continued product of the three sides. 10. Resolve into their simplest real factors the expressions
(1) 2,5 + 45 – 4y – xyt; (2) 24 + 6444 II. Given
x = 2a – b-C, y = 26 – 6–a, z = 26 - a - b; find the value of
203 + 73 + 28 – 3xyz. 12. Gunter's chain is 4 perches long, and is divided into 100 links ; show that any number of square links may be converted into acres by simply moving the decimal point a certain number of places which it is required to determine.
MR. F. PURSER.
13. Through the vertex B of a quadrilateral ABCD inscribed in a circle is drawn, as in Ptolemy's theorem, a line making with BC an angle equal to the angle ABD; show that if this line is parallel to AD, then BD = CD.
14. State accurately Euclid's definition of equality of ratios, and prove from it that two ratios which are equal to the same are equal to one another.
15. Construct a rectilinear figure similar to another rectilinear figure, and having a given ratio to it. 16. Given a + b +c=p, bc + ca + ab = 9, abe = r, prove that
a3 + b3 + c3 = p3 – 3P9 + 3r. 17. Reduce to its simplest form
V3+V2 - 1
18. Explain clearly what is meant by the division of one vulgar fraction by another, and hence deduce the rule for obtaining the result.
1. Describe a circle passing through two given points, and cutting a given circle orthogonally.
2. A, B, C, and Dare four collinear points : find the locus of a point P if the angles APB and CPD be equal.
3. Prove that the product of the perpendiculars from the extremities of a chord of a circle on a variable tangent is in a constant ratio to the square of the perpendicular on the tangent from the pole of the chord.
4. Find the locus of points having a common conjugate with respect to three given circles.
5. Being given the sum of any number of magnitudeš, prove that their product is a maximum when they are all equal, and apply this to find a point within a triangle such that the product of the perpendiculars from it on the three sides may be a maximum.
6. If three points P, Q, R taken arbitrarily on the three sides BC, CA, AB of a triangle ABC, lie in a right line ; prove that the three circles QAR, RBP, PCQ intersect at a common point situated on the circle ABC.
7. If the line joining the centres of two circles meet them in the points A, B, C, D, and if any circle cutting both orthogonally meet them in the points A', B', C', D'; prove that the four lines À A, BB, CC", DD meet in a point.
8. Show how to place a given triangle so that each of its sides shall touch one of three given circles.
9. For any pair of conjugate points with respect to a circle, prove(1) that the square of the distance between them is equal to the sum of the squares of the tangents from them to the circle; and (2) that the semi-distance between them is equal to the length of the tangent from its middle point to the circle.
10. Given the hypothenuse of a right-angled triangle, construct it so that the sum of one side and twice the other shall be a maximum.
MR. F. PURSER.
II. Determine a point on a given line the sum of the distances of which from two fixed points shall be given. Under what circumstances will the problem become impossible ?
12. Through a fixed point Pwithin the legs of a fixed angle, show how to draw a line forming with them a triangle of minimum area, and prove that the triangle so drawn is a minimum,
13. Show that the ratio of tangents to two circles drawn from any point on a fixed coaxal circle is constant.
14. Prove that the square of the distance of any point on a circle, from either of two inverse points 0,0ʻ, varies as its distance from their axis of reflexion.
AB is a chord of the circle the length of which is proportional to the length OM of the line drawn from 0 to its middle point M; find the locus of M.
15. The vertex C of a right-angled triangle is fixed, and the base AB produced passes through a fixed point 0, X0 being bisected in B; find the locus of the point B.
1. Prove that the feet of the perpendiculars from the vertices of a triangle on the bisectors of the angles lie in fours on the three lines joining the middle points of the sides.
2. Perpendiculars are drawn from the middle point of each side of a quadrilateral to the opposite side, and from the middle point of each diagonal to the other diagonal ; prove that these six perpendiculars intersect in threes at four points.
3. Prove that the centre of the nine-point circle of a triangle is the mean centre of the vertices and intersection of perpendiculars.
4. Three circles are in external contact with each other; show how to describe a pair of circles touching them.
5. From a given point draw two lines cutting a given circle, such that the quadrilateral formed by joining the points of intersection shall be a
6. If two circles be so placed that a quadrilateral may be inscribed in one and circunscribed to the other, prove that the diagonals of the quadrilateral intersect in one of the limiting points.
7. Determine the position of the centre of a circle of given radius, so that the area of the polar triangle with respect to this circle of a given triangle shall be a minimum.
8. Show how to find the position of a point, such that it will be the centre of mean distances of the feet of the perpendiculars let fall from it on the sides of a given triangle, and prove that the sum of the squares of the perpendiculars from such a point is a minimum.
MR. F. PURSER,
9. Show how to construct a triangle PQR of given magnitude and species, whose vertices P, Q, R lie on the sides of the triangle ABC. Prove also that in general-(a) if the species of PQR be the same as that of ABC, the point O is the orthocentre of PQR, and the centre of the circle circumscribing ABC; (6) if o be the centre of the circle inscribed in PQR, it is the orthocentre of ABC.
10. Prove that any two conjugate points on the axis of reflexion of two inverse points 0, 0, with respect to any circle, subtend a right angle at O and O.
11. Through a variable point P on one of three fixed concurrent lines, and a fixed point 0, two circles are described touching the other two lines in T, T'; prove that the ratio of the rectangles PT. OT', PT'. OT is constant.
12. Prove that a circle which cuts two given circles at constant angles touches two fixed coaxal circles.
Hence prove that if the distance of a point P from a given fixed point o exceed by a constant difference the length of the tangent from P to a given circle, the sum or difference of the distances of P from two fixed points will be given.
MR. M'CAY. 1. Solve the equations
(y – a)(2 – a) = bc,
(x – c)(y - c) = ab. 2. Solve the equation
Va+r+ Va-x=b. 3. Resolve into factors the expression
a3 (b + c) + 63 (c + a) + c3 (a + b) + abc (a + b + c). 4. Find the (5 + r) th term of
5. Find a factor which will rationalize V2 - 3.
6. Find the simplest form of the fraction
(62 – 62)8 + (62 – az): + (a2 – 62) 8
be fractions, which are not all equal; prove by 2 b 2
bn that the fraction
ar + a2 +...
lies between the least and greatest of them.
Ta — «)2 + (6 - )?