(c) If two chords intersect within a circle the angle which they include is measured by half the sum of the intercepted arcs. (d) How will the result be changed in (c) if the two chords intersect without the circle? 6. (a) If two chords of a circle cut one another within the circle, the rectangle contained by the segments of the one is equal to the rectangle contained by the segments of the other. III. 35. (b) Enunciate and prove this proposition when extended to n chords and intersecting on the circumference or outside the circle. 7. (a) To describe an isosceles triangle having each of the angles at the base double of the third angle. IV. 10. (b) In your figure to IV. 10, show that the side of a decagon inscribed in the larger circle is equal to the side of a pentagon inscribed in the smaller circle. (c) Determine the magnitude of the angle of a regular pentagon.. (d) What is the purpose of Euclid's Bk. IV. 8. (a) To inscribe a regular hexagon in a given circle. IV. 15. (b) Join the alternate points of the hexagon and prove that the triangle formed is equilateral. (c) The area of a regular hexagon is twice that of an equilateral triangle inscribed in the same circle. 9. Explain the terms multiple, submultiple, ratio, duplicate ratio as applied to magnitudes. 10. (a) Triangles and parallelograms of the same altitude are to one another as their bases. VI. 1. (b) Triangles and parallelograms that have equal bases are to one another as their altitudes. (c) Where are these properties discussed in Algebra ? Time-Three hours. Algebra. 1. (a) State in the form of an equation the relation between the terms of a division. (b) Examine if an+bn be divisible by a + b. (c) Divide an+bn by a+b to four terms. (d) From the result in (c) write the mth term of the quotient and the mth remainder, m being less than n. 2. (a) Resolve any two of the following into factors: (ii) a1+b+c1—2a2b2—2b2c2—2c2a2. (iii) a3 (b—c)+b3 (c—a)+c3 (a—b). (b) From inspection write H.C.F. and L.C.M. of: (c) State the relation between the H.C.F. of two expressions and all their other common factors. Also between their L.C.M. and all their other common multiples. (d) Find the H.C.F. of 3a3+23a-13a2-21 and 21+a2-44a +6a3. 3. (a) Give reasons for or against considering a fraction and a ratio identical. (b) Show that a fraction or a ratio is made more nearly equal to unity by adding the same number to each term. (x+a)l+(y+b)m+(z+c)n=p, show that each fraction 4. (a) Solve: (i) x2+y2+z2—a3—b2—c2 x+√x2—1 x—√x2-1 + -34 (ii) 16x(x+1)(x+2)(x+3)=9 (b) Solve ax2+bx+c=0. From your result what is your inference regarding surd roots? (c) If 2+√3 be one root of xa—5x3—15x2+79x=20 find the other roots. (d) A drover bought oxen for $1,120. Had he paid $10 less for each he could have purchased 2 oxen more for the same money. Interpret the negative result in your solution. 5. (a) Explain the phrases "one number varies directly as another"; "one number varies inversely as another." (b) The surface of a sphere varies as the square of the radius, and its volume varies as the cube of the radius. Find the radius of a sphere whose volume equals the sum of the volumes of spheres whose radii are 3, 4 and 5 feet respectively, and compare its surface with the sum of the surfaces of the three spheres. 6. (a) Find the sum to n terms of a Geometrical Progression when a, r and n have their usual meaning. (b) Find the form this sum takes when r is less than unity and n is infinite. (c) Show that the product of any two terms equi-distant from any given term is always the same. 7. (a) Distinguish between a permutation and a combination. (b) Find the expression for the number of permutations of n different things taken r at a time, where r is any integer not greater than n. (c) At an election there are 5 candidates and 3 members to be elected, and an elector may give one vote to each of not more than three candidates. In how many ways can an elector vote? 8. (a) Explain what is meant by the Binomial Theorem. (b) Assuming the Binomial Theorem true for a positive integral exponent prove it true for a positive fractional exponent. (c) Write the rth term of (1+x)", r being less than n. (d) Deduce an expression for the sum of the squares of the coefficients in the expansion of (1+x)n. 9. (a) Define the logarithm of a number to any base. (b) Point out the advantages of the common system of logarithms. (c) Prove any two of the principles by which logarithms shorten arithmetical processes. (d) Show that the characteristic of a logarithm depends only on the position of the decimal point. (e) Find the log of V3÷2V 6 having given log 2-30103 and 12 V •6 1. (a) What is meant by a system of logarithms ? (b) State the advantages of the common system. (c) "In the common system the mantissa is always kept positive." If log 25380-4 40449 determine the log of 002538 illustrating this principle. (a) Find the log of .6 V27 ×54×(5·76) having given log 2 = 3010300; log 3=4771213 and log 2.79865=4469478. 2. (a) Define a Trigonometrical angle. From your definition draw an inference as to the magnitude of the angle. 90°. (b) Define the ratios for the sine and cosine of an angle less than (c) By what conventions are these ratios applied to angles between 90 and 360°? (d) Represent by a diagram the angles having sines equal to 3.. Determine algebraically the other ratios of the same angle. 3. (a) What is the relation between the ratio of an angle and its complement? (b) Show how this influences the arrangement and extent of the trigonometrical tables. 4. (a) Prove that tan A+cot A=V (sec2A+cosec2A). (b) Find cos 2212° from cos 45°. 5. (a) Assuming the formulae for the sum and difference of two angles deduce any two of the following: (i) Cos 2A in terms of Tan A. (ii) Tan (A+B). (iii) Cos A+cos B. (b) Express as a product Cos A+cos 3A+cos 5A+7A. 6. (a) Deduce either the Law of Sines or Tangents. (b) State each in words. (c) Solve the triangle ABC given: a=8ft. c=10ft. angle ABC =47°. (Cos. 47°6820. 7898 sin 52° 10′ and 9873 sin 80° 50'. 7. Derive the formulae for the area of a triangle (a) in terms of the three sides, (b) in terms of two sides and their included angle. 8. In any triangle where S area, s-semiperimeter, R=radius of circumscribed circle, r-radius of inscribed circle prove the following relations: 9. At a distance of 200 yards from the foot of a church tower, the angle of elevation of the top of the tower was observed to be 30°, and of the top of the spire of the tower 32°. Find the height of the tower and of the spire. Log 2=30103 L. tan 32 9.79579 10. (a) Find a general expression for all angles which have the same sine. (b) Solve the equation sin x cos x√3 (c) What is the general value for the angles which have the sines determined from the equation? Latin Authors. Time-Two and one-half hours. 1. Vestris amicum fontibus et choris Nec Sicula Palinurus unda. Utcunque mecum vos eritis, libens Visam Britannos hospitibus feros, Visam pharetratos Gelonos Et Scythicum inviolatus amnem. Vos Caesarem altum, militia simul Fessas cohortes addidit oppidis, Pierio recreatis antro: Vos lene consilium et datis et dato (a) Translate. (b) Parse: Philippis, visam, quaerentem, almae, dato. (c) What incidents in the life of the poet are indicated in: Vestris..... ....unda? (d) Explain the allusions in: litoris Assyrii, Britannos hospitibus feros, Fessas cohortes addidit oppodis, Pierio antro. 2. Prudens futuri temporis exitum Fas trepidat. Quod adest memento. Componere aequus: cetera fluminis |