IO. A carrier charges 3d. each for all parcels not exceeding a certain weight; and on heavier parcels he makes an additional charge for every 7 lbs. above that weight. The charge for half a cwt. is Is. 3d., and the charge for 9 stones is five times that for I qr. What is the scale of charges? II. Investigate a formula for the sum of any number of quantities in Arithmetrical progression. 12. To each of three consecutive terms of a geometrical series the second of the three is added. Shew that the three resulting quantities are in Harmonical progression. 13. Find the number of permutations of n things taken all together which are not all different. How many different permutations can be made of the letters of the word essences, and how many of these will begin with n and end with s? 14. Expand (1 – 3x)3 to 5 terms, and write down the general term in its simplest form. Also if a, denote the coefficient of x" in the expansion of (1 − x)2m-1, shew that a-1+A2m-r=0. IV. PLANE TRIGONOMETRY AND LOGARITHMS. (Obligatory.) [N.B. Great importance will be attached to accuracy in working.] I. State briefly the advantages or disadvantages of the English or French methods, respectively, of measuring angles. If an angle be expressed in French minutes, shew that it will be transferred to English minutes when multiplied by 54. Verify this by first expressing an angle of an equilateral triangle in French measure. Divide the English angle 77° into two parts, so that the number of English seconds in one part may equal the number of French seconds in the other part. 2. Define the sine and cosine of an angle. Prove that the sine of any angle is the cosine of its complement. If A be greater than two and less than three right angles, represent geometrically the complement of (A – 180o). If (n) be a positive whole number, shew that the angles (2n. 180° + A) and {(2n+1) 180o – A} have the same sine as A. Given the sine of an angle, find the tangent. Prove tan 60°= √3. == 3. Without assuming the formula for sin (A+B), prove geometrically sin 24 = 2 sin A. cos A, A being less than 45°. Obtain the equation 4. Express sin A in terms of sin 2A. In the general expression how many values of sin A are thus obtained? Shew in any particular case how the correct value is to be selected. Assuming the equation cos 34=4 (cos A)3 – 3 cos A, find sin 18o, and hence exhibit the true value of sin 9o without reducing the surds. 5. Prove (1) (cos A) - (sin A)1 = cos 2A. (2) sin A+ sin (72o + A) + sin (36o – A) = sin (72° – A) + sin (36o + A). (3) cos-11+2 sin-11=120°. 6. Given the three sides of a triangle, find the cosine of one of the angles, and hence express the sine of half that angle in terms of the sides and in a form adapted to logarithmic computation. If a, b, c be the sides subtending the angles A, B, C respectively of the triangle ABC, 7. Find the radius of the escribed circle of the triangle ABC, when the circle touches the side BC and the sides AB and AC produced. 8. In a triangle ABC, given BC=2AC, A=3B, find the angles of the triangle and the ratio of AB to BC and AC. 9. State the property which furnishes the principle for constructing the proportional parts in the ordinary tables of logarithms. Given log10 60389='7809578, log10 6·0390=7809650, calculate the corresponding proportional parts, and find the number of which 7809601 is the logarithm. Given log10 2='3010300, find log10 250 and the tabulated logarithms of sin 30° and cosec 30°. 10. Explain when the solution of a triangle is said to be ambiguous. Supposing the data for the solution of a triangle to be as in the two following cases (a), (B), point out in each case whether the solution will be ambiguous or unambiguous. Find the angle C in the case not ambiguous, and the third side of the obtuse angle in the ambiguous case. (a) A=30°, AB=250 feet, BC=125 feet. (B) A=30o, AB=250 feet, BC=200 feet. Log sin 8° 41'=9'1789001. Note. The logarithms of Question 9 to be used, when required. II. A church tower BCD with a spire above it stands on a horizontal plane, B being a point in its base, and BC being 9 feet vertically above B. The height of the tower is 289 feet and of the spire 35 feet; from the extremity A of a horizontal line BĀ, it is found that the angle subtended by the spire is equal to the angle subtended by BC; prove that BA=180 feet nearly. FURTHER EXAMINATION. V. PURE MATHEMATICS (1). I. Draw a straight line perpendicular to a given plane from a given point without it. 2. The semi vertical angle of a conical box is 30°, and the length of its slant side is 10 inches. Find the volume of the largest sphere which can be placed inside it. 3. If a solid angle be formed by three plane angles, any two of them are together greater than the third. 4. Shew how to find the area of the portion of a spherical surface included between two small parallel circles. 5. Find the length of the perpendicular from the point (4, 5) on the straight line 3x+4y=10. 6. Find the area of the triangle included between the three straight lines x+y=5, 2x- -3y+5=0, x+6y=5 and find the radius of the circle inscribed in the triangle. 7. Find the radius and co-ordinates of the centre of the circle represented by the equation x2+ y2+6x-4y=12 and find the length of a tangent drawn to this circle from the point (-3, -5). 8. Shew that if two tangents be drawn to a parabola from an external point they will subtend equal angles at the focus. 9. Shew that the locus of points from which the tangents drawn to an ellipse are at right angles to each other is a fixed circle. IO. Shew that the tangent at any point of a hyperbola makes with the asymptotes a triangle of constant area, and that the part intercepted `on the tangent is equal to the diameter conjugate to that passing through the point of contact. II. Find the equation to the normal at any point of an ellipse in terms of the tangent of the angle which it makes with the major axis. 12. Assuming Demoivre's Theorem, prove that 13. Find an expression for tan-1x in a series of ascending powers of x; and hence shew how to approximate to a value of π. VI. PURE MATHEMATICS (2). [Full marks may be obtained without doing the whole of this paper. Great importance will be attached to accuracy in numerical results.] 3. Between two quantities A and B a harmonic mean H is inserted. Between A and H and between H and B geometric means G1 and G, are inserted, and it is found that G1, H, G2 are in A.P. Find the ratio of A to B. 4. The duration of a railway journey varies directly as the distance and inversely as the velocity. The velocity varies directly as the square root of the quantity of coal used per mile and inversely as the number of carriages in the train. In a journey of 25 miles in half an hour with 18 carriages 10 cwt. of coal is required. How much coal will be consumed in a journey of 15 miles in 20 minutes with 20 carriages? 5. The difference of two numbers that are expressed by the same digits is 35453221; in what scale are the numbers expressed? 6. The base of a right circular cone is 6 inches in diameter and its height is 8 inches. Find the area of the conical surface. Find also the weight of the cone (to 5 places of decimals), supposing it to be constructed of cast iron, of which one cubic foot weighs 440 lbs. 7. If the radius of the circumscribed circle of a triangle be equal to the diameter of the inscribed circle, shew that the triangle is equilateral. 8. If any hexagon, inscribed in a circle, has two opposite angles equal, shew that it will have two opposite sides parallel. 9. A sphere whose radius is 2 inches rests on three wires in the form of a plane triangle, the sides of which are 3, 4, and 5 inches respectively. What will be the height of the top of the sphere above the plane of the wire? |