(7) 10. The following is an extract from a logarithm book: N. 01 23 4 5 6 7 89 Dif. 1905 2798950 9178 9406 9634 9862 0090 0317 0545 0773 1001 228 What is the logarithm of 190 and of 190595 7? Divide the product of these two quantities by 19057.5, using logarithms to obtain the result to two places of decimals. 11. What are the characteristics of the logarithms of 10 and to the bases 3, 3, 3, and respectively? 12. What is the cube root of 1 to six places of decimals, having 19.053 given log 3.744085733455, and the logarithms in Question 10? 13. An officer proceeding to India in a troop-ship is allowed 60 cubic feet of baggage. A lieutenant had two cases and a basket, one case being 4 feet 2 inches long, 2 feet 3 inches wide, and 2 feet 4 inches deep, and the other being half those dimensions in length, width, and depth, and the cylindrical basket being 3 feet high by 2 feet in diameter. By how much did he exceed or fall short of his allowance, actual contents being taken for the cylinder, and fractions of a cubic foot being omitted in the result? 14. The chord of an arc is 8 feet and the height of the arc is 2 feet, what is the radius of the circle? 15. The great pyramid of Egypt was 481 feet high, when complete, and its base was 764 feet in length; find the volume to the nearest number of cubic yards. 1. If a straight line be divided into any two parts, the square of the whole line is equal to the squares on the parts and to twice the rectangle contained by the parts. Write down the corresponding algebraical formula. If a straight line be divided into any three parts, exhibit a geometrical construction from which it may be demonstrated that the square of the whole line is equal to the squares on the three parts, together with twice the rectangle contained by every two of the parts. 2. When are circles said to touch one another? If two circles touch each other internally, the straight line which joins their centres being produced shall pass through the point of contact. Draw a circle of given radius which shall touch internally a given circle at a given point. 3. The opposite angles of any quadrilateral inscribed in a circle are together equal to two right angles. If a circle be inscribed in a quadrilateral ABCD, show that the sum of the opposite sides AB and CD together, is equal to the sum of BC and DA together. 4. Inscribe a square in a given circle. If AB be the side of an equilateral triangle inscribed in the circle, and AD be a side of the inscribed square, prove 3AD2 = 2AB2. 5. Inscribe a circle in a given equilateral and equiangular pentagon. 6. If a straight line be drawn parallel to one of the sides of a triangle, it shall cut the other sides or those produced proportionally. Construct a triangle similar to a given triangle, such that each of its sides shall be one-fourth of each of the sides of the given triangle. What will be the ratio of the areas of the two triangles? 7. If four straight lines be proportional, the similar rectilineal figures similarly described on them shall be proportional. Show that if the condition in italics were disregarded the proportion would not necessarily be true. 8. Explain how the foreign or French method of measuring angles differs from the English method. If an angle is given in one measure, investigate a rule for finding its value adapted to the other measure. Express in foreign measure each angle of the isosceles triangle described in Euclid's 10th proposition, Book IV. 9. Show how to express the cosine, tangent, and secant of an angle in 3 terms of the sine. The sine of an angle less than 90° is find 5' cosine, tangent, and secant. 10. Investigate the formula for sin (A - B) in terms of the sines and cosines of (A) and (B); express the formula also in terms of the cosines of A and B only. If A be the angle of an equilateral triangle, find tan A from the above formula. 12. If A, B, C be the angles of a plane triangle subtended by the sides a, b, c respectively, prove the formula tan = 2 sin B sin C B-C b -c A cot show how this is applib+c = and thence deduce 2; cable to the solution of a triangle of which the data are two sides and the included angle. 13. The base of an isosceles triangle is 100 feet and the vertical angle is 125°; solve the triangle. angle Ă = 14. In a triangle ABC, given AC = 166.5 feet, BC = 162.5 feet, the 52° 19′, solve either of the triangles to which the data belong. SET II I. MATHEMATICS. (1.) 1. Prove that am × a" = a”+” when m and n are positive integers : m and hence deduce the meaning of a-" and a". 2. Reduce to their simplest forms— 3. Find the greatest common measure of 5×3 მეც3 13x2 4. Extract the square root of 1 29x2+19x+5, and 15x + 25. 5. Find the fourth power of 1± √1 and the square root of 1 + a2 + {1 + a2 + aa}*. 6. Prove that an equation of the second degree has two roots and no more; and show in what cases the roots are equal, rational, irrational, or imaginary. (3.) xy 165 x2 = y3. What value of a gives the value 3 (21) ̄* to the quantity 8. Show that numbers are divisible by 11, when the sum of their odd digits equals the sum of their even; and by 9, when the sum of their digits is divisible by 9. 9. If 2 cubical blocks of stone contain together 8 cubic feet, and the side of the less is to that of the greater as 3 to 4; find the side of each. 10. A rectangular garden contains 1200 square yards, and the length is to the breadth as 4 to 3; what will the fencing cost at 3s. 6d. per yard? 11. What is the weight of a hollow sphere of metal, whose inside diameter is 18 inches, and the thickness 2 inches? 12. The area of a triangle is 6 acres 2 roods and 8 perches, and a perpendicular from one angle on the base measures 524 links; find the length of the base in chains. 22 yards. 13. Find the area in acres of a triangle, whose sides are 25, 20, and 15 chains. 14. Explain the following terms in logarithms: base, mantissa, characteristic, modulus, and prove that log b x loga = 1. 15. Prove that log, a = log,alog,b, and that log, (ab) Find log,3125, and having given log 2 = 3010300, find the number of digits in 264. |