8. Find the equation of the tangent to the curve 9. Find any expression for the radius of curvature at any point of a curve, and apply it to the curve in the last question. 10. Prove the formula 2.2 dA = where A is the sectorial area subtended by a curve at the origin. If r be the distance of any point of an ellipse from one of the foci, and a be the semi-major axis, prove that Σα (2) ds along any arc is proportional to the γ sectorial area subtended by that arc at the other focus. 11. Integrate the expressions and find a formula of reduction for the integration of (a2 + x2). 12. Find the equation of a plane referred to three rectangular axes. The planes 2x + 3y+4za and 5x+4y+3z = b intersect the plane, x = c. Find the cosine of the angle between the lines of intersection. 13. Interpret the equations (1) (x − a)2 + (y − b)2 + (≈ − c)2 =k2 {(x − a) 1 + (y — b) m + (z — c) n}2. (2) (x − a)2+(y− b)2 + (z − c)2 = {xl+ym + zn}2, where +m+n2 = 1 in each case. Prove that the surfaces thus represented intersect in plane curves, and find the equations to the planes of such intersection. Any of the following questions may be substituted for an equal number of the above. A. If the curve in question 8 have contact with a circle of radius r, whose centre is in the axis of y, and if the angle of inclination of the common tangent to the axis of a be 45°, prove that Determine also whether the radius of curvature at the point of contact is greater or less than that of the circle. B. How are solid angles measured? Prove that the solid angle subtended by a square whose side is 2a at a point in the perpendicular to its plane through its centre and at a distance b from its plane is to the solid angle subtended by any closed surface surrounding the same point as sin1 to π. a2 is a2 + b2 2 2 WOOLWICH. September, 1873. ARITHMETIC AND LOGARITHMS. 1. Add together of 3, 7 of, and 5 off. 3. Multiply together 5, 34, 235, and §. 4. Divide 514 by 1i. 5. Add together 7131, 754.5, 63, 07385, and 891.062. 6. Subtract 378-0695 from 400.021. 7. Multiply 57.01 by 00284. 8. Divide 80 14662 by 236.7. 9. Reduce 12s. 114d. to the decimal of £2. 11s. 9d. 10. Add together 7, 8, and 14. 11. Subtract 10 from 12. 12. Multiply together 7 of 31, 51 of 4, and . 13. Divide 87 by 15. 14. Add together 15.63 of a day and 401625 of an hour, and give the answer in minutes and the decimal fraction of a minute. 15. Subtract 5.72 of a pole from 256 of a mile, and give the answer in yards and the decimal fraction of a yard. 16. Multiply 314.5 by 6.28. 17. Divide 410.9 by 3.26 to 3 places of decimals. 18. Reduce 5'056 dwts. to the decimal of a pound troy. 19. Reduce 5 m. 2 fur. 25 p. 4 yds. to inches. 20. If 7 cwt. 1 qr. of sugar cost £26. 10s. 4d., what will be the price of 2 tons 3 cwt. 2 qrs.? 21. Find (by Practice) the dividend on £6,052. 13s. 4d. at 17s. 93d. in the pound. 22. Find the simple interest on £2,350 for 6 years at 7 per cent. per annum. 23. In 5 acres 3 roods 25 square yards how many square feet? 24. If 5 acres contain grazing for 20 sheep for 7 days, how many will be required for 52 sheep for 35 days? 25. Find (by Practice) the value of 79 tons 8 cwt. 3 qrs. at £18. 3s. 4d. per ton. 26. Find the compound interest on £7,850 in 3 years at 4 per cent. (neglecting fractions of a penny). 27. Compute by means of the tables the value of (3.05) 61.28 × √(3056.25) to five places of decimals. 28. Given log2=3010300, log3=4771213, log 13 = 1∙1139434; 24 log 01625, and find how many figures " 135 find log 3 the number represented by 23 is composed of. 29. Extract the square root of 255200625 and of 37217. 30. How much money must be invested in the 3 per cent. consols at 941, so as to yield an income of £225 per annum? EUCLID. 1. If a side of any triangle be produced, the exterior angle is equal to the two interior and opposite angles; and the three interior angles of every triangle are together equal to two right angles. Prove that the angle between the bisectors of two consecutive angles of a quadrilateral is half the sum of the two remaining angles. 2. Parallelograms-on equal bases, and between the same parallels, are equal to one another. 3. Divide a given straight line into two parts, so that the rectangle contained by the whole and one of the parts may be equal to the square on the other part. If on the larger of the two parts into which the given straight line is divided, a portion be marked off equal to the smaller of these parts, prove that the larger part will be divided in the same way as the given straight line. 4. If two circles touch one another internally, the straight line which joins their centres, being produced, shall pass through the point of contact. 5. The angles in the same segment of a circle are equal to one another. Of all the triangles upon the same base, and having the same vertical angle, prove that the isosceles triangle has the greatest area. 6. If AD, BE, and CF be perpendiculars from A, B, and C respectively, upon the opposite sides of the triangle ABC, prove that AD bisects the angle EDF. 7. If from any point without a circle two straight lines be drawn, one of which cuts the circle, and the other touches it; the rectangle contained by the whole line which cuts the circle, and the part of it without the circle, shall be equal to the square on the line which touches it. 8. Inscribe a circle in a given triangle. If the triangle be right-angled, prove that the two sides containing the right angle are together equal to the sum of the diameters of the inscribed and circumscribing circles. 9. If the exterior angle of a triangle, made by producing one of its sides, be bisected by a straight line which also cuts the base produced, the segments between the dividing straight line and the extremities of the base shall have the same ratio which the other sides of the triangle have to one another. 10. Divide a given arc of a circle into two parts whose chords shall be in a given ratio. |