If ABCD be a quadrilateral inscribed in a circle, diagonals intersect in O, and if, while A, B, and D remain fixed, C approaches B, find the limiting value Since AC and BD (fig. 18) chords of a circle intersect in O, we have 40.OC=DO.OB; and therefore when C moves up to and coincides with B, 9. Find the equation to the tangent through a given point to the curve y=f(x). Todhunter, Diff. Calc., Art. 257. The equations to the tangents drawn from the origin to the parabola ay = x2+ bx + c are y = b+2c a Now, because the tangent passes through the origin, which are the equations to the two tangents through the origin. 10. If the logarithmic curve y=ae has contact of the second order with the curve p(x, y) =0, shew that the subtangent of the latter curve has a maximum or minimum value at the point of contact. If y = (x) be the equation to the curve, value of the subtangent, and the condition for maximum or minimum value of this is which condition is satisfied if (x), p′(x), p′′ (x) have 11. Find the radius of curvature at any point of the polar curve r = ae", and shew that it makes a constant angle with the radius vector. H If r = ae", we may readily prove_r=p√√/(1+n2), hence dr p=r = a √√/(1+n2) eno. dp Also, since is a constant, we see that the angle r between the radius vector and the tangent is constant, and therefore also the angle between the radius vector and the radius of curvature. 12. Prove the rule for integration by parts, and integrate the following expressions: 13. If f(x), (x) be rational functions, f(x) being of higher dimensions than $ (x), and if the roots of the equation f(x) = 0 are all real and unequal, shew how 1 14. Find the length of the arc of a parabola from the vertex to the ordinate y. Let equation to parabola be y2 = 4ax, then ROYAL MILITARY COLLEGE, SANDHURST. Papers in Competitive Examination for Admission to. ARITHMETIC. WEDNESDAY, 29TH NOVEMBER, 1876. 2P.M. TO 4.30 P.M. 1. Add together 4, 1, 3, and 4. Ans. 518}. 2. Subtract 113 from 12. Ans. 7. 3. Multiply 2 by 13. Ans. 43. 4. Divide 53 by 3. Ans. 74. 5. Add together 570 638, 80506, 5437·07, and 0005. Ans. 6008.51356. 6. Subtract 389-762 from 1036′5. Ans. 616.738. 7. Multiply 85.7629 by 7.0063. Ans. 600-88060627. 8. Divide 3.420 by 45. Ans. 7·6. 9. Assuming that a year = 365 days, express § of a year in days and the decimal of a day. Ans. 228.125. 10. In 5207653 inches how many miles, furlongs, &c.? Ans. 82 m. 1fur. 21 p. 1yd. 1ft. 7 in. |