3. Having given log102,=*301030 and log107=*845098, find log1014, + + = ... I+ + 1.3.5 I I.3.5.7 I + 6 1.2 62 1.2.3 63 I.2.3.4 64 5. On a certain road the number of telegraph posts per mile is such that if there were one less in each mile the interval between the posts would be increased by 21 yards. Find the number of the posts per mile. 6. Prove that and that cos 30° + cos 60° + cos 210° + cos 270° = }}, sin 70 sin 30+2 sin 50+ sin 70 sin 50+2 sin 70+ sin 90 7. If A+B+C=90°, prove that tan B tan C+tan C tan A+tan A tan B=1. Hence shew that 2 (tan2A +tan2B+tan2C) - 2 = (tan B - tan C)2 +(tan C-tan A)2 + (tan Atan B)", and that the expression tan2A +tan2B+tan2C is never less than unity. 8. Expand each of the fractions in a series of ascending powers of x, and find in each case the coefficient of xn. 9. A length of 300 yards of paper, the thickness of which is the hundred and fiftieth part of an inch, is rolled up into a solid cylinder; find approximately the diameter of the cylinder. IO. Find the centre of a circle cutting off three equal chords from the sides of a triangle. II. In any triangle the straight line bisecting an angle, and the straight line passing through the middle point of the opposite side, perpendicular to it, meet on the circumscribing circle. 12. Having given the base of a triangle, the vertical angle, and the ratio of the sides, construct the triangle. 13. A man, walking along a straight road at the rate of three miles an hour, sees in front of him, at an elevation of 60°, in the vertical plane through his path, a balloon which is travelling horizontally in the same direction at the rate of six miles an hour; ten minutes after he observes that the elevation is 30°; prove that the height of the balloon above the road is 440√√3 yards. 14. A quadrilateral ABCD can be inscribed in a circle; if E, F, G, H be the centres of the circles which circumscribe the triangles ABC, BCD, CDA, DAB, prove that the quadrilateral EFGH is also inscribable in a circle. [= 3'14159.] VII. PURE MATHEMATICS. (3.) I. Find the relation between the coefficients and roots of an equation. One root of the equation x3- 13x2+15x+189=0 exceeds another root by 2; solve the equation. 2. Prove that incommensurable roots enter by pairs into an equation with rational coefficients. If one root of such a biquadratic equation be √a+√b, where √a and √ are dissimilar quadratic surds, prove that the other roots are - √a+√b, Ja-√o, - √a- √b. 3. Prove that an odd number of roots of the equation ƒ'(x)=。 lies between every two adjacent real roots of the equation f(x)=0. of Prove that the equation ay2+3 - bx2=0 has always one positive value for every value of x, and that it has two negative or two impossible y values of y according as x is numerically greater than 4. 2α 3 Find the differential coefficient of x" for all values of n. Differentiate 6. State, without proving, the method of finding the limiting value of ( dy\2 y-x dx f(x) for values of x which make f(x) and F(x) zero. F(3 7. What is meant by the statement that a function of x is capable of expansion in positive integral powers of x? Prove that Maclaurin's theorem holds in all such cases. Find the first 5 terms in the expansion of y=log. (1+x sin x). 8. Prove that if a is such a value of x as to make ƒ(x) a maximum or minimum, then f'(x) must change sign when x=a. Hence deduce a method for determining all such values of x. If SP and SQ be two focal distances in an ellipse inclined to each other at the angle a, find the greatest and least values of the area of the triangle PSQ. 9. Find expressions for the perpendicular on the tangent and for the radius of curvature in polar curves. 10. 2r p= sec. 3 Find the asymptote of, and trace, the curve whose equation is prove that if x=a cos30, then y=a sin30, and that the equation of the tangent at the point determined by 0 is y cos 0+x sin 0 = a sin 0. Find the locus of intersection of tangents at right angles to one another. 12. Find the following integrals: 13. Find the area of the loop of the first of the two curves in Ques tion 10. VIII. STATICS. I. When any number of forces act upon a particle in one plane, explain generally how their resultant may be found by the application of the parallelogram of forces to these different forces in succession. If the forces are in equilibrium, what does the resultant become? Assuming the parallelogram of forces, prove the equilibrium property of the triangle of forces. A picture of given weight hanging vertically against a smooth wall is supported by a string passing over a smooth peg driven into the wall; the ends of the string are fastened to two points in the upper rim of the frame which are equidistant from the centre of the rim, and the angle at the peg is 60°; compare the tension in this case with what it will be when the string is shortened to two-thirds of its length. 2. State (without proof) the condition of equilibrium when two forces act on a straight lever in one plane to turn it round a fulcrum. Assuming this property, prove the condition of equilibrium of a lever acted on by any number of forces at different arms in one plane to turn it round a fulcrum. If any number of forces represented by the sides of a regular hexagon taken in order act along the sides to turn the hexagon round an axis perpendicular to its plane, shew that the moment of the forces is the same through whatever point within the hexagon the axis passes. Is this true if the hexagon is not regular? 3. If any system of forces acting in one plane on a rigid body have a single resultant, the moment of the resultant round any point in the plane is equal to the algebraical sum of the moments of the forces. If the forces not in equilibrium cannot be reduced to a single resultant, they will be equivalent to a single couple. ABCD is a rectangle; AB, BC, adjacent sides, are 3 and 4 feet. Along AB, BC, CD, taken in order, forces of 30, 40, 30 lbs. act respectively; find their resultant. 4. Define the centre of gravity of a body. In what sense can a plane area be said to have a centre of gravity? Given the centre of gravity of a given area, and also of a portion of it, find the centre of gravity of the remainder. ABC is a triangle, DE a line drawn within the triangle, parallel to the base BC intersecting the other sides in D and E, DE and BC are equal to (b) and (a) respectively; if (1⁄2) be the line drawn from A bisecting BC, prove that the distance of the centre of gravity of the trapezoid BCED from A is 5. Describe any arrangement of three equal moveable pulleys by which a mechanical advantage is obtained, and in the arrangement described find the relation of the power to the weight when there is equilibrium, neglecting the weight of the pulleys. Find also the whole pressure sustained by the upper block. 6. Describe the common balance, and find its position of equilibrium when loaded with unequal weights. State the requisites of a good balance, and shew how its sensibility may be secured. 7. State the principle of virtual velocities as applicable to the simple mechanical powers. Shew that the principle holds good on a straight lever loaded with unequal weights. If the weights hang from the ends of the lever by strings, examine how the position of the centre of gravity of the weights is affected as the lever is turned round the fulcrum in a vertical plane. 8. Explain generally how a rough surface affects the direction of the resistance of a pressure applied to it. What is meant by the limiting angle of resistance of such a surface? A weight (P) hangs over the top of a rough inclined plane by means of a string attached to a weight (W) on the plane, find the relation of (P) to (W) when (P) is just on the point of descending, and shew that if the inclined plane make an angle (24) with the vertical, (W) will just be moved when it is equal to (P), (ø) being the limiting angle of resistance. 9. A homogeneous sphere of given radius rests at the bottom of a hemispherical bowl of larger radius, examine the conditions of stable or unstable equilibrium, the surfaces being such as to prevent sliding. If the sphere is so loaded that the height of its centre of gravity above the lowest point is of its radius, determine the radius of the hemisphere when the equilibrium is neutral. IO. How is work mathematically estimated? Distinguish between the useful work done by a machine and the work applied to it. How is the modulus of a machine expressed with reference to the useful work and the work applied? When weights are raised through different heights, prove that the whole work expended is equal to the work that would be expended in lifting a weight equal to the sum of the weights through the same height as that to which the centre of gravity of the weights has been raised. |