IO. A particle is projected in a given direction with a given velocity, show how to find its position and the direction in which it is moving at the end of a given time. A stone is thrown from the top of a tower with a velocity of g feet per second in a direction making an angle a with a line drawn vertically upwards through the point of projection; prove that at the end of two seconds the line joining the stone to the point of projection will make an angle of (+a) with the vertical line. FURTHER EXAMINATION. V. PURE MATHEMATICS. No [Full marks may be obtained for about three-fifths of this paper. Candidate must attempt more than ten questions. Great importance will be attached to accuracy.] I. Describe a circle touching three given straight lines. If a circle be inscribed in an isosceles right-angled triangle, show that its diameter is equal to the excess of the sum of the equal sides over the hypotenuse. 2. Under what circumstances will the solution of two simultaneous equations in two unknown quantities depend upon the solution of a quadratic equation? Two trains are proceeding in the same direction upon the same line of rails, each with uniform speed. The quicker train, which is in front, gains of a mile every minute, and also of a minute every mile upon I n I n the slower train. Determine the speed of each train in miles per hour, and show that, n being a positive quantity, the speed of the quicker train cannot be less than 60 miles an hour. 3. Find the sum of ʼn terms of a geometrical progression. terms, and also to infinity, the series 2 (1 + x^) ̄}+x(1 + x2) ̄ + x2(1 + x2) ̄å + .... 4. State the circumstances under which (I + x)n is expressible in the form of a convergent infinite series. Sum to n by the binomial theorem, and show that the coefficient of x32% is equal to the coefficient of the same term in the development of the product ( y − z)3 (z − x)2(x − y). 5. What is the characteristic of the logarithm of a number? Prove the rule for assigning the characteristic of a given decimal number. Expand log in a series proceeding by ascending powers of x-I x + I 6. One angle of a triangle is given, and the ratio of the sines of the other angles. Show how to find these angles by a formula adapted to logarithmic computation. 7. Express the area of a triangle in terms of its sides. Show that the formula involves Euclid's theorem that two sides of a triangle are together greater than the third side. Find an expression for the area in terms of the sums of first, second, and third powers of the sides. If the sum of the sides is 12 feet, and the sum of the squares on the sides is 50 square feet, and the sum of the cubes on the sides is 216 cubic feet, find the area of the triangle. 8. Find expressions for the curved surface and volume of a right circular cone. A conical tent is five feet high. Find the radius of its base so that the number of square feet of canvas may be equal to the number of cubic feet of space inside the tent. 9. Show how to transform from rectangular coordinates to oblique, and from oblique to rectangular. The equation of a straight line referred to axes inclined at 30° is y = 2x + 1. Find its equation referred to axes inclined at 45°, the origin and axis of x remaining unchanged. 10. Define an ellipse, and prove geometrically that the sum of the focal radii of any point upon it is equal to the major axis. II. Points (1, 0), (2, 0), are taken on the axis of x, the axes being rectangular. On the line connecting the points an equilateral triangle is described so that the coordinates of its vertex are both positive. Find the equations of the circles described upon its sides as diameters. 12. Find the equation to the tangent at any point of an ellipse, and the lengths of its intercepts upon the coordinate axes. In an ellipse of eccentricity tan ß, a focal chord is inclined at an angle a to the major axis; show that the tangents at the extremities of the chord include an angle tan (tan 2ẞ sin a). 13. If a small portion of a hyperbola be given, show how to verify its hyperbolic nature geometrically. If two sides of a triangle be given in position, and its perimeter given in magnitude, show that the locus of the point which divides the base in a given ratio is a hyperbola. VI. MECHANICS. [Full marks may be obtained for about two-thirds of this paper. Great importance is attached to accuracy. N.B.-g may be taken = 32]. I. Forces are represented in magnitude, direction and position, by the sides of a polygon taken in order; prove that they are equivalent to a single couple, and that the magnitude of the couple is proportional to the area of the polygon. 2. A uniform beam, 12 feet in length, has a fixed hinge at one end, and is supported by a cord, 13 feet long, attached to the other end and to a fixed point situated 20 feet vertically above the hinge. Find the tension of the cord, assuming the weight of the beam to be 140 lbs. 3. A uniform beam AB, 20 feet long, is supported by props at C and D, two points at distances 4 feet from A and 6 feet from B, respectively. If a load of I ton be placed at each extremity of the beam, calculate the magnitude of the moment which tends to break the beam at the middle point of CD: assuming the weight of the beam to be 2 tons. 4. A ladder is placed with one end on a rough horizontal plane and the other against a rough vertical wall; find, by geometrical construction, or otherwise, the limiting position of equilibrium, being given the coefficients of friction and the centre of gravity of the ladder. If an additional load be placed at any point on the ladder, in this limiting position, find whether the equilibrium will be disturbed or not. 5. Three forces acting on a rigid system are in equilibrium, prove that their directions lie in the same plane, and either pass through a common point or are parallel. A uniform cubical block is sustained on a rough inclined plane by a rope, which is parallel to the plane and is attached to the middle point of the upper edge of the cube, which is horizontal. The rope lies in the vertical plane, which contains the centre of the cube and is perpendicular to the inclined plane. Determine the greatest inclination of the plane consistent with equilibrium. |