. Find the value of the expression -- (eyni dadi wodą r (cos ↳1⁄2 cos ↳ – sin 17 sin la cos L1-Lg) where r 4000 1=36° 45′ 10′′ 19=72° 15′ 17 L1=65° 27′ 38′′ne anvig ynivelị 35 8. From each of two ships a mile apart the angle is observed which: 35, is subtended by the other ship and a mark on shore: these angles are found to be 52° 25′ 37′′ and 75° 9′ 45′′ respectively: find the distances of the mark from each ship correct to five places of decimals. 9. Find a general expression for the area of the segment of a circle. Find the area of the minor segment cut from a circle of radius 1 ft. eduk 3 in. by a chord which subtends at the centre an angle of 23° 40′ 10. Prove that the volume of a frustum of a cone is 1 2 40 k (E1+ EEE) cubic feet, where the thickness of the frustum is k feet, and the ends contain E and Eg square feet respectively. The volume of the frustum of a cone is 407 cubic inches, and its thickness is 10 inches. If the diameter at one end is 8 inches, find the diameter of the other end. 11. A solid sphere fits closely into the inside of a closed cylindrical 40 box, the height of which is equal to the diameter of the cylinder. Having given the radius of the sphere, write down the expression for the volume of the sphere, the surface of the sphere, and the volume of the empty space between the sphere and the cylinder. Find the values of the above expressions in cubic feet for a diameter assumed by yourself. MATHEMATICS.. SECOND PAPER. *** ALGEBRA, TRIGONOMETRY AND PLANE ANALYTI CAL GEOMETRY. Examiner C. LITTLE, ESQ., M.A. The figures in the margin indicate full marks. W 1. Find the sum of n terms of a series in Arithmetical Progression, 30 If the pth term of the arithmetical progression is q, and the qth term is p, find the mth term. 2. Write down the general term and find the sum of the following: 30 series to n terms : 3. Prove the Binomial Theorem when the index is a positive integer. 40 If c c......c be the coefficients in the expansion of (1+), prove that show how the above formula may be used to find log. 10 and find it correct to the 6th decimal place. 10 30 into partial fractions. 6. Prove De Moivre's theorem ; and use the theorem to obtain an 30 expansion of sin ne in ascending powers of sin e. 7. Find the sum of the cosines of a series of angles, the angles being 40 in arithmetical progression. e 9. Find the equation of a straight line, in any form; and show how 30 the equation in that form can be best used for tracing the straight line." Find the equations of the sides of the triangle whose angular points are (1, 2), (2. 3), and (−3,-5) 10. Find the equation of the circle in its simplest form. Find the equation of the circle, passing through the points (9, 4), (0,8), (5,-4); and show that its radius is 5'1 approximately. 11. Prove that the polar of any point on the circle + y2 — 2ɑ®—3u2 ≈0. 40 with respect to the circle 2+ y2+2a8—3u2=0. will touch the parabola y2+4ɑs=0. : Draw the circles and the parabola, and show in the figure one of the poles and the corresponding poiar 12. Find the equation of the ellipse in the form 30 Find the equation of the points in which the straight line 5x+2y➡2 meets the ellipse 25′′2+2y2=100, and the length intercepted. 1. Explain how uniform and uniformly accelerated motion are measured. Prove that when a point is moving with uniformly increasing velocity 45 its average velocity for any given interval is equal to the velocity at the middle instant of that interval. Hence or otherwise find the space passed over by a falling body in time t 2. A point has 32 units of acceleration: it has at a certain point a 30 certain velocity in the opposite dir etion to its acceleration : it returns to that point after an interval of 4 seconds; find how far it travels from that point and its velocity at that point 3 Find the range the time of flight and the greatest height of a 25 particle projected from a point in a horizontal plane. 4. From the foot of an inclined plane whose rise is 7 in 25 a shot is pro- 30 jerted with a velocity of 600 ft. per second at an angle of 30' with the horizontal, (1) up the plane, (2) down the plane Find the range in each case 5. Enunciate Newton's First Law of Motion: and state what infer- 80 ence it enables us to draw in the case, f uniform motion in a circle. 6. Find the resultant of two parallel forcs acting on a rigid body. 40 A uniform beam 4 feet long. is supported in a horizontal position by two props, which are three feet apart. so that the beam projects one, foot beyond one of the props: show that the pressure on one prop is double that on the other. 7. Find the centre of mass of a triangular lamina of uniform density. 30 8 Find the conditions of equilibrium of a coplanar system of 30 forces. 9. A uniform ladder rests in limiting equilibrium with one end, on a 40 rough floor, whose coefficien' of friction is μ and with the other against a smooth vertical wall: show that its inclination to the vertical is tan - 1·2μ 14 10 Find the relation between the power and the weight in the case 30 of a screw, when friction is taken into account. 11. A smooth sphere of mass m impinges directly with velocity woń 40 another smooth spiere, of mass m′ moring in the same direction with velocity u'. If the spheres be perfectly elastic, find their velocities after impact. ,:;,:;: 12. State the principle of work and energy and show that it holds 30 in the case of a body falling under gravity. MATHEMATICS. FOURTH PAPER. DIFFERENTIAL AND INTEGRAL CALCULUS. Examiner C. LITTLE, ES, MA. The figures in the margin indicate, full marks. 1. From the definition of a differential coefficient deduce its value for .45 the following functions :— i: Apply Leibnitz's theorem to the above expression to find the relation C day connecting and lower differential co-efficients of y with respect to z. 4. Enunciate Taylor's theorem, and by means of it expand loge (a) (40 in ascending powers of a. 5. Prove that if (a)=0 and (a)=0, the limiting value of If f(x)=5x+125 → 154–40≈3 + 15x2+60x +17, find whether the values of the function corresponding to a values +1 and -2 are maximum or minimum values or neither. 7. Find the equations of the tangent and normal to the curve y❤f().° - Find the subtangent and subnormal in the case of the parabola y2=4ax. 9. Prove the following formula for integration by parts : 40 11. Find the moment of inertia of a semi-circular lamina about an extremity of the bounding diameter. Write down the moment of inertia of a circular lamina about a point in the circumference. 3 Examiner E. R. GARDINER, Esq. in The figures in the margin indicate full marks. 1. Construct a scale of 2 miles to the inch to read furlongs. What is 20 its representative fraction ? 2. Describe the work of a road survey with a prismatic compass; and 30 note the checks you would apply to your work. 08. The survey of a large plot is to be done with the chain. Describe 30 the process, and detail all the steps you would take to ensure its accuracy |