a . Find the value of the expression 10.8.2001 - . 66 doh! would 1 (cos le cos, la – sin o sin la cos L1-L2) where yo 4000 l1=36° 45' 10" LI=65° 27' 38" * = convincin le-72° 15' 171 'Iq=15° 4' 22" : : 19. * * * * * * * * * สี 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 anglos 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 ont from a circle of radins l'ftsi 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 18" 1:40. ** (E;+ VÉTELI E) cubic feet, where the thickness of the frustum is k feet, and the ends contain E, and Eg square feet respectively, The volame of the frnstum of a cone is 407 cubic inches, and its thickness is 104 inches. If the diameter at one end is 8 inches, find the diameter of the other evd. 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 rading of the sphere, write down the expression for the volume of the sphere, the snrface of the sphere, and the volume of the empty space between the spliere and the cylinder. Find the valges 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. wi The figures in the margin indicate full marks, 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 nth term. @6".;. Shin!: 2. Write down the general term and find the sum of the following: w 30 series to n terms : 1+3+7+13+21 ........ 3. Prove the Binomial Theorem when the index is a positive integer. 40 If Ca... ...C be the coefficients in the expansion of ( 1 + o)*, prove that CI +2c2 + 3cg + ... +n enen 2n-1. a6 ) 90 : +008 + 008 4 Show that loge(*+a) - logn-a) a8 3n3srb Having givon that Jogo 2-6931373 show how the above formula may bo used to find log. 10 and And i8 cor- 10 rect to the 6th dooimal place. 5. Resolve 683 +672 + 50 (82-1) (•+1) into partial fractions. 6. Prove De Moivro's theorem ; and 1180 the theoram to obtain ap 30 ou pansion of sin ne in ascending powers of sin e. 7. Find the sum of the cosines of a series of anglos, the anglos boing, 40 in arithmetionl progression. O. sam to n terms the following series 37 51 2n + 1 2n +1 2n + 1 8. Prove that 1) cosh Rassinh 2a =1 (2) cosh (a-B) = cosh a cosh B - sinh a sinh B. 9. Find the oquntiori ofin straight line in any form ; And show how 30 the equation in tune form cho be bent used for racing the straight line.'*?". Find the equations of the sides of the triangle whose nngulir points are (1, 2), 12. 3), and (-3,-5) 10. Find the egnation of the cir•le in its simplest forın. 40 Find the eqnation of the circle, pissing throngh the points (9, 4), (0,3), 15. -4) ; and show that its radius is 5*! apiroximately. 11. Prove that the polar of any point on the circle +7% -228 - 342=0. with respect to the ciroio 202 + y2 + 200 - 3ul=0. will toach tho parabola y2 + 40% = 0. : Draw the oiroles and the parabola, and show in the figure one of the poles and the corresponding par 12. Find the equation of the ellipse in the form 30 Find the eqnation of the points in which the straight line 52+ 2y = 9 woote the ellipse 2582 +2y2 = 100, and the length interceptod. 1 HATHEMATICS. THIRD PAPER STATICS AND DYNAMICS. Esaminer-C. LITTLR, Esq., M.A. The figures in the margin indicate full marks. 1. Bxplain' bow uniform and oniformly accrlerated motion are measured. Prove that when a point is ryoying with uniformly increasing velocity 46 its average pulocity 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 :& certnin point a 30 oert in velocity in the opposite dir .ction to itu acceleration : it returns to that point after an interval of 4 grconds; find how far it travels from that point and its velocity at that point 8 Find the ringe the time of light and the greatest height of a 25 particle projucted from a voint in a horizontal pline. 4 From the foot of an inclined plane wiore rise in 7 in 25 a shot is pro 30 jerted with a pelo.ity of 600 ft. ver we oond at an angle of 30' with the horizontal, (1) up the plane, (2) down the plane Find the range in each case 5. Emanciate Newton'x First. Law of Motion : and state whit infer. 80 ence it on blog us to draw in the case, if uniform unorior: in s cirr.le. 6. Find the resultant of two parallel forc .n scting on a rigid body. 40 A Quiforın beam 4 feet long. 18.,map: vorteil in a b rizontal prisition by two prope, which are three fert aport. so that the beam projects one, foot beyond one of the prop* : show that the pressuire on one prop is donble that on the other. 7. Find the centre of mugg of a triangular lavnina of nniform density. 30 8 Fid the conditions of equilibrium of a coplanar system of 30 forces. 9. A uniform ladder rest.8 in limiting eqnilibrinm with one end, on a 40 rough floor, whose coefficiens, of friction, in to and with the other againsta om oth vertical wall: 'show that its inclinarion to the vertical is tun - 1:24 10 Find the relation betwuen the power :ind the weight in the case 30 of a sirew, when friction is taken ipio ncronnt. 11. A smooth sphere of wars m impingon directly with velncity u on 40 another, smooth spiere, of mano m' moring in the same direction with velocity u'. If the spheres be perfectly elartic, find their velocities after impact. 12. State the princime of work and energy : and show that it holds 20 in the case of a body falling under yravity. MATHEMATICS. FOURT PAPER. DIFFERENTIAL AND INTEGRAL CALCULUS. Foaminer---. LITTLE, Esq.,. .M.A. The figures in the margin indicate, full marks. 1. From the definition of a differential coefficient plednoe its valtip for .45 the following functions : :, sin ,'ex and m. Differentiate (1). sin:(1+x2) ...؟ 3. Enunciate and prove Leibnitz's theorem. 40 If y=d sin æ prove that dly dy +(22+2): y=0. dx2 i: Apply Leibnitz's theorem to the above expression to find the relation dty connecting and lower differential Co-efficients of y with respect to *. dx4 4. Enunciate Taylor's theorem, and by means of it expand logela'#*) ;40in ascending powers of φία) 5. Prove that if p.: (a)=0 and ' (a)=0, the limiting value of 30 $(*) :.::id/fal). when x is equal to a is 4!(9) Find the value of jih ::.in 203-3002 + 1 when æ=1. 3x6-503 + 2 6. If y=f(x), show how maximum and minimum values of y can be determined and identified. Illustrate the theory by means of a diagram. 30 ) ? If I 56 +1246 - 1596-4003 + 15x2 +60x + 17, find whether the values of the function corresponding to a values Fland-2 aramaximum or minimum values or neither. 7. Find the equations of the tangent and normal to the carve y-f(w). Find the subtangent and sabnormal in the case of the parabola 42 - 4ax. Trace the carve aly=(2-6) 8. Explain the connection between the series ho(a)+ hp (a+h). hø{a+n–1n} in and the integral ♡ (2),da., S 40 Write down the value of -w). , stan Sa Su seco: da, tan, a da, da and a2 + a2? 9. . . Prove the following formula for integration by, parts : fain sin må sin näda. 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 cirçalar lamina about a point in the circumference. * kg 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. Deporible the work of a road burney withi i přismatic compass; sizid 30 note the checks you would apply to your work. 08. Prie šurvey of large plőt is to be done with the chafi: Describe 30 the process, and detail all the steps you would take to ensure its acouracy: |