9. Find the H.C.F. of 2 + ax" — any — yo and 24 + 2.2%y — aʻx? + xʻy– 2axy– y4. 10. If a and B be the roots of ax2+bx+c= 0, then ax? +bx+c = a(x – a) (x – B). 2x - 7 222 — 2.0 11. If x be real, prove that value between 1 and 1. can have no real 5 12. If 3x – 2y_3y – 2z_32 – 2x 3a + 2636 + 2c3c + 2a prove that (x+y+z) (191a+1786+2010)=95(a+b+c) (x+2y+32). 13. If y is equal to 1 + ma + nx? where l, m, n are constant, and x variable, and if when x=a, y= 0, when x=2a, y=a, : and when x=3a, y = 4a, show that when x= na, y=(n − 1)'a. 14. There are 3 numbers in Harmonical Progression. The greatest is the product of the other two, and if unity be added to each, the greatest becomes the sum of the other two. Find the numbers. 15. Find the coefficient of an in (1 + x + 3 + .....). ? ANSWERS TO JANUARY PAPER. 1. 10. 3. (a + 3b) (a-1). 5. - 4. 6. x' – 10.x2y2 +9y. 7. (i) x=5 or }; (ii) x=y=a +b; (iii) V1 — =-1V3. 16.15 21 9. 491 min. past 9. 9. 10. 1-26 y 11. 262 =9ac. 12. x=y=z=0, or x = y=z= -1. a + 3c. Higher Mathematics. Junior and Senior. TRIGONOMETRY; STATICS. 1. Describe the method of circular measurement of angles, and find the number of degrees in the unit angle. 2. The radius of a circle is 18 feet ; find the length of an arc which subtends an angle of 10° at the centre. 3. Find the values of the following ratios :-sin 30'; sin 210°; tan 300° ; sin 225o ; sec 315o. 4. Solve the equations : (i) sec 0 cosec 0 + 2 cot 0 = 4. 0 5. Show that the following is an identical relation :- ө 6. Find an expression for all the angles which have a given tangent. Solve the equation : 2 sind = tano. 7. O is a point within a triangle ABC, and D, E, F, the middle points of the sides. Show that the forces represented by 0A, OB, OC are equivalent to those represented by OD, OE, OF. 8. Find the magnitude and direction of the resultant of two parallel forces. Can a resultant always be found ? 9. Define the moment of a force about a point. Describe the lever, and give the conditions of equilibrium. Higher Mathematics. Women. STATICS, DYNAMICS, ASTRONOMY. 1. A circular disc is kept at rest by three forces acting perpendicularly to the circumference at three points in it. Show that they are proportional to the sides of the circumscribing triangle through the points. 2. Parallel forces act at the angles A, B, C of a triangle, proportional to the sides a, b, c. Show that their resultant acts at the centre of the inscribed circle. 3. A rough plane is inclined to the horizon at an angle of 60° ; find the magnitude and direction of the least force which ° will prevent a body weighing 100 lbs. from sliding down the 1 plane (v = 13). 32, in 4. The unit of force is the weight of 1 oz., and the mass of a cubic foot of the substance of unity density is 162 lbs. What is the unit of length, the unit of time being 1 sec. ? (g the foot-second system.) 5. A body is projected horizontally with velocity 4g from a point 169 feet above the ground ; find the direction of motion (i) when it has fallen half-way to the ground, (ii) when half the whole time of falling has elapsed. 6. Two equal balls start at the same time with equal velocities from the opposite angles of a square along the sides, and impinge ; find the angle between their directions after impact. 7. Give two methods of finding the latitude of a place. Find the latitude of a place whose longest day is 16 hours. 8. What is meant by direct and retrograde motion of planets ? Illustrate by diagram the apparent motions of an inferior planet. 9. When Venus is a morning star and stationary, does it begin to move backwards or forwards among the stars ? Higher Mathematics. Women. TRIGONOMETRY AND CONICS. 1. Divide the angle A into two parts whose sines shall be as a:b. 2. ABC is a triangle, right-angled at C. The side AC = 100 feet, and the hypothenuse AB = 196 feet. Solve the triangle having given log 2 = -30103 L sin 30° 40' = 9.70761. = log 7 = •84510 L sin 30°41' = 9.70782. 3. If sin (Trcos) = cos(asing) show that 0 = + sin = = 13 = = 4. If the co-tangents of the angles of a triangle be in A. P., the squares of the sides will also be in A. P. 5. Show that in an ellipse, PN: AN.NA' = BC: AC2; and ; that if the ordinate NP be produced to meet the auxiliary circle in Q, PN:QN= BC: AC. 6. If the tangent at P of a parabola meet the axis in T, and PN be the ordinate, show that NT= 2AN. 7. If the tangent at any point P of an ellipse, or hyperbola, meet the major axis in T, and PN be the ordinate, show that CN.CT= CA. 8. If the tangent and normal at any point of an ellipse meet the major axis in T and G respectively, CG.CT= CS? 9. Find the condition that the line læ + my = n should touch the conic axa + 2bxy + by* = e. 10. Define the eccentric angle of an ellipse or hyperbola. Find the equations of the tangent and normal to the ellipse 72 1 at the point whose eccentric angle is 0. Natural Philosophy. Junior and Senior. (a) CHEMISTRY ; (6) PRACTICAL CHEMISTRY ; (c) STATICS, DYNA MICS, AND HYDROSTATICS EXPERIMENTALLY TREATED ; (d) THE Junior Students will only be examined in three of the subjects (a), (b), (c), (d), (g), (h). Senior Students will only be examined in three of the subjects (a), NOTE.—(6) cannot be taken with (a), nor (9), nor (k), without ($). (a) 1. Describe clearly how you would prepare Hydrogen gas, giving equations of the reactions taking place. 2. How would you determine the composition of water by weight? 3. What volume (in cc.) will •845 grammes of Hydrogen occupy at normal pressure and temperature ? (6) 4. How is carbonic dioxide made? How would you prove its presence ? 5. Show clearly how carbon mon-oxide is prepared from formic and oxalic acids. How much should be obtained from 24 grammes of oxalic acid ? 6. What is “Bleaching Powder"? How is it made ? (c) 7. Show in what direction a body will move if acted on by two forces at right angles to each other. If one force is equal to 20 lbs. and the other to 10 lbs., in what direction will the body move ? 8. Neglecting the weight of a lever which is 5 feet long, find where the fulcrum must be placed in order that 2 lbs. and 8 lbs. may balance at its extremities. |