11. Explain Newton's method of approximating to a root of an equation. On what does its success depend? Ex. (1) 2a+152+16x-3=0 (2) Algebraic Geometry. 3+48x2+261x-16-0. 1. y=ax+b being the equation to a straight line, explain the geometrical signification of the constants. Trace the lines included in the above equation, supposing a and b to remain the same in absolute value; but to be, one or both, positive or negative. Discuss the equation y = ax + b when a= =0. 2. Transform the equation referred to rectangular axes to another referred to rectangular axes making a given angle with the former, the origin remaining the same. 3. Find the polar equation to the circle, the origin being the pole. 4. Define the centre and diameter of a curve. Find the co-ordinates of the centre, and the magnitude and position of the axis, of the curve whose equation is 5y2+6xy+5x2-22y-26x+29=0 the axis of co-ordinates being rectangular. (which is referred to rectangular co-ordinates) to its simplest form. Trace the curve it represents. 6. Define the tangent of a curve; and, consistently with your definition, find the equation to the tangent of an ellipse which is drawn through a given point in the curve. 7. Find algebraically the locus of the intersections of the perpendicular from either focus upon the tangent of the ellipse and the tangent itself. 8. The equation to the hyperbola being Find the equation to its asymptotes. 9. Find the equation to the normal of a parabola at a given point in the curve. 10. Find the intersection of two given straight lines in space. What relation must subsist amongst the constants of the equations to the straight lines that they may meet one another? 11. Define a plane, and from your definition deduce its equation. Reduce the equation to the form trical interpretation of a, b and c. X y 2 + + 1 and give the geome Mechanics. Mixed Mathematics. 1. In what respect do the fundamental principles of Statics differ from those of Dynamics? 2. What is Weight? under what limitation may it be taken as a measure of mass? 3. State the three laws of motion. What is the nature of the evidence on which they rest? Show its application in the case of the second law. 4. Prove the proposition called the parallelogram of forces. 5. Find the relation of P. to W. in the system of pulleys in which each pulley hangs by a separate string, the strings being parallel. Explain clearly the means by which the smaller weight is made to balance the larger. How is the difference of the weights of P. and W. disposed of? The answer to the preceding query is to be given numerically as well as in general language. 6. Define virtual velocity. Prove the principle of virtual velocities in the case of the inclined plane, the string connecting P. and W. being parallel to the base. It being granted that what is gained in power is lost in time, show in what the practical advantage of using machinery consists. 7. Find the centre of gravity of a pyramid on a triangular base. 8. Find the equation to the curve described by a body projected from a given point in a given direction with a given velocity; find also the horizontal range and line of flight, and the focus and latus rectum of the parabola. Account for the discrepancies between practice and theory. 9. Explain the construction of the common balance. What are the requisites of a good balance? How are they secured? 10. State clearly what is meant by the isochronism of the pendulum. In what path does the extremity of the common pendulum move? What is the moving power in the commonest household and turret clocks, and in watches? A pendulum clock, which has been going correctly in Calcutta, is removed to Simlah at an elevation of 7000 feet above the former station, and thereby exposed to a much lower temperature. What will be the effects of this change on the clock's rate? will they conspire, or oppose each other? 11. Find a general expression for the effect of small contemporaneous alterations in 7 (the length of a pendulum) and g (terrestrial gravity) on the time of an oscillation, t. 12. A person, on being weighed in an ordinary pair of scales, with a stick in his hand, (1) against the ground, (2) against the presses What is the effect produced in each case? beam. 13. The force which a man can exert in the tension of the cord in 1 3 X his own weight. the single moveable pulley is Show how by means of a fixed pulley and a bucket attached to the moveable pulley he may raise himself from the floor towards the ceiling of a room, considering the strings (1) parallel; (2) not parallel. Is there any limit to the possibility of this problem in the second case? State your opinion of the effect produced by the angle which (in the second case) the string in the man's hand makes with the vertical. Hydrostatics. 1. Forces which have any assignable ratio to each other may by the transmission of fluid pressure be made to keep each other in equilibrium. What is the manner of applying this principle in Bramah's press? 2. Find the pressure on a portion of a plane immersed in a fluid. 3. What is a level surface? Prove that the surface of a fluid at rest is a level surface. What modification does the expression of this principle undergo, when the surface is of limited extent? What is the meaning of the expression level of the sea'? 4. If a body floats in a fluid, it displaces as much of the fluid as is equal in weight to the body. A cube whose specific gravity is 1 m is placed in a vessel containing a fluid whose specific gravity is 1 n. If it remains at rest, find the ratio of the portions of the cube that are in and out of the fluid: if not, find the moving force with which it descends. 5. Explain the construction of a common hydrometer, and show the manner in which it indicates the specific gravity of a fluid. 6. State Mariotte's law, and detail an experiment by which it may be proved. 7. The mercury is sustained in the barometer by the pressure of the air on the surface of the mercury in the basin; and the pressure of the atmosphere is accurately measured by the height of this column. 8. The specific gravity of sulphuric acid is 1.8; of mercury 13.5. What variation in a sulphuric acid barometer corresponds to one of a tenth of an inch in the mercurial column? What are the advantages and disadvantages of a sulphuric acid barometer? What resemblance is there between a pump and a barometer. What would be the length of the water barometer, the average height of the mercurial column being 30 inches? 9. Explain the manner in which water is raised by the screw of Archimedes. English Essay. The influence of climate on national character. Latin Essay. "Nemo illic vitia ridet; nec corrumpere et corrumpi sæculum vocatur. plusque ibi boni mores valent, quàm alibi bonæ leges." TACITUS, Germania. Vernacular Essay. On Truth. সত্যের মহিমা বর্ণনা কর । JUNIOR SCHOLARSHIPS 1848. English Grammar, 1. How many sounds has the letter A? Give two examples of each sound. 2. Why ought W and Y to be considered consonants when they begin a word, and vowels when they end one? 3. "Consonants are divided into mutes and semivowels." What letters are mutes? Explain clearly why they are called mutes. 4. Define the words Substantive, Adjective, Mood, Tense. 5. Mention the different kinds of Adjective Pronouns, and define and give examples of each kind. 6. Write out the Imperfect Tense and Perfect Participle of the following verbs : : Bid, Bite, Choose, Clothe, Do, Forsake, Go, Let, Mow, Sit, Write. 7. Parse "A time there was ere England's griefs began, When every rood of ground maintained its man.” 8. Define the following words of like meaning : Accent, Emphasis. Language, Dialect. Doubt, Suspense. Cheerfulness, Mirth. 9. Compose a sentence or sentences in which the following phrases shall be correctly applied: "Cloud of words." "Stubborn facts." "Break loose." 66 Play the fool." "Set the teeth on edge." 10. Correct the following sentences: "I do now publish my Essays, which of all my other works have been most current." 11. Conjugate the Indicative Mood of the verb "to work." |