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Pure Mathematics. 1. What is Euclid's definition of proportion, and what is the common arithmetical definition ? Shew that any four quantities which are proportional according to the latter definition will satisfy one of the tests of the former.
2. Give a proof of the following proposition :
“If two triangles have one angle of the one equal to one angle of the other, and the sides about two other angles proportionals; then, if each of the remaining angles be either less, or not less than a right angle, or if one of them be a right angle, the triangles shall be equiangular."
And show clearly by a diagram the necessity for the introduction of the condition, “ if each of the remaining angles be either less, or not less than a right angle, or if one of them be a right angle.”
3. Shew how to describe a parallelogram the area and perimeter of which are respectively equal to the area and perimeter of a given triangle.
4. Divide by , and explain fully what is sought to be effected by the operation, and shew that the ordinary rule furnishes a correct solution.
5. If a person lend me 1000 rupees for 60 days, when the rate of interest is 91 per cent. per annum, what sum must I lend him for 25 days, when the rate of interest is 10 per cent. per annum, in order to return the obligation ?
6. If the French kilogramme weighs 2.205 pounds avoirdupois nearly, and the pound avoirdupois contains 7,000 grains troy, of which 180 make the Indian standard tola; what is the constant multiplier to be used for converting any given number of seers into kilogrammes ?
7. Find the circulating period of the decimal fraction which the vulgar fraction is.
8. Apply the Binomial theorem to calculate accurately, to the second decimal part of a pound, the amount of £5,000 put out at compound interest for 11 years at 6 per cent. per annum.
9. What annual payment (P) must I make on this day, and on the same day of each of the (n) years next ensuing, in order to entitle me to receive an annuity (A) to continue for (m) years, to be paid to me by equal quarterly instalments, the first instalment to be paid at the end of the quarter next after the last of the annual payments made by me?
10. Find the square root of 101—28 13—and shew that the square root of a quantity of the form a + bv-1 may be expressed by a quantity in the same form.
11. What is the number of balls in a broken rectangular pile, the edges of whose lowest tier contain 150 and 50 balls respectively, and of which only the 10 lower tiers remain ?
12. Solve the Equation
28+3x7—22c6—74x + 105x4 +443.25–36x*—7562—432=0 which has equal roots.
13. Prove the truth of the following formula : sin A + sin (36°—A) + sin (72° + A) = sin (36° + A) + sin (72o —A), and explain its use in verifying tables of trigonometric functions. 14. Shew how to adapt the expressions.
a + val_12 and a tva? + 02. to logarithmic computation by means of subsidiary angles.
15. How many acres of ground are contained in a triangular field whose sides are in length respectively 15 chains 14 links, 12 chains 20 links, and 9 chains 8 links ? 16. Prove that
0 = tan 0 - * (tan 0)' + } (tan 0)5 - &c. and hence deduce a rapidly converging series expressing the ratio which the circumference of a circle bears to its diameter.
17. The tops of two vertical rods on the Earth's surface, each of which is 10 feet high, cease to be visible to one another when 8 miles distant. Prove that the Earth's radius is nearly 4224 miles. 18. Give a description of the Theodolite: and shew how it may
be used to determine the zenith distance of an object of observation, so that any errors arising from the axis of rotation of the vertical circle of the instrument not coinciding exactly with the centre of graduation may be avoided.
19. Prove that in a spherical triangle whose sides are a, b, c, and angles A, B, C,
cos } (a - b) tan (A + B) =
cos } (a + b) 20. Shew that, if each of the angles of a spherical triangle, whose sides are small when compared with the radius of the sphere, be diminished by one-third of the spherical excess, the triangle may be solved as a plane triangle whose sides are equal to the sides of the spherical triangle, and whose angles are these reduced angles.
- sin l.
21. Find the position of the centre, the magnitude of the axes, and the inclination to the axis of x of the axis major of the ellipse whose equation, referred to rectangular co-ordinates, is
5y' +6xy +5x22y—262 +29 = 0 22. Find the length of the perpendicular let fall from the focus upon the tangent to the hyperbola ; and prove that the locus of the points in which such perpendiculars intersect the tangents is a circle described upon the transverse axis.
23. Shew how the conchoid of Nicomedes may be applied to solve the problem of the trisection of any proprosed angle; and give a description of an instrument by which the conchoid may be drawn.
24. Having given the Equations to 2 planes referred to rectangular co-ordinates, find the angle of inclination of the planes to one another,
25. Shew, from geometrical considerations, that the first differential co-efficient of cos x with regard to x is equal to
26. Find the second differential co-efficients with regard to x of the following expressions :
a tan x t-V7? a?
a tan tát v B qe ) 27. Shew that for those values of x which render u=f(x) a maximum or minimum the first differential co-efficient of u with regard to x which does not vanish must be of an even order; and that u =
= f (x) is a maximum or minimum, according as the value of the first differential co-efficient which does not vanish is negative or positive.
28. Find the Equation to the evolute of an ellipse.
29. Ifu=f (x y) be a function of two independent variables; shew how to find the new value of Uy when z & y receive finite increments h & k respectively.
30. Having given the expression for the radius of curvature at any point of a curve R=
+ (d y) };
find the expression for the same radius, when the length s of the arc of the curve is made the independent variable.
31. Integrate the following expressions :
(x + 1) (x + 2) (x + 3)
(cos x)6 (u? + r)n
(sin x)* 2x
+ 1 (4)
1 32. Find the value of v« (sin x)" (between the limits x = 0 and
ht, the index n being a positive whole number. 33. Find the volume of an oblate speroid and the area of its surface.
34. Find the length of the arc of the Lemniscata whose equation referred to polar co-ordinates is
pe = al cos 20
Natural Philosophy. 1. What branches of knowledge and investigation are comprehended in the term Natural Philosophy.
2. If two weights balance each other upon a uniform lever, they must be inversely proportional to their distances from the fulcrum. Prove this. If the lever be not of uniform thickness, how do you proceed to find the proportion of the weights at its extremities ?
3. Show that the principles of the pulley and the wheel-and-axle are involved in that of the lever; and those of the wedge and screw in that of the inclined plane.
4. Explain the mechanical action of wheels in assisting the motion of a carriage; and of springs in relieving the shock.
5. Prove that if two forces acting upon a point are represented in magnitude and direction by two sides of a parallelogram, their resultant force will be represented by the diagonal of the parallelogram which passes through the point upon which the forces act.
6. Find the centre of gravity of a given triangular board of uniform thickness and density. By what property is the centre of gravity determined ? Does the centre of gravity always coincide with the centre of figure ?
7. A ladder of given weight and dimensions rests with one end on the ground and the other against a wall, and the friction against both the wall and ground prevents its sliding down. Find the least angle with the horizon at which it can be placed without falling
8. If a uniform chain be suspended between two piers, of the same or different heights, show that the difference of the tensions at any two points of its length equals the weight of a piece of the chain of which the length equals the vertical distance by which one of the points is higher than the other.
9. What is meant by the Laws of Motion ? How many are there, and what are they?
What use do philosophers make of them ?
10. The first practical step which Newton made in his great discovery of the Universal Law of Gravitation was the determination, that the space through which the Moon, in moving round the Earth, is deflected from a straight course during every second of time, is equal to the space through which a body, carried up from the earth to the elevation of the Moon and left to fall back, would descend in the same time: the attraction of the Earth being supposed to vary inversely as the square of the distance. Show from the following data that these spaces are numerically equal. The distance of the Moon from the Earth's centre 239,000 miles : the length of a degree on the Earth's surface 694 miles : the mean length of the sidereal month 27.32 days: the space through which a body at the Earth's surface falls in the first second of time 16.1 feet.
11. Two rectangular folding flood-gates are closed against a stream which just overflows them, and are held together by a chain connecting them at a point in the vertical line where they meet. How low down should the chain be, that the hinges of the gates may suffer no twisting strain ?
12. Draw as accurate a diagram as you can (describing the principles upon which it is drawn) to show the course of a single pencil of light proceeding from an object through a magnifying glass to an eye looking at it: take for your illustration a pencil which does not pass through the centre of the magnifying glass. How must the distances of the eye, the object, and the glass be modified for a short-sighted person ?
13. As you attentively examine an object (say two inches long and wide) by a magnifying glass, and vary the distance, you will observe two defects in the objects as seen through the glass—Distortion and Confusion. Explain distinctly and separately the causes of these.
14. The Astronomical Telescope, in its most simple form, consists of two-glasses—one an object-glass, the other an eye-glass. Describe the action of this simple instrument, and give a diagram showing the course