Page images
PDF
EPUB

10. At one corner of a parallelopiped the plane angles are all acute. One of these angles is 2ẞ, and each of the others is equal If the edges are of lengths a, b, c, prove that the volume of the parallelopiped is 2abc sin ẞsin (a + B) sin (a-ẞ) }.

to a.

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

2. Find such a value of x as will satisfy the equation

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors]

3. If the roots of the equation a cos 2 considered as an equation to find sin 0, be sin

[blocks in formation]
[merged small][ocr errors][merged small]

+ b sin 20
1, and sin 02, then

[ocr errors]

4. Show that if ax + by + cz = 0, the value of

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small]

6. Show how to find whether 1/3

2/2) is positive or

negative. Rationalise the denominator of the fraction. 7. Eliminate x between the equations

x + 1/x = y; x2 + 1/x2 = 2.

Also between the equations

ax2 + bx + c = 0; cx3 + bx2 + ax + d = 0.

8. The area of the whole surface of a right cone, including that of the circular base, is equal to that of a circle of 6-inch radius. The height and radius of base of a second cone are equal to the radius of base and height of the first cone respectively: but the whole surface of the second has only two-thirds of the area of the whole surface of the first. Find the dimensions of the cones.

9. Imagine a square to be inscribed within a circle of given radius; within that square a circle: within that circle a square, and so on indefinitely. Find the sum of the areas of all the

circles.

10. There are eight guests at a dinner-party, four ladies and four gentlemen. The host and hostess take the two ends of the table, and the guests are so arranged that ladies and gentlemen alternate round the table. In how many ways can this be done ! In how many of these ways will a certain pair of the guests (lady and gentleman) sit next to each other?

APPLIED MATHEMATICS.

Time: three hours.

1. Assuming the triangle of force, state and prove the polygon of forces.

Three forces, each perpendicular to the plane of the other two, act on a particle; their values are 8, 11, and 13 respectively. Find the value of the resultant and the cosines of its angles of inclination to the direction of each of the three forces.

2. The arms of a balance, the mass of whose beam is negligible, are unequal in length. A body placed successively in each scale is balanced by masses of 13 and 15 lbs. Find the actual mass of the body and the ratio of the lengths of the arms of the balance.

3. Prove that if any number of forces in one plane, acting on a rigid body, have a resultant, the algebraic sum of their moments about any point in their plane is equal to the moment of their resultant.

Prove that the forces represented by the sides of a triangle taken in order are equivalent to a couple, and find its moment.

4. Given the centre of gravity of the whole body and the centre of gravity of a portion, find the centre of gravity of the remainder.

A line parallel to the diagonal of a square is drawn cutting off one quarter of the square. Find the distance of the centre of gravity of the remainder from the centre of gravity of the square.

5. Define the terms coefficient of friction and angle of friction.

A body of mass 7 lbs. rests on a table, and connected to it by a string passing over a pulley at the edge of a table is a mass of 6 lbs. If the coefficient of friction between the weight and the table be 53, find how far it is pulled along the table in 2 secs., and the velocity acquired.

6. Show that the change of kinetic energy per unit space is equal to the acting force.

A train weighing 150 tons is set in motion at the rate of 53 miles an hour by a constant force acting for five minutes. What is the amount of this force?

7. Define the terms density and specific gravity.

Two bodies,

when suspended in water from the arms of a balance, are in equilibrium. The mass of the one body is 27 gramines, and its density 5.8. If the mass of the other is 35 grammes, find its density.

8. State "The Laws of Gases."

52 c. cms. of a gas are collected at a temperature 21° C, the barometer reading 74 cms. The volume, on cooling down to 0° C, becomes 47.6 c. cms., the barometer being at 75 cms. Find the coefficient of expansion of the gas.

9. Describe a Smeaton's air pump.

If at the end of the fourth stroke the density of the air in the receiver is to its original density as 256 is to 625, find the ratio of the volume of the receiver to that of the barrel.

PHYSICS.

Time: three hours.

1. State Newton's Laws of Motion. What is the distinction between mass and weight? Give their respective units; also those of velocity, acceleration, and work in the British and C.G.S. systems.

2. State the laws of fluid pressure. The arms of a U tube are 12 inches high, and are filled to a height of 9 inches with water. As much liquid as possible of Sp. gr. 0-8 is now poured into one What length of tube does the liquid occupy?

arm.

3. Describe a mercury barometer suitable for accurate measurements. What corrections must be applied to its readings?

4. Distinguish between a gas and a vapour. What is meant by maximum vapour pressure? Does it depend on temperature! If so, how?

5. Describe an experiment for the determination of the mechanical equivalent of heat. A mass of 100 kilograms falls through a distance of 300 metres. If all the energy developed during descent be converted into heat, and supplied to 20 gr. of ice at 30°F., what will be the result?

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small]

6. An organ pipe when sounded emits notes other than the fundamental. To what subdivision of the column into nodes and segments do each of these correspond, and what is the relative pitch of the corresponding note in

a. closed pipes ;

b. open pipes.

7. Explain what is meant by the angle of minimum deviation. Knowing its value for a prism, also the angle of the prism, deduce the value of the refractive index of the prism.

8. How is the intensity of a magnetic field defined? Hence show how the magnitude and direction of a field may be stated in terms of lines of force.

9. State the laws of electro-magnetic induction, describing experiments to illustrate the effects. Also explain the occurrence of eddy currents.

INORGANIC CHEMISTRY.

Time: three hours.

[N.B.-Candidates are expected to write formulae and equations wherever possible.]

1. You are given some nitrogen in a tube, graduated in c.c., and standing over water. State how you would proceed to determine the weight of the nitrogen.

2. How would you determine the composition of ammonia by

volume ?

3. Write what you know of the physical and chemical properties of carbon dioxide.

4. Write what you know about the luminosity of flame and the non-luminosity of a Bunsen burner.

5. Write an account of the preparation and properties of phosphine, and compare it with ammonia.

6. How would you prepare

a. Potassium ferrocyanide, and from it potassium cyanide. b. Mercurous chloride.

c. Ultramarine.

d. Fine porcelain.

e. Stannous chloride.

Explain the reaction between the latter and a solution of mercuric chloride.

7. How much potassium bichromate, mixed with hydrochloric acid, will be sufficient to convert one gramme of ferrous chloride into ferric chloride?

[blocks in formation]

incorrectly written? Explain fully what is involved in the equation when properly written.

« PreviousContinue »