7. A Bill before Parliament was lost on a division, there being 600 votes recorded. Afterwards, there being the same voters, it was carried by twice as many votes as it was before lost by, and the new majority was to the former as 5: 4. How many members changed their minds?

8. Find the sum of n terms of the Arithmetical progression 17, 117, 21......

If G is the Geometric mean between two quantities A and B, shew that the ratio of the Arithmetic and Harmonic means of A and G is equal to the ratio of the Arithmetic and Harmonic means of G and B.

and insert two geometric means between 1 and - I.

IO. If y is the sum of two numbers, of which the first varies directly, and the second inversely as x, and if y=7, when x=2, and y= −1, when x=1, shew that

II. Given that the number of combinations of n things taken 4 together is to the number of combinations of n-1 things taken 5 together as 5 : 3, find n.

12. Assuming the form of the continued product of any number of binominal factors, deduce the truth of the binomial theorem for a positive integral value of the index.

Shew that three consecutive terms of the expansion of (1+x)" can be in continued proportion only when n+1=0.

13. Expand (1 − 4x) to five terms; and shew that the general term may be thrown into the form

[N.B.-Great importance will be attached to accuracy in numerical results.]

I. In plane trigonometry state how a right angle is usually divided by English and French mathematicians respectively. If the circumferences of the quadrants of two circles be divided similarly to the right angles they subtend, what would be the radius of a circle divided according to the French scale, in which the length of the arc of one grade would be equal to the length of the arc of one degree on a circle whose radius was 18 feet? Express in the French scale (1) the sum of the angles of a quindecagon, (2) one of the angles when the quindecagon is equiangular.

2. According to the ratio definition, point out which of the elementary trigonometrical functions are never less than unity, and which may be either less or greater than unity.

Prove (sin A)2 + (cos A)2 = 1, and express the numerical values of sin 135° and tan 150° with their proper signs.

3. If (A+B) is an angle in the first quadrant, obtain by means of a geometrical figure the formula cos (A + B) = cos A cos B-sin A sin B, and deduce from it the expression for cos (A – B).

Find sec (A + B) in terms of sec A and sec B, and prove

5. Prove that in every circle the angle subtended by an arc equal to the radius is an invariable angle. Express that angle in degrees and decimals of degrees, and find the circular measure of one minute. On a circle 10 feet in radius it was found that an angle of 22° 30' was subtended by an arc 3 feet 11 inches in length; hence calculate, to four decimal places, the numerical ratio of the circumference of a circle to its diameter. 6. If a, b, c be the sides subtending respectively the angles A, B, C of a plane triangle,

7. If (r) be the radius of the circle inscribed in the triangle ABC, prove

8. In a right-angled triangle ABC, C being the right angle, find AB, if B=30° and BC= 100 feet.

A'B'C' is also a right-angled triangle, C' the right angle and B'= 30°, find A'C' when the area of the triangle A'B'C' is three times the area of ABC.

9. When two sides and an included angle of a triangle are given, investigate a formula for determining the other two angles. Shew that for this determination it is not necessary to know the absolute lengths of the two sides, provided their ratio is given.

Ex. The included angle is 70° 30′, the ratio of the containing sides is 5 3, find the other angles.

IO. A tower stood at the foot of an inclined plane whose inclination to the horizon was 9°; a line was measured straight up the incline from the foot of the tower of 100 feet in length, and at the upper extremity of this line the tower subtended an angle of 54°; find the height of the tower.

Note. For sin 54° see question 4.

II. Shew that the logarithms of proper fractions are negative. Express the true value of the logarithm of to the base 10. How would it be expressed with a negative characteristic?—Since sin 30°=1, explain why the logarithm of sin 30 is tabulated 9'6989700. Given that log tan 38° 16′ is tabulated 9.8969714, determine log cotan 38° 16′.

Given

Log tan 38° 16' 10"=9.8970147,

Log tan 38° 16′ 20′′=9.8970580,

find the angle whose logarithmic tangent is 9.8970365. Note.-Log102 is given in question 9.

I.

FURTHER EXAMINATION.

V. PURE MATHEMATICS. (1.)

How do you measure the inclination of a straight line to a plane, and of a plane to a plane?

line.

2.

3.

If two planes cut one another, their common section is a straight

Name and define the five regular solid figures; and shew that there cannot be more than five.

4. If two straight lines be cut by three parallel planes, they shall be cut in the same ratio.

5. Find the roots of the equation

x3+3x+36=0,

and shew that every equation of the form x3+qx+r=o has two impossible roots and one negative root.

and prove that they are at right angles to each other.

7.

Prove that the subnormal in the parabola is constant; and shew how to draw a normal to the curve at any given point.

8. A quadrilateral is inscribed in a circle, one of its diagonals coinciding with the diameter of the circle; find in terms of its sides (1) the radius of the circle, (2) the other diagonal, (3) the area of the quadrilateral.

9. Draw the circle represented by the equation

x2=2ay-y2,

and transform the equation into polar co-ordinates.

10. Prove (1) geometrically, (2) analytically, that the perpendiculars dropped from the foci upon the tangent to any point of an ellipse intersect the tangent on the circumference of the circle described on the axis major as diameter.

II. Find the equation to the ellipse referred to a pair of conjugate diameters as axes, and shew that equal conjugate diameters are parallel to the lines joining the extremities of the major and minor axes.

12. Find the equation to the tangent to an hyperbola, and the locus of its intersection with the perpendicular upon it from the centre.

13. Prove the following trigonometrical formulæ (ʼn being a positive integer):

14. Explain the method of computing the value of π=3'1415926..., and calculate e=2'7182818...

VI. PURE MATHEMATICS. (2.)

[N.B.-Great importance will be attached to accuracy in numerical results.]

I. Find the algebraical expression which, when divided by x2+x-1, gives x3-3x2+4x-7 for the quotient and 11x- 7 for the remainder.

2. Prove that

x(y+ 2)2+y(z+x)2 + (x+y)2 - 4xyz=(y + z) (≈ +x) (x+y),