13. If n be a prime number, and N be not divisible by n, ove that N 1 is divisible by n.

Prove also that Nn"-n-1 – 1 is divisible by n".

Shew also that any number in the scale whose radix is 2n, ds in the same digit as its nth

power.

14. Find the value of the series

n" − n . (n − 1)" +

n. (n - 1)

(n-2)" - &c.

1.2

all values of r from 0 to n + 1 inclusive, and prove that

15. A digit from 1 to 9 inclusive is raised to the nth power, being any integer greater than unity. Shew that the chances 13 to 5 that the digit in the tens' place is even.

VIII. TRIGONOMETRY.

1. Compare the measures in degrees and in grades of e angle whose circular measure is ; and find the circular asure of an angle the complement of which contains as any degrees as the supplement of an angle nine times as ge contains grades; find the circular measure of 39°.23′.7′′, ing for the value of the first of its convergents.

2. From a point A outside a circle two lines of equal gth AB, AČ, drawn to the ends of a diameter BC, cut e circumference in D, E; if BE, CD meet in O, the area the quadrilateral ABOC BC cot BAC. Compare that rt of the area of the circle which is outside the triangle BC with that which is included within it.

=

3. Trace the changes in sign and magnitude of sin A - cos A A changes from zero to four right angles, pointing out e values of 4 for which the expression is numerically greatest, d least, and also those for which it vanishes. Give also e greatest and least values which the expression assumes. 4. Define the sine and cosine of an angle. If a side BC jacent to the largest angle C of an obtuse-angled_triangle BC be produced to N, prove from your definitions that

sin ACN = sin ABC.cos BAC+ cos ABC. sin BAC.

If one of the acute angles be double of the other, and the e of the least angle be, shew that the tangent of e exterior angle

1 λ 3

5. Express sinA and cos4 in terms of sin A, and determine the proper signs of the radicals when A 150°.

=

If sin A be expressed as a function of sin 6A, determine by à priori reasoning how many values the resulting expression ought to have for a given value of sin64.

(2)

=

(seco seco+tan tanp)3-(tan@secp+sec@tanø)2 _ sec20sec2p 2 (1 + tan30 tan3Ø) – sec❜0 sec3Ø

sec❜0 sec2p

(3) {sec + cosec ◊ (1 + sec 0)} (1 – tan2 10) (1 – tan2 10)

7. Find the distance between the centers of the inscribed and circumscribed circles of a triangle having one angle = 17, and the opposite and one adjacent side known.

Find the side opposite to the given angle that this distance between the centers may be to the given adjacent side in the ratio of sin - sin : sin 17.

8. If the axis of a sail of a windmill be supposed to move in a plane inclined at a given angle to the horizon, find the inclination of this axis to the horizon when it is midway between its horizontal and its highest position.

If the axis of the sail be supposed to describe a given conical surface about an axis inclined at a given angle to the horizon, shew how to find the inclination of the sail's axis to the horizon when midway between its horizontal and highest position.

9. Assuming De Moivre's Theorem for positive values of the index, prove its truth when the index is fractional or negative. Shew that one side does not possess more values than the other.

Find the value of

without limit.

+ √/(-1) tan}" when n increases

{1+

What

10. Shew how to expand sin x in powers of x. would be the expansion if x were measured in seconds?

Thence find to three places of decimals sin 30".

Shew that the cube of the expansion for sin x is

11. Find the sum to n terms of the following series :

sin 0 + sin (0 + a) + sin (0 + 2a) +.....

1 + cosa.x + cos 2a. x2 +...

12. Prove Wallis' Theorem, and thence deduce an expression for 1.2.3.............n where n is very large.

13. What geometrical interpretation can be given to √(− 1)? Find all the logarithms of 1 + √(− 1), and represent their positions in a figure.

IX.-CONIC SECTIONS.

N.B.-The first seven questions must be proved geometrically.

1. Define a parabola, its focus, directrix, and axis; and prove that the tangent at any point makes equal angles with the axis and the focal distance.

The tangent at any point P of a parabola, cuts the directrix in Z, and any line is drawn through Z cutting the curve in Q and Q'; prove that, if S be the focus, SQ and SQ' are equally inclined to SP.

2. In the parabola, prove that the sub-normal is equal to half the latus-rectum. ̧.

3. If from an external point O, a pair of tangents OP, OQ be drawn to a parabola, and the focus S be joined to the points of contact P, Q; prove that the triangles SOP, SOQ will be similar.

Describe a parabola touching three given straight lines and having its focus in another given line.

4. If from the foci S, H of an ellipse, SY, HZ be drawn perpendicular to the tangent at any point P, prove that Y and Z lie on the circumference of the circle described on the axis major as diameter, and that the rectangle under the perpendiculars is equal to the square on the semi-axis minor.

If CD be the semi-diameter conjugate to CP, and if DQ be drawn parallel to SP, and CQ perpendicular to DQ, prove that CQ is equal to the semi-axis minor.

5. Define conjugate diameters in the ellipse, and prove that if CD be conjugate to CP, then CP shall be conjugate to CD. 6. Given an hyperbola traced upon the paper; shew how to find its foci and asymptotes by a geometrical construction. 7. If through any two points in an hyperbola, a chord be

drawn and produced to meet the asymptotes, prove that the portions of the chord intercepted between the curve and the asymptotes are equal.

8. Explain how a plane curve can be represented by an equation between two variables.

What is the nature and position of the curves represented by the following equations?

(1) xy (x-y)=0, (2) (A + By + C) + (A'x + By + CY=0, (3) (Ax+By+C)3+(A'x+By+C′′)3±0. (4) (r cos0-a)(a cost–r)=0.

9. Investigate the equation to a straight line in the form y = mx + c, and explain the geometrical signification of the constants m and c, both when the axes are rectangular and oblique.

A straight line QM moves parallel to a given line, one extremity M being terminated in a given straight line AB of constant length, and the other Q in another straight line given in position. If a point P be taken in Q in QM such that QP: PM :: AM: MB, shew that the locus of P is a parabola whose axis is parallel to QM.

10. Find the expression for the length of the perpendicular drawn from a given point upon a given line, and explain from your figure when the positive and when the negative sign is to be used. 11. If a = 0, B O be the equations to two straight lines in the form x cos a + y sin a - - p=0, &c. interpret the equations la + mB = 0, ta – B= 0, mà + B = 0, ma – 7B = 0, for a given position of the origin.

=

CAB is a triangle, and P and Q are points such that

prove that the locus of P is a straight line, and that the locus of Q is a conic touching the sides of the triangle at A and B and passing through the centre of the circle inscribed in the triangle ABC.

12. Find the equation to the tangent at any point of a parabola.

Find the locus of the points of intersections of tangents to a parabola which include a constant angle.

13. Tangents are drawn to a parabola from an external point (h, k); find the equation to the chord of contact.

From every point of the parabola y = 4bx, tangents are drawn to the parabola y2 = 4ax, prove that the locus of the

4ab

middle points of the chords of contact is y2:

=

14. Write down the relation connecting conjugate diameters n an ellipse, and find the equation to an ellipse referred to semi-conjugate diameters as axes.

15. Prove that the polar equation to a line passing through wo points P, Q, of an ellipse or hyperbola is

the focus being pole, the axis major prime radius, and (a – ß), a + B) the angular co-ordinates of the points P and Q.

P is a point on a conic, Q and Q' points on another conic having the same focus S, and such that the angles QSP and Q'SP are constant; prove that the locus of the intersection of the tangent at P and the chord QQ is another conic.

16. Find the condition that the general equation of the second degree may represent (1) a parabola, and (2) an ellipse. Prove that the co-ordinates of the focus of the parabola, whose equation is A2x2 – 2A Cxy + C2y2 + 2Dx + Ey + 1 = 0, Cs+ C2E+A2C+ 2ADE A3 + A2D + AC2 + 2CDE 2 (A3 + C2) (AE+ CD) 2(4+ C) (AE+ CD)

are

the axes being rectangular.

and

17. Find the equation to a conic inscribed in a triangle. Two conics are inscribed in a triangle CAB, each touching the base AB in the same point D, and intersecting in the points P, Q; prove that the chords of contact of the conics with the other sides, the common chord PQ, intersect in a fixed point in AB.

Find also the equations to CP and CQ and shew, that in the particular case when CD bisects the angle ACB, CP and CQ are equally inclined to the sides.