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(d) Wie still, wie dämmerig, wie rein war Alles,
was ihn umgab! Wunderbare Ruhe umfing ihn, süsze Mattigkeit beschwichtigte jede stürmische Regung seines Herzens. So oft er das Auge
. aufschlug, begegneten ihm zärtliche, sorgende Blicke. Selbst wenn der Schmerz sich erneute, genosz er stilles, tröstliches Seelenglück. Auch das fühlte sie und empfand es als einen Lohn sondergleichen.
PURE MATHEMATICS.-Part I.
The Board of E.caminers.
1. If two circles touch one another externally, the straight line which joins their centres shall
pass through the point of contact.
If the distance between the centres of two circles is equal to the sum of their radii, then the circles must meet in one point but in no other.
2. Inscribe a regular hexagon in a given circle.
AB, BC, CD, DE, . .... are consecutive sides of a regular polygon, and AD, BE intersect in X; shew that AB = BX.
3. If two triangles be equiangular to one another, the
sides about the equal angles shall be proportionals, those sides which are opposite to equal angles being homologous.
If one of the parallel sides of a trapezium is n times the other, shew that the diagonals intersect one another at one of the points in which each diagonal is divided into n + 1 equal parts.
4. If four straight lines be proportional, the rectangle
contained by the extremes is equal to the rectangle contained by the means, and conversely.
On a given straight line construct a rectangle equal to a given rectangle.
5. Shew how to solve two simultaneous equations
when all the terms which contain the two un-
x2 + y2 = a2 + b2, xy = ab.
6. State the meanings given to all when mn is
fractional or negative, and explain why such
a3 + b3 + o brot - chat – atbt
at + b + ch.
7. Define an arithmetical progression, and find the
sum of any number of terms of an arithmetical progression.
The sum of three quantities in arithmetical progression is 3a, and the sum of their squares is 3a2 + 25; find the quantities.
8. State and prove the formula for the number of
combinations of n things r at a time.
In how many ways can two elevens be chosen
out of 23 players to play a match ? 9. Define the secant, cosecant, and cotangent of an angle of any magnitude, and prove that
sec (- A) = sec A.
10. Prove that in any triangle
a = b cos C + c cos B,
+ c(a? + b2) cos C= 3abc. 11. Shew how to solve a triangle, having given two
sides and the included angle.
If a = 135, b = 105, C = 60°, find A, having given
= .3010300 L tan 13° 12' = 9.3348711, log 3 = .4771213 L tan 12° 13' 9.3354823. 12. Find an expression for the radius of an escribed
circle of a triangle.
log 2 =
PURE MATHEMATICS.-PART II.
The Board of Examiners.
1. Find the coordinates of the point which divides in
a given ratio the straight line joining two given points.
Find the ratio in which the line joining the points X1, Yı; X2, Yz is cut by the line
ax + by +c=0.
2. Find the general polar equation of a circle.
r = a cos 0, r = b sin 0,
3. Obtain the equation of the normal at any point of a parabola in the form
y = mx 2am amu. Shew that from a given point three normals can be drawn to a parabola.
4. Find the condition that the line
x cos a + y sin a= :P may touch the ellipse
1. a? 72 Find the locus of the intersection of perpendicular tangents to an ellipse.
5. State and prove the rule for differentiating x”. Differentiate
x2)* + x
6. State and prove a formula for the nih differential
coefficient of eax cos (bx + c).
eau cos pu cos qx cos rx. 7. Assuming that f(x + h) can be expanded in a series
of positive integral powers of h, shew that the
Expand sec x in ascending powers of x as far as the term in 26.
8. Investigate a rule for finding maxima and minima
values of a function of one independent variable.
9. State and prove the rule for integration by sub
ea + e
10. Shew how to find the partial fractions correspond
ing to a repeated factor of the first degree in the decomposition of a rational fraction.