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3. Prove that the foci of a plane section of a circular

cone are the points at which the plane of the section is touched by two spheres inscribed in the cone.

Shew that the locus of the vertices of the circular cones which stand on one conic section is a conic section of the other species in a perpendicular plane, whose vertices are the foci and whose foci are the vertices of the former; and deduce that the sum or difference of the distances of a variable point on one of these conics from

any two fixed points on the other is constant. 4. A curve is such that the ordinate PN varies as the mth power

of the abscissa ON; find the area contained between the curve, two ordinates, and the line of abscissæ.

PNP is a double ordinate of a parabola whose vertex is A, and PM is drawn parallel to the axis. Find the volume generated by the

revolution of PAP' about PM. 5. If s, denote the sum of the peh powers of the roots of the equation

20

2* + poch- + ... + Pn= 0, and r be a positive integer not greater than n,

prove that

Sy + P18-1 + P28-2 + + Pr-181 + rp, = 0.

Prove that the sum of the nth powers of the roots of the equation x2

px + y = 0 is equal to p* np*-*9+

n (n − 3)

p"-79

1.2 ——r

IL

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+(-1)=n(n=r-1)(n=-— 2)...(n–2r+1) pu-tq*+&c.

n

6. Find the general term in the expansion of Ax + B

in ascending powers of x. axé + bx + c

A plant produces p seeds at the beginning of the second year of its life, again at the beginning of its third year, and dies at the end of its third year. Prove that, if all the seeds come to maturity, the number produced from one seed in the year (n > 1) from planting that seed is

2 g 1{(2 + x)(p -q)"}

where q = pa + 4p.

nth

7. State and prove Fermat's theorem.

If N be prime to n and L (n) be the number of positive integers which are less than n and prime

to n, prove that Nl(n) – 1 is a multiple of n. 8. A pack of cards consists of p suits of q cards, each

P

? numbered from 1 up to q: A card is drawn and turned up, and r other cards are then drawn at random. Prove that the chance that the card first drawn is the highest of its snit among all the cards drawn is p

(-9

Pq91pq -r-12 1

P9 P9-9-1-15

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r

9. The sides of a quadrilateral are a, b, c, d, and the

sum of two opposite angles is 28. Shew that
the square of the area is equal to
(8 – a) (8 b) (8 c) (8 d)
a)

d) abcd cos’e, where 8 is balf the perimeter.

The area of a quadrilateral field ABCD is

8

determined by chaining the four sides and measuring one angle; prove that, on the hypothesis that the error in a side is proportional to the length of the side, and the error in an angle is the same for all angles, the angle A is preferable to the angle C, provided the area BAĎ is less than the area BCD.

10. Shew that if n be a positive integer cos no can be

expressed in the form

2n-1 cos”+ A cosn–20 + B cos-40 + &c., where the coefficients A, B, &c., do not contain 0. Hence shew that

cos na = 2n-1 (cos e cos a) x 27

27 A- at

cos no

n

{cos 8–cos(x+3)}...{cos –cos (a+n-1)}

-
(a + %) (a

and hence or otherwise shew that

27

cos”a + cos"

+

+ cos" ( a +n

-1

n

is independent of a provided r be less than n.

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prove that

1 1 A12

Ax +

+ + Q(x) aj? -

x2

a 2 - x2 where A, is equal to the value of a;? – x2

хф (х) when x = Ayo

Hence shew that 1 1 23c

2x +

&c. sin a 72 2272

3272

+

[ocr errors]

x2

x2

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ADVANCED MATHEMATICS.

Professor Nanson. 1. Expand

1

log (x + V1 + x2) ✓1 + x2 in ascending powers of x, and find the general term.

2. Define a homogeneous function, and prove that if

u be a homogeneous function of m dimensions in

two variables x, y, dau

du
+ +

x"y"
dx”
n-rou

dx®-dy"
= m (m - 1)(m — 2) (m m + 1) .

n

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+

dru + ya

dy"

a

3. If x, y, z denote the distances of a point P from

the vertices of a triangle ABC, prove that when f (x, y, z) is a maximum or minimum

1 df 1 df 1 df sin BPC dx sin CPA dy sin APB dz

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4. Prove that the elimination of the constants a, b, c, f, g, h from the equation

ax2 + 2xy + by2 + 2gx + fy +c=0 leads to the equation

d day d.x3 m2

= 0.

5. ABC is a triangle, and D, E, F are the middle

points of its sides. If P be any point, prove

L

that parallels to PD, PE, PF through A, B, C meet in a point.

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6. If the equations of two circles whose radii are

r, qo be S = 0, S' = 0, prove that the circle
whose equations are

S'
+ =k(r + r)

gia
S
on=k (r — m')

r
will intersect at right angles.

r

7. A, B are two fixed points on a conic, and P, Q are

two variable points on the conic. If the locus of the intersection of AP, BQ be a straight line, prove that the envelop of P Q is a conic whose centre is the pole of the straight line.

8. PP', QQ, RR are three chords of a conic meeting in 0. Prove that 1

1 1

+
P
ОР

OQ

1 sin ROP +

sin POQ = 0. OR 0

(op+ OP) sin QOR +

» (oe+oa)

+

OR)

9. Shew how to find the rectilineal asymptotes of a curve whose equation is of the form y

= 0.

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+

Find the equation of a curve of the fourth degree which has two coincident asymptotes x + y = 1, an asymptote x - y =1, a fourth

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