Puslapio vaizdai
PDF
„ePub“
[ocr errors]

TRIGONOMETRY I.

1. (a) Prove that cos(A-B)=etc., and deduce from it the rest of the set.

(b) Prove geometrically that

tan Atan B

tan (A+B)=

I-tan A tan B'

2. (a) Obtain an expression for cot (4+B+C) in terms of cot A, cot B, cot C.

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

sin 0

3. Draw the graphs of sin 0+sin 30, and

between

[merged small][ocr errors]

limits of zero and 2π.

4. (a) Find sides and area of the pedal A to ABC in terms of the parts of ABC.

(b) Prove that the area of a quadrilateral inscribed in a circle is (s—a) (s—b) (s—c) (s—d).

5. The distance between centres of two circles is 18 inches d their radii are 6 in. and 3 in. Find length of a string i will reach around the two and cross between them.

[ocr errors]

Solve generally sin 30-sin 50 V2.cos 0.
Prove 8( tan-1%+tan-1)+4 tan-11/..

[ocr errors]

7. Develop cos 0 as a series and then prove ei✪

cos 0+i sin 0.

8. Observations to find height of a mountain are taken at two points A and B which are at same height and 4000 feet apart; the elevation of the top from A is 59° 37', and angles PAB, PBA are 74° 56', and 61° 19', respectively, P being at the top. Find height of mountain, using logarithms through

out.

INTERMEDIATE HONOURS.

SPHERICAL TRIGONOMETRY AND ASTRONOMY.

1. (a) Prove that cos a=cosb cosc+sinb sinc cosA, and in any way render it logarithmic for finding A.

(b) Two courses of 5000 miles and 3000 miles respectively start at an angle of 60°. Find the distance between their terminal points.

2. If r be the inradius of a spheric triangle, show that tan r sin a sin B sin C/4 cosA cos B cos C.

3. Prove any one of Napier's analogies.

4. Define meridional parts, and obtain a formula for their computation.

5. If m is the number of miles in 1° of longitude at the equator, find the number of miles in 1° of longitude at latitude, the earth being taken as an oblate spheroid with a and b as radii.

6. Explain how to find the time by equal altitudes of the sun; and calculate the correction for change of declination.

7. Show how to calculate the obliquity error in the equation of time.

8. On May 20th the siderial time at Greenwich mean noon is 3h 50m 448.6; the local longitude is 8h 44m W.; and the local mean time is 3h 18m p.m. Find the siderial time.

9. Give the expressions for the moon's parallactic displacement in right ascension and in declination, and show how these must be treated in predicting an occultation of the

moon,

I. and

INTERMEDIATE HONOURS.

ALGEBRA II.

Examine the convergence of the following series, prove the correctness of the tests you apply :Σ1/n, 21/n(n+1)!, Σ1/n(log n)”.

[blocks in formation]

3. If u is a positive integral function of n of degree , find the generating function of

[blocks in formation]

4. Find the fraction, greater than 580/1949, which is nearest to it in value and has a less denominator.

4

5. Find the nth convergent to --....

6. Show that 10.7"+5(3.5"-1 − 1)+16.4" is divisible by 24.

7. If m is prime, 1+(m−1) !=o (mod m).

8. Two persons make the same statement in testimony. Obtain a formula for the probability that the statement is (a) true, (b) false.

9. Three points are taken at random on the circumference of a circle. Find the probability that they are the vertices of an acute-angled triangle.

« AnkstesnisTęsti »