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University of New Zealand.

B.A. Pass Examination, 1885.

Mathematics.

Paper a. Geometry and Trigonometry.

Examiner: Prof. C. Niven, E.R.S.

1. Any two sides of a triangle are together greater than the third. If a triangle and quadrilateral stand on the same base and none of the angles of the latter are reentrant, prove that the perimeter of the triangle is greater than that of the quadrilateral.

2. If a straight line AB be divided into any two parts at the point C, prove that the squares on AB, BC are together equal to twice the rectangle under AB, BC, together with the square on AC.

3. The angles in the same segment of a circle are equal to one another. State and prove the converse of this proposition.

4. If a chord of a circle and a tangent be drawn from an external point, the square on the tangent is equal to the rectangle under the segments of the chord.

Describe the circles which pass through two given points and touch a given straight line.

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5. About a circle describe a triangle equiangular to a given triangle.

6. Find a third proportional to two given straight lines.

7. Prove, from the definitions of the sine and tangent of an angle, that

Find the sines of 30° and of 675°.

8. If

sin A =tan b tan c, sin B = tan a tan c, tan A tan B = sin c, prove that

sin c=cos a cos b.

9. Prove the formula;

sin 2A = 2 sin A cos A,

sin⁴ (A + 45°) − sin⁴ (A−45°) = sin 2A.

10. Define a logarithm, and prove that log a/b = log a−log b. Find x from the equation being given

11. Find formula; for solving a triangle when the three sides are given.

The sides of a triangle are 2/3, 5/6, 11/8; find the greatest angle, being given