I. IV. PLANE TRIGONOMETRY. (Obligatory.) (Including the Solution of Triangles.) [N.B.-Great importance will be attached to accuracy in results.] Shew how to express in degrees, minutes, and seconds an angle whose circular measure is known. Find, correct to three places of decimals, the radius of a circle in which an arc 15 inches long subtends at the centre an angle containing 71° 36′ 36′′. (π=3*1416.) 2. Define the sine of an angle, and prove that sin A=sin (180o – A)=sin { − (180°+A)}. Write down formulæ including all angles which satisfy (1) 2 sin A = I, (2) 2 sin2A = I. 3. Prove that cos (A+B)=cos A cos B-sin A sin B, and deduce expressions for cos 2A, cos 3A in terms of cos A. 2 4. Given cos A = 28, determine the value of tan and explain fully the reason of the ambiguity which presents itself in your result. 6. State and prove the rules by means of which you can determine by inspection the integral part of the logarithm of any given number. Given log 4'96='6954817, log 4'9601='6954904, find the logarithms of 496010, 000496, and 49600*25. 7. Shew that in any plane triangle a=b cos C+ccos B. Hc=√2, A=117°, B=45°, find all the other parts of the triangle. 8. Find the greatest angle of the triangle whose sides are 50, 60, 70 respectively, having given log 6=7781513, L cos 39° 14'-9.8890644, diff. 1'=1032. 9. Express the area of a triangle in terms of one side and the two angles adjacent to it. Two angles of a triangular field are 223o and 45° respectively, and the length of the side opposite to the latter is a furlong. Shew that the field contains exactly two acres and a half. 10. Find an expression for the diameter of the circle which touches one side of a triangle and the other sides produced. If d1, d2, da be the diameters of the three escribed circles of a triangle, shew that d1d2+dd2+d ̧d1 =(a+b+c)3. II. A man standing at a certain station on a straight sea-wall observes that the straight lines drawn from that station to two boats lying at anchor are each inclined at 45° to the direction of the wall, and when he walks 400 yards along the wall to another station he finds that the former angles of inclination are changed to 15o and 75° respectively. Find the distance between the boats, and the perpendicular distance of each from the sea-wall. FURTHER EXAMINATION. V. PURE MATHEMATICS. (1.) [Great importance will be attached to accuracy in results.] I. Prove that planes which have a common normal are parallel. 2. Prove that in any trihedral angle the sum of any two of its angles is greater than the third. 3. Prove that if two straight lines in space be parallel each to a third they are parallel to one another. 4. Prove that a section of a right cone by a plane which cuts all the generating lines is an ellipse and that the two spheres inscribed in the cone so as to touch the plane touch it in the foci of the ellipse. 5. Prove that if two tangents to a parabola are at right angles the chord of contact passes through the focus. 6. Shew how to describe an ellipse which shall have a given point for one of its foci and shall touch each of three given straight lines. 7. Shew that ax2 + bxy+cy2=o in general represents two straight lines, and find the condition that they shall be inclined at an angle a to one another. Find the conditions that the equation ax2+by2+cx+cy=0 should represent two straight lines. 8. Find the condition that y=m1x+a1, y=mx+a2, y=m3x+a3 shall all meet in a point. 9. Define an ellipse, and from your definition deduce its equation in the form ax2+by2+c=0. IO. Define conjugate diameters of an ellipse. Given that 11 are the co-ordinates of the extremity of a semidiameter of the ellipse ax2+by2+c=o, find the co-ordinates of the extremity of the conjugate semidiameter. II. Prove that if an ellipse and hyperbola have the same foci they intersect one another at right angles. 12. Prove that the area is constant of the triangle contained between the axes of x and y and the tangent to the hyperbola xy=c2. I. VI. PURE MATHEMATICS. (2.) [Great importance will be attached to accuracy in results.] Find all the values of x and y which satisfy the equations 2. Eliminate x and y between x-a y+b x2-12 and a + b ̄a−b ̄b (x + b) − a ( y − a) 3 (ax+by) (bx+ay) = c2 (a - b) ( y − x). 3. Prove that the sum of the first and last of four quantities in Geometrical Progression will be greater than the sum of the two intermediate terms, unless the sum of the first two terms is negative. 4. Obtain an integral solution of the equation and in the particular case in which c=5, find the least positive integral values which x and y can have. 5. State and prove Fermat's Theorem respecting prime numbers. If n be not a multiple of either 3 or 5, prove that n1- I must be a multiple of both. 6. Obtain the coefficient of xn-1 in the expansion of series of powers of x. Shew also that the coefficient of x-1 in the similar expansion of = Also if sin 2 sin (0–4), find the limiting value of Ф approach to п. 8. Assuming the truth of Demoivre's Theorem, express the cosine of an angle in a series of powers of its circular measure. Shew that the series is convergent. 9. The circumference of a semicircle of radius a is divided into n equal Shew that the sum of the distances of the several points of section from either extremity of the diameter of the semicircle is equal to arcs. 10. a (cot-1). Prove that the number of real roots (if any) of the rational integral equation f(x)=o, which lie between a and b, will be even or odd, according as f(a), f(b) have the same or contrary signs. In what cases can you see by inspection that there must be some real roots lying between a and b? II. Form an equation whose roots shall be the products of every two of the roots of the equation x3-ax2+ bx+c=0. 12. Solve completely the equation 2x5+x2+x+2=12x3+12x2. |