6. Find the sum of (a) 2:3·4+3·4·5+4·5·6+.... 7. What is interpolation? Interpolate two mean terms between the successive terms in the series 7-8-5+ 16+55+... ; and show that the new series satisfies the definition. 8. Examine the convergency of (a) the binomial series, (b) the series 1+3x+5x2+7x3+...... After what term does this latter series begin to converge when r=·75. PRELIMINARY HONOURS. CONICS I. 1. Write four important equations of the straight line, and the general equation of the circle, and explain the meaning of the parameters in each. 2. (a) Find the discriminant of the general quadratic equation. (b) Find the equation of the bisectors of the angles between the lines ax2+2hxy+by2=0. 3. The chord of the circle 7(x2+y2)-17x-23y+6=0 intercepted on the line 3x+2y=6 subtends a right angle at the origin. 4. Show that the locus of the point of intersection of perpendicular tangents to a circle is a concentric circle, and find its radius. 5. (a) Develop the condition that two circles cut orthogonally. (b) Find the equation of the circle which cuts orthogonally the three circles x2+y2-4, x2+y2-4x+2y+4=0, x2+y2+9x+3y=37. 6. A parabola bisects that part of any diameter intercepted between any point on it and the polar of the point. 7. From definition develop the equation of the general conic, and from it deduce the equation of the standard hyperbola. 9. Find the equation of the tangent and normal at the point 4, where & is the eccentric angle of the ellipse. 8. The focal lines to any point on an ellipse are equally inclined to the normal at that point. PRELIMINARY HONOURS. TRIGONOMETRY I. 1. Draw the graph of sin+cos 30 from 0=0 to 0=2π. 2. Apply the principle of projection to prove (a) sin(A+B) etc., (b) cos A+cos B = etc. from a figure how the four values of 2 sin A 2 A 2 4. By means of an auxiliary angle, change the cosine formula to a form adapted to logarithms; and use the result to find c when a = 20, b=30, C=42°18′. 5. The sides of a quadrilateral in order are 10, 17, 14, 16; and the angle between the diagonals opposite the side 10 is 63°33'. Find the area.. 6. Find the smallest positive value, and the general value, of 0 that will satisfy 4 sin0+3 cos 0=2. 7. If cos-x+cos1y+cos-12=0, prove x2 +y2 +22 2xyz=1. |