3. In any triangle (i.) a3 cos A+b3 cos B+c cos C=abc [1+4 cos A cos B cos C]. 5. Enunciate and prove De Moivre's Theorem. Find all the roots of the equation π 4 6. Expand cos ne in a series involving products of powers of sin and cos◊. Hence (or otherwise) expand cose in a series of ascending powers of 0. 7. Prove that 128 sin cos 0=14 sin 20-14 sin 40+6 sin 60—sin 80. 8. Sum the series (i.) sin @sin 20+ sin 20sin 30+ to n terms. (ii.) 1—3ccos0+6c2cos2 0 ... +(−1)»(n+1)(n+2) "cosne 2 + .. too, c being positive and less than unity. 9. Prove that the expression 3 sin 20 differs from by 5 nearly, when is a small angle. 4 45 STATICS AND DYNAMICS. HONOURS. TWO HOUrs. 1. Enunciate the theorem, called the parallelogram of forces. P, Q are two forces acting at a point Ŏ, and R is their resultant; a transversal cuts the lines of action of these forces in L, M, and N; shew that Hence shew that, if forces m. OP and n.OQ act at O, their resultant is (m + n)OG, where G lies in PQ such that m. GQ=n. GP. 2. Find the centre of gravity of a triangle. Three uniform rods of lengths a, b, c are placed in the form of a triangle, and each carries a weight w at a fixed point in its length. If these weights are moved along the sides of the triangle the same way round through distances ka, kb, kc, shew that the position of the centre of gravity is unaltered. 3. A uniform bar is placed in a sloping position leaning against a smooth peg, the lower end being on a rough horizontal plane, and its upper end in the air. Shew that it is just on the point of slipping when it makes an angle a with the plane given by the equation I sin 2a sin(a+6)=2h sin e where 27 is the length of the bar, h the height of the peg, and the angle of friction. 4. At what angular distance from the highest point can a particle rest on a rough sphere ? 5. A revolving wheel is throwing off mud; from what point on the wheel will the mud thrown off reach the highest point? 6. State and discuss the second law of motion. M and m are in equilibrium when suspended from a wheel and axle. If M and m are interchanged find the accelerations with which they move, the mass of the wheel and axle being ignored. 7. A particle slips down a smooth inclined plane of angle a in the same time as it slips down a rough inclined plane of equal length and angle B. Shew that the coefficient of sin ẞ-sin friction is and that the velocities at the cos B bottom are the same. a 8. A particle is moving with uniform velocity v in a circle of A wet ring of diameter one yard does not throw off drops of water until it is made to revolve three times a second. What is the force of adhesion between the ring and the drops? 9. A body of mass m at the top of a smooth vertical circular tube slips down, and strikes a mass 3m lying at rest at the lowest point. Find the height to which each body will rise after the impact. ANALYTICAL GEOMETRY. HONOURS. TWO HOURS. 1. Find the condition that two straight lines whose equations are given may be at right angles to one another. Given the base of a triangle in magnitude and position, and the altitude in magnitude, shew that the locus of the orthocentre is a parabola. 2. Prove that a homogeneous equation of the nth degree in a and y represents n straight lines through the origin. Find the equation to the straight lines joining the points of intersection of ax2+by2+2gx+2fy+c=0 and la+my=1 to the origin, and shew that they will be at right angles. if a+b+2(fm+gl)+c(l2+m2)=0. 3. Find the polar equation of a straight line. Prove that the equation of the normal to the circle r=2a cos at the point where 0=a is a sin 2a=r sin (2a—0). 4. Define the polar of a point with regard to a given circle x2+y2+2gx+2fy+c=0, and find its equation. Prove that each of the straight lines y-x-1=0,y+2x-7=0, 2y+x-50 is the polar of the intersection of the other two with respect to the circle 9 (y2+x2)—42y—30x+78=0. 5. Draw a diagram representing the straight lines of question 4. 6. Find the equation to the normal at any point of the parabola y=4ax. Find the locus of the intersection of normals at the extremities of a focal chord of a parabola. 7. Determine the locus of the middle points of a system of parallel chords of an ellipse. Find also the locus of the middle points of a system of chords passing through a given point. 8. Trace the following ellipses(i.) x2+4y2=1. (ii.) x2+4y2-8y=0. In each of the above curves find (i.) The eccentricity; (ii.) The coordinates of the ends of the major and minor axes; (iii.) The coordinates of the foci. ELEMENTARY INFINITESIMAL CALCULUS. HONOURS. TWO HOURS. 1. Find from the definition of the differential coefficient those of into its Partial Fractions and thus obtain its nth differential coefficient. 4. With the aid of the tables and the squared paper provided, plot the curve y=log10. dy d'y Write down the values of and and state what they tell dx de2 us about the shape of this curve. 5. The tangent at the point P (x, y) to the curve y=log, meets the ordinate at a neighbouring point Q (x+h, y+k) in T. Prove that, if x is great, the length of QT approxi 6. Show, without quoting a general theorem, that the curve is a continuous curve. Discuss its shape, and show that there is a minimum ordinate at = and a maximum one at x=0. 7. Obtain the values of the Indefinite Integrals of the functions 8. Starting from the definition of the Definite Integral as the limit of a sum, obtain the value of and interpret your result geometrically in the case of the parabola 2=2ay. 9. Obtain an expression as a Definite Integral for the volume of that portion of the Solid of Revolution, obtained by making the curve y=f(x) revolve about the axis of x, cut off by two planes perpendicular to this axis. Hence prove that the volume of a slice of the right cone cut off by planes perpendicular to the axis is where a, b are the radii of the ends, and h is the distance between the planes. |