subtending equal angles at his eye. How far off is he in a horizontal direction from the object? 12. A regular hexagon, each side of which is 10 ft., revolves about a line which joins the points of bisection of two opposite sides; find the whole surface of the solid thus generated. 13. Determine the geometrical signification of the equations (1) 10x2-xy - 21y2 — 9x−y+2=0. Write down the first and second differential coefficients of (1); (2) xetan; (3) tan-x. 15. Prove the formula of integration by parts. 16. Find the area of a portion of a common parabola. 17. Prove that the greatest rectangle which can be inscribed in an ellipse is half of that contained by the axes. 18. If r be the radius vector at any point of a curve, and p the perpendicular from the pole upon the tangent, prove rdr o the radius of curvature equal to dp In the ellipse, whose semi-axes are a, b given p2= b2r find the radius of curvature drawn at the extremity 2a-r of the major axis. CL. 1. Explain the rule for multiplying a number consisting of several digits by a like number, by means of the example 356 x 125. 2. What is meant by a proper fraction rule for dividing it by a whole number c. a ? Explain the 5. Find 3 fractions converging to √26. 6. Show that impossible roots enter equations in pairs. What assumption is here made respecting the coefficients of the equations? In what case will irrational roots enter in pairs? 7. If a be an arc of a circle subtending an angle 0, and r the radius of the circle, show that Ꮎ ; 2 right-angles α π X- where 3.14159. r What will this become if we take an angle one-sixth of a right angle as the angular unit? 8. If tan 20√3, express all the positive values of which satisfy the equation. 9. If A, B, C be the angles of a spherical triangle, and A+B+C=2s, show that tan α 2 a being the side opposite the angle A. cos S cos (S-A), cos (S-B) (S-C) Show that this expression always expresses a possible quantity. 10. Differentiate y=! (x+1)2) ·; y=esin x; y=log,sin ̄1x. x2+1 11. Determine the maxima and minima of f (x), when f(x)=(x-2) (x-4)2. 12. Determine the ratio between the height and radius of the base of a cylindrical quart cup so that its surface may be a minimum. 13. Show that if y'=f (x') be the equation to a curve, the equation to the tangent is y-y' =ƒ' (x') (x—x'). 14. Expand as far as 24— (1) log (x+√x2 + a2); (2) (ex +e-x)n. 15. Prove the following differential formula which occur in the theory of plane curves:— (1) Subtangent=y ; (2) Subnormal=y dx dy. dy dx 16. Integrate with respect to x, the functions: 18. Find the relation between a and b, that the envelope of MIXED MATHEMATICS. FORMULE IN STATICS. R=resultant of P and Q acting at an angle 0, Р R2=P2+Q2+2 PQ cos 0; if P=Q, R=2P cos 1 0. Q. If P, Q, R are in equilibrium at a point o, PQR=sin QOR : sin POR : sin POQ. R=resultant of any number of forces acting upon a body in one plane. R2= {Σ (x) } 2 + {2(x)}2 : tan 0=2(r) ; in case of equilibrium Σ (x)=0: (Y)=0. 2(Yx-xy)=0. R=resultant of any number of forces acting upon a body in any directions, in case of equilibrium Σ (x)=0, Σ (y)=0 : Σ (z)=0; Σ (zy-yz)=0; Σ(xz-zx)=0; 2(xx—xy)=0. If x, y, be the coordinates of the centre of gravity of a system of bodies, Pressure on fulcrum=[p2+w2 - 2 pw cos (a+ß)]§. Direction of pressure=0; tan 0= In the wheel and axle Single moveable pulley, the strings. P sin a+w sin B P cos a-w cos System of pulleys each hanging by separate string 1 P= {w+(2-1)w}, n being no. of moveable pulleys. 2n System of pulleys, same string passing round all the pulleys, W+B=n P. B=weight of block. System of pulleys, when all the strings are attached to the weight, w=(2-1) P+(2′′-n-1) w; n being no. of strings attached to weight. P_vertical distance between two threads W circumference of circle described by P Screw, with friction FORMULE IN DYNAMICS. Motion uniformly accelerated v=ft ; s= }} ft2; v2=2fs; u=v±ft; s=vt±}ft2; u2=v2±2 fs. |