Two numbers \(p\), \(q\) are given. It is required to form a cubic equation such that, if the roots are \(\alpha\), \(\beta\), \(\gamma\) (not necessarily distinct) then \(p\alpha + q\), \(p\beta + q\), \(p\gamma + q\) are also the roots. Find the cubic equation (i) for general values of \(p\), \(q\), (ii) when \(p = +1\), (iii) when \(p = -1\).
Sketch the curve \[y = \frac{(x-2)(x-3)}{(x-1)(x-4)}.\] Prove that \[\frac{dy}{dx} = \frac{-2(2x-5)}{(x-1)^2(x-4)^2}, \quad \frac{d^2y}{dx^2} = \frac{12(x^2-5x+7)}{(x-1)^3(x-4)^3},\] and deduce that the radius of curvature at the point where the curve is parallel to the \(x\)-axis has the value \(3^4/4^3\). Find all the points both of whose coordinates \(x\), \(y\) are integers, positive, negative or zero.
Given that \(s^2 + c^2 = 1\), prove that \[(4c^3 - 3c)^2 + (4s^3 - 3s)^2 = 1.\] Given, conversely, that \[(4c^3 - 3c)^2 + (4s^3 - 3s)^2 = 1,\] prove that \(s\), \(c\) satisfy one or other of three relations of the form \[ps^2 + qsc + rc^2 = 1,\] where \(p\), \(q\), \(r\) are numbers (not necessarily rational), to be determined.
Two circles \(\alpha\), \(\beta\) are each of unit radius and their centres \(A\), \(B\) are three units apart. The inverse of \(\alpha\) with respect to \(\beta\) is a point distant \(\frac{8}{9}\) from \(A\); the inverse of this point with respect to \(\beta\) is a point distant \(\frac{8}{9}\) from \(B\); and so on. The distances from \(B\), \(A\), \(B\), \(A\), \ldots are \[\frac{8}{9}, \frac{8}{9}, \frac{64}{81}, \ldots,\] the \(n\)th such distance being \[\frac{u_n}{a_{n+1}}.\] Establish the relation \[u_n - 3u_{n-1} + u_{n-2} = 0,\] and derive a formula for \(u_n\).
Two triangles \(ABC\), \(PQR\) are inscribed in a conic. The lines \(BC\), \(QR\) meet in \(L\); \(CP\), \(AR\) meet in \(M\), \(AQ\), \(BP\) meet in \(N\), so that (Pascal's Theorem, which you are not asked to prove) \(L\), \(M\), \(N\) lie on a line, meeting the conic in two points \(I\), \(J\), assumed distinct. The figure is such that \(L\) lies on \(AP\), \(M\) lies on \(BQ\), \(N\) lies on \(CR\). Prove that \(I\), \(J\) are the self-corresponding points of each of the (1, 1) correspondences
Prove that the equation of the chord of the conic \[ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0\] with middle point \((x_1, y_1)\) is \[(ax_1 + hy_1 + g)(x - x_1) + (hx_1 + by_1 + f)(y - y_1) = 0,\] the axes not necessarily being rectangular. A conic \(S\) and two lines \(p\), \(q\) are given, with the property that every chord \(UV\) of the conic meets the lines in two points \(P\), \(Q\) such that \(UV\), \(PQ\) have the same middle point. Determine whether \(p\), \(q\) are necessarily the asymptotes of the conic.
Show that necessary and sufficient conditions for the equilibrium of a system of coplanar forces are that the resultant forces in any two directions are zero, and that the sum of the moments of the forces about any point in the plane is zero. A uniform ladder, weight \(W\), leans against a vertical wall and makes an angle \(\theta\) with the horizontal ground. The vertical plane through the ladder is perpendicular to the wall. The coefficient of friction between the ladder and both the ground and the wall is \(\mu\). A force \(P\), perpendicular to and away from the wall, is applied to the top of the ladder. Obtain an expression in terms of \(P\) for the minimum value of \(\mu\) which will prevent the ladder from slipping. Show that if \(\mu > \frac{1}{2}\cot\theta\) it will not be possible to make the ladder slip for any value of \(P\).
A thin uniform plank, length \(2l\) and weight \(W\), rests on a fixed circular radius \(a\), whose axis is perpendicular to the length of the plank. The plank is horizontal position by a vertical string, under tension \(T_0\), attached to one end of the plank and to a point \(P\) above the plank. \(P\) is then moved so that the plank makes an angle \(\alpha\) with the horizontal, the string remaining vertical. Assuming that the plank is sufficiently long to prevent slipping, find an expression for the tension in the string. If the coefficient of friction between the plank and the cylinder, \(\mu\), is very much less than one, and maximum and minimum possible values of the tension, before slipping occurs, differ by \[2\mu(T_0 - W)^2/W].\]
A rigid pendulum, mass \(m\), is attached to a point \(A\), which is in turn connected to a fixed point \(O\) by a light elastic spring for which the restoring force is \(\lambda\) times the displacement. The point \(A\) is constrained to remain on a horizontal line containing \(O\). The distance to the centre of mass \(G\) of the pendulum is \(l\) and the moment of inertia about any axis perpendicular to \(AG\) and passing through \(A\), is \(m(l^2 + k^2)\). The displacement of the pendulum from the vertical, in the plane containing \(OA\), is \(\theta\) (positive when the spring is extended). Obtain equations describing the motion of the system and show that \[\ddot{x}\cos\theta + (l^2 + k^2)\ddot{\theta} + gl\sin\theta = 0.\] Show that if \(\lambda(l^2 + k^2) = glm\) then \((l^2 + k^2)\ddot{\theta} - x\) executes simple harmonic motion for small \(\theta\). What is the period of this motion?
A man, whose height can be ignored, stands on a hillside which may be taken as a flat surface making an acute angle \(\alpha\) with the horizontal. He can throw a ball, mass \(M\), with initial velocity \(V\). Derive the complete specification of the direction in which the man should throw the ball in order that its range should be a maximum. Air resistance can be ignored. For a general throw express the range in terms of the angle \(\beta\) between the direction of throw and the horizontal, and the angle \(\gamma\) between the vertical plane containing the direction of throw and that containing the line of greatest slope. Show that, for \(\gamma = \alpha\), the maximum ranges, up and down the slope, are \[\frac{V^2}{g}[(1+\sin\alpha)^{\frac{1}{2}} \pm \sin\alpha](1+\sin\alpha)^{\frac{1}{2}}\] respectively. What is the angle between the directions of throw corresponding to these two ranges?