Problems

Prove that, if \(a>0\) and \(ac-b^2>0\), the expression \(ax^2+2bx+c\) is positive for all real values of \(x\). Prove that the expression \[ 10x^2 - 6xy + y^2 - 8x + y + 7 \] is positive for all real values of \(x\) and \(y\).

If \(P(x)\) is a polynomial, state what can be asserted about the number of (real) roots of \(P'(x)=0\) lying between two successive roots of \(P(x)=0\). Discuss also the number of roots of \(P(x)=0\) lying between two successive roots of \(P'(x)=0\). If \[ P_n(x) = 1 + x + \frac{x^2}{2!} + \dots + \frac{x^n}{n!}, \] prove, by considering \(P_1(x), \dots P_n(x)\) successively, that the equation \(P_n(x)=0\) has no root when \(n\) is even and exactly one when \(n\) is odd.

The diagram shows a horizontal plank, of weight \(W\), which is supported at \(B\) on a rough plane inclined at an angle \(\alpha\) to the horizontal, and rests at \(A\) on a uniform rough circular cylinder, of weight \(W'\), which in turn rests at \(C\) on the inclined plane, so that the generators are horizontal. The centre of gravity \(G\) of the plank bisects \(AB\). Show that a necessary condition for equilibrium is that \(W \sin\alpha + 2(W+W') \sin\alpha \cos\alpha\) should be less than \(\mu\{W(1+\cos\alpha) - 2(W+W')\sin^2\alpha\}\), where \(\mu\) is the coefficient of friction at \(B\). [A diagram shows a horizontal plank AB with center of gravity G. End A rests on top of a circular cylinder of weight W'. The cylinder rests on an inclined plane at point C. End B of the plank rests directly on the same inclined plane. The plane is inclined at an angle \(\alpha\) to the horizontal.]

Prove Pascal's theorem that, if a hexagon is inscribed in a conic, the meets of pairs of opposite sides are collinear. Two coplanar triangles \(ABC\) and \(XYZ\) are such that \(ABC\) is in perspective with \(YZX\) and also with \(ZXY\) (the vertices being associated in the order written). By considering the two axes of perspective as a conic, or otherwise, prove that \(ABC\) is also in perspective with \(XYZ\).

Show that the equation \[ \sin x = \tanh x \] has infinitely many real roots and that, if \(n\) is a large positive integer, approximate values of a pair of roots are \[ x = (2n + \tfrac{1}{2}) \pi \pm 2e^{-(2n+\frac{1}{2})\pi}. \]

Show that for two values of \(\lambda\) the equations \begin{align*} (2+\lambda)x + 4y + 3z &= 6, \\ 2x + (9+\lambda)y + 6z &= 12, \\ 3x + 12y + (10+\lambda)z &= A \end{align*} have no solution unless \(A\) has a definite value (not necessarily the same for the two values of \(\lambda\)). For each of these values of \(\lambda\) find the value of \(A\) for which the equations have a solution, and obtain the general solution in each case.

A light inextensible string is in contact with a rough cylinder of any convex section, and is in a plane perpendicular to the generators. If the string is about to slip, show that the difference of the logarithms of the tensions at any two points is proportional to the angle between the normals at these points. A rough horizontal shaft of radius \(a\) is rotating about its axis, and carries a loop of light inextensible string, to one point of which a mass \(M\) is fastened. If the coefficient of friction is \(\mu\) and the mass remains at rest, so that the two straight portions of the string enclose an angle \(2\alpha\), show that the plane through the mass and the axis of the shaft is inclined at an angle \(\theta\) to the vertical, where \(\tan\theta = \tan\alpha \tanh\mu(\frac{1}{2}\pi+\alpha)\). Show also that at the highest point of the string the normal reaction, per unit length, exerted on the string by the shaft is \[ M(g/a) \sin(\alpha-\theta) \operatorname{cosec} 2\alpha \, e^{\mu(\frac{1}{2}\pi+\alpha+\theta)}. \]

A square \(PQRS\) lies in a given plane, and the sides \(PQ, QR, RS\) (produced if necessary) pass, respectively, through three fixed points \(A, B, C\). Prove that the fourth side \(SP\) passes through one of two fixed points. Hence, or otherwise, show how to construct a square whose sides \(PQ, QR, RS, SP\) shall pass, respectively, through four given coplanar points \(A, B, C, D\). How many solutions has this problem in general?

Prove, as simply as you can, that of the three following equations there are two which cannot be satisfied by any real value of \(\theta\). Find for what values of \(\theta\) the remaining equation is satisfied: \begin{align*} \cos \theta + 98 \cos 2\theta &= 100, \\ \cos^2\theta + \cos^2 2\theta - \cos\theta \cos 3\theta &= 1, \\ (1-2\cos\theta)^2 + (1-2\cos 2\theta)^2 &= 0. \end{align*}

Define limit and convergent series. Taking \(a\) to be positive, discuss the limits of \[ \text{(i) } a^n, \quad \text{(ii) } \frac{a^n-1}{a^n+1} \] as the positive integer \(n\) tends to infinity. Discuss as completely as you can the convergence of the series \[ 1 + z + z^2 + z^3 + \dots \quad (\text{\(z\) real or complex}). \]