Prove that if the two equations \begin{align*} ax^2+2bx+c &= 0 \\ a'x^2+2b'x+c' &= 0 \end{align*} have a single common root, then \[ 4(b^2-ac)(b'^2-a'c') - (ac'+ca'-2bb')^2=0. \] Show that the condition \[ 4(b^2-ac)(b'^2-a'c') - (ac'+ca'-2bb')^2 \ge 0, \] is necessary for the fraction \[ \frac{ax^2+2bx+c}{a'x^2+2b'x+c'} \] to assume all real values for real values of \(x\), and that if this condition is not fulfilled, the range of inadmissible values of the fraction will either be entirely between or entirely outside the roots of the equation \[ x^2(b'^2-a'c') + x(ac'+ca'-2bb') + b^2-ac = 0. \]
Show that the conditions that an algebraic equation \(f(x)=0\) has a double root at \(x=a\) are that \(f(a)=f'(a)=0\). If the equation \[ x^4 - (a+b)x^3 + (a-b)x - 1 = 0 \] has a double root, prove that \[ a^\frac{2}{3} - b^\frac{2}{3} = 2^\frac{2}{3}. \]
Prove Leibniz' theorem for the \(n\)th differential coefficient of the product of two functions. By using this theorem, or otherwise, prove that if \(n\) is a positive integer the polynomial \[ \frac{d^n}{dx^n}(x^2-1)^n \] is a solution of the differential equation \[ (1-x^2)\frac{d^2y}{dx^2} - 2x\frac{dy}{dx} + n(n+1)y=0. \]
(i) Evaluate the integral \(\displaystyle\int \sec^3\theta \,d\theta\). (ii) The mass per unit area \(\sigma\) at any point of a square lamina is proportional to its distance \(r\) from one corner, that is \(\sigma=kr\), where \(k\) is a constant. Find the total mass of the lamina in terms of \(k\) and \(a\) the length of side of the square.
Show that the function \[ \frac{x^3+x^2+1}{x^2-1} \] of the real variable \(x\) has only two critical values one of which is at \(x=0\) and the other at a certain value of \(x\) lying between 2 and 3. Establish which of these is a maximum and which a minimum value.
Prove that if \[ I_{p,q} = \int_0^{\pi/2} \sin^p\theta \cos^q\theta \,d\theta, \] where \(p>1, q>1\), then \[ (p+q)I_{p,q} = (p-1)I_{p-2,q}, \] and find the corresponding reduction formula involving \(I_{p,q-2}\). If the function \(\Gamma(n)\) is defined to have the following properties \[ \Gamma(n+1)=n\Gamma(n), \quad \Gamma(1)=1, \quad \text{and} \quad \Gamma(\tfrac{1}{2})=\sqrt{\pi}, \] verify that for all positive integral values of \(p\) and \(q\) greater than unity \[ I_{p,q} = \Gamma\left(\frac{p+1}{2}\right)\Gamma\left(\frac{q+1}{2}\right) / 2\Gamma\left(\frac{p+q+2}{2}\right). \]
A family of ellipses, all having eccentricity \(e\), have for their major axes parallel chords of a fixed circle. Show that their envelope is an ellipse of eccentricity \[ \sqrt{(1-e^2)/(2-e^2)}. \]
A flat strip of wood, of mass \(M\), lies on a smooth horizontal table; a particle, of mass \(m\), rests on the strip, the upper surface of which is rough, and the coefficient of (dynamical) friction between the strip and the particle is \(\mu\). If a velocity \(U\) is suddenly given to the particle so that it moves along the strip (which may be assumed not to rotate), find for how long a time the particle slips on the strip; show also that at the instant when slipping ceases the distance through which the strip has moved is \[ m M U^2 / 2\mu g (m+M)^2. \] Find how far the particle slips along the strip, and verify that the loss of kinetic energy is equal to the work done against the frictional force.
A particle is projected with velocity \(V\) under gravity from a point \(O\) of a plane inclined at an angle \(\alpha\) to the horizontal. The direction of \(V\) makes an angle \(\beta\) with the upward direction of a line of greatest slope of the plane and lies in a vertical plane through that line. Show that (i) the time taken for the particle to attain the maximum distance from the plane is one-half the time elapsing before it strikes the plane, (ii) the particle strikes the plane normally if \(2\tan\alpha = \cot\beta\). The particle is projected as before with an assigned velocity \(V\) so as to strike the plane at a point \(P\) above \(O\) and distant \(a\) from it; show that if \(P\) is within range there are two possible values of \(\beta\), given by \[ \sin(\alpha+2\beta) = \sin\alpha + (ag/V^2)\cos^2\alpha. \]
Water, of density \(\rho\) lb./ft.\(^3\), is pumped from a well and delivered at a height \(h\) ft. above the level in the well in a jet of cross-section \(A\) sq. in. with velocity \(v\) ft./sec. Find the horse-power at which the pump is working. If the water strikes a vertical wall normally and falls to the ground without recoil, find in lb. wt. the force exerted on the wall.