Find the conditions that the roots of the cubic \(a_0x^3+3a_1x^2+3a_2x+a_3=0\) should satisfy the relation (i) \(\alpha\beta + \beta\gamma = 2\alpha\gamma\), (ii) \(\alpha^2\beta + \beta^2\gamma = 2\alpha^2\gamma\).
Define the differential coefficient of a function. Has \(x\sin\dfrac{1}{x}\) a differential coefficient at \(x=0\)? Find the \(n\)th differential coefficient of \(x\tan^{-1}x\).
State McLaurin's theorem on the expansion of a function of \(x\) in ascending powers of \(x\). Prove that, if \(a_0+a_1x+a_2x^2+\dots\) is the expansion in ascending powers of \(x\) of \(\{\cosh^{-1}(1+x)\}^2\), \((n+1)(2n+1)a_{n+1} + n^2 a_n=0\).
Find the equation of the tangent at any point of the curve given where \(x=f(t), y=\phi(t)\). Prove that, if \(\theta_1, \theta_2, \theta_3\) are the vectorial angles of three points on the curve \(r(\cos^3\theta+\sin^3\theta)=3a\cos\theta\sin\theta\) at which the tangents are concurrent, then \(\Sigma\cot\theta_1=0\).
Integrate: \(\sec x, \quad \dfrac{1}{(x^2-x-6)\sqrt{1+x+x^2}}, \quad \dfrac{\sqrt{a^2+b^2\cos^2\theta}}{\cos\theta}\). Find a formula of reduction for \(\displaystyle\int_0^{\frac{\pi}{2}} x^n \cos^m x dx\).
Interpret the expressions \(\displaystyle\int x \frac{dy}{ds} ds\) and \(\displaystyle\int y \frac{dx}{ds} ds\) when taken round the boundary of a closed plane curve. Trace the curve \[ y^4 - 2xy + x^3 = 0 \] and prove that the area of a loop is \(\frac{2}{5}\).
Forces \(P, Q, R\) acting at a point \(O\) are in equilibrium and a straight line meets their lines of action in \(A, B, C\) respectively; shew that, with certain conventions of sign, \[ \frac{P}{OA} + \frac{Q}{OB} + \frac{R}{OC} = 0. \]
Shew that any system of co-planar forces, not in equilibrium, may be reduced to a single force or a couple. \(D, E, F\) are the middle points of the sides \(BC, CA, AB\) respectively of a triangle \(ABC\). Forces of 1, 2, 3 lbs. act along \(BC, CA, AB\) and forces of 2, 3, 5 lbs. act along \(FE, ED, DF\). Find the magnitude of the resultant force, the inclination of its line of action to \(BC\) and its perpendicular distance from \(A\).
Two equal cylinders lie in contact on a horizontal plane and an isosceles triangular wedge is placed symmetrically upon them so as to touch each cylinder along a horizontal line. If the angle of the wedge is \(\alpha\) and the angle of friction is greater than \(\frac{1}{4}(\pi-\alpha)\), shew that the cylinders will not move, no matter however great the weight of the wedge may be.
[A diagram shows a simple truss A-C-B, with C above the line AB, and a vertical member from C to the midpoint of AB.] The figure represents a framework of rigid rods, supposed to be loosely jointed at their intersections, and to be of negligible weight. A given weight is suspended from \(C\) and the framework is kept in equilibrium by vertical forces at \(A\) and \(B\). Draw a stress diagram for the figure shewing the tensions and thrusts in the rods.