Expand \(\cos x\) in ascending powers of \(x\), and prove that \[ \cos x \cosh x = 1 - \frac{2^2x^4}{4!} + \frac{2^4x^8}{8!} - \dots. \]
Find the maximum and minimum values of \(y\), where \(y^2=x^2(x-1)^3\).
If \(y\) is a function of \(x\) and \(x\) is a function of \(t\), express \(\frac{dy}{dx}\) and \(\frac{d^2y}{dx^2}\) in terms of differential coefficients with respect to \(t\). Change the independent variable from \(x\) to \(t (=\log x)\) in the equation \[ x^2 \frac{d^2y}{dx^2} + x\frac{dy}{dx} + y = 0. \]
Find the equation of the normal at any point of the curve \(x=f(t), y=F(t)\). Shew that the centre of curvature at the point \(t\) on the curve \(x=at^3, y=at^2\) is \[ x=\frac{1}{2}at(1+3t^2), \quad y=-\frac{1}{6}at^2(2+9t^2). \]
Shew how to find the envelope of the curves \(f(x,y,\alpha)=0\), where \(\alpha\) is an arbitrary parameter. Find the envelope of the circles which pass through the vertex of the parabola \(y^2=4ax\) and have their centres on the curve.
Find formulae of reduction for \[ \int (1+x^2)^n dx, \quad \int e^x \sin^n x dx. \]
Shew that the area bounded by the parabola \(ay=x^2\) and the lines \(y=x, y=2x\) is \(\frac{7}{6}a^2\).
State the laws of (i) limiting friction, and (ii) rolling friction. A uniform rod \(AB\) of weight \(W\) and length \(l\) lies on a horizontal plane whose coefficient of friction is \(\mu\). A string is attached to \(B\) and is pulled in a horizontal direction perpendicular to the rod. As the tension is gradually increased shew that the rod begins to turn about a point in it whose distance from \(A\) is approximately \(\frac{3l}{16}\), and the tension of the string is then about \(\frac{2\mu W}{5}\).
A uniform rod, of length \(c\), rests with one end on a smooth elliptic arc whose major axis is horizontal and with the other on a smooth vertical plane at a distance \(h\) from the centre of the ellipse; the ellipse and the rod both being in a vertical plane. Prove that, if \(\theta\) is the angle which the rod makes with the horizontal, and \(2a, 2b\) are the axes of the ellipse, \[ 2b\tan\theta = a\tan\phi, \] where \[ a\cos\phi+h=c\cos\theta; \] and explain the result when \[ a=2b=c, h=0. \]
A regular pentagon \(ABCDE\), formed of light rods, jointed at the angles, is stiffened by two light jointed bars \(AC, AD\). Two equal and opposite forces, each equal to 3 lbs. weight, are applied at \(B\) and \(E\): find graphically or otherwise the stress in each bar of the framework, stating whether it is tensile or compressive.