Problems

For a lamina in motion in its own plane define the instantaneous centre \(I\), and prove that the motion of the lamina can be reproduced by rolling the locus of \(I\) in the lamina (the body centrode) on the locus of \(I\) in space (the space centrode). Find the instantaneous centre for the motion of the rod \(AB\) in which the end \(A\) is moving along the given line \(CA\) and the end \(B\) at a constant distance from the given point \(C\).

A uniform rod \(AB\) of length \(2a\) can turn without friction about the end \(A\) in a vertical plane. A light elastic string of natural length \(a\) connects the end \(B\) to the point \(C\) vertically above \(A\) such that the length of \(AC\) is \(2a\). When the system is in equilibrium \(ABC\) is an equilateral triangle. Prove that the period of small oscillations about this equilibrium position is \(2\pi\sqrt{\frac{8a}{3g}}\).

Solve the set of equations: \begin{align*} x + y + \lambda z &= 2, \\ x - 3y + 7z &= 0, \\ \lambda x + 7y - 8z &= 5, \end{align*} for general values of \(\lambda\). Shew that there is a value of \(\lambda\) for which the equations have an infinite number of solutions, and give a formula for these solutions. Determine also the value of \(\lambda\) for which these equations have no finite solution.

The sum of two roots of the equation \[ x^4 - 8x^3 + 19x^2 + 4\lambda x + 2 = 0 \] is equal to the sum of the other two. Determine the value of \(\lambda\) and solve the equation.

Sketch the curve \[ y = x^2 / (x^2 + 3x + 2). \] By means of the line \(y+8=m(x+1)\), or otherwise, find the number of real roots of the equation \[ m(x+1)^2(x+2) = (3x+4)^2, \] when \(m\) is a real constant which is (i) positive, (ii) negative.

Prove that the geometric mean of \(n\) positive numbers does not exceed their arithmetic mean. Prove that, if \(p\) and \(q\) are positive integers, \[ \sin^{2p}\theta \cos^{2q}\theta \le \frac{p^p q^q}{(p+q)^{p+q}}. \]

Express \(\tan 5\theta\) in terms of \(\tan\theta\). (If a general formula is quoted, it must be proved.) Prove that the roots of the equation \[ t^5 - 5pt^4 - 10t^3 + 10pt^2 + 5t - p = 0, \] where \(p\) is real, are all real and distinct. Evaluate \(\tan \frac{\pi}{20}\).

State and prove De Moivre's theorem for \((\cos\theta + i\sin\theta)^n\), when \(n\) is (i) a positive integer, (ii) a negative integer, (iii) a fraction of the form \(p/q\), where \(p,q\) are integers. Express \((1+i)^n\) in the form \(A+iB\), where \(n\) is a positive integer and \(A, B\) are real.

Sketch the curve \[ xy = x^3 + y^3, \] and find (i) the radii of curvature at the origin of coordinates, (ii) the area of the loop.

Find the indefinite integrals \[ \int \left( \frac{1+x}{1-x} \right)^{\frac{1}{2}} dx, \quad \int \cos^3 3x dx, \quad \int \tan^{-1} x dx. \] Evaluate \[ \int_0^1 \frac{dx}{(1+x^2)^{\frac{3}{2}} - x}. \]