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 \[ \int_0^1 \frac{dx}{(1+x^2)^{\frac{3}{2}} - x}. \]
Find the values of \(x\) for which the function \(e^{mx} \cos 3x\), where \(m\) may be positive or negative, has a maximum or a minimum value, and distinguish between them.
If \(f(x,y)\) is a function of \(x,y\) which takes the form \(g(u,v)\) when \(x,y\) are transformed by the relations \(x=\phi(u,v), y=\psi(u,v)\), prove that, in the usual notation, \[ \frac{\partial g}{\partial u} = \frac{\partial f}{\partial x}\frac{\partial \phi}{\partial u} + \frac{\partial f}{\partial y}\frac{\partial \psi}{\partial u}. \] If \(f(x,y)\) is a function of \(x,y\), and \(x,y\) are functions of \(t\) defined by the relations \(u(x,t)=0, v(y,t)=0\), and if \(f(x,y)\), when expressed as a function of \(t\), takes the form \(g(t)\), prove that \[ \frac{dg}{dt} = - \frac{ \left( \frac{\partial f}{\partial x}\frac{\partial u}{\partial t} \frac{\partial v}{\partial y} + \frac{\partial f}{\partial y} \frac{\partial v}{\partial t} \frac{\partial u}{\partial x} \right) }{ \frac{\partial u}{\partial x} \frac{\partial v}{\partial y} }. \]