Find the condition that the equations \[ ax^2+2bx+c=0, \quad a'x^2+2b'x+c'=0 \] may have a common root. Prove that, if \(a, c, a', b', c'\) are given so that \(b'^2 > a'c'\), two real values \(b_1, b_2\) of \(b\) can be found to ensure a common root; and form the equation whose roots are the other roots of the equations \[ ax^2+2b_1x+c=0, \quad a'x^2+2b_2x+c'=0. \]
Prove that, if \(A-x^2=u\), (\(x>0, u>0\)), then \(\sqrt{A}\) lies between \(x\) and \(x+u/2x\). Hence prove that, if \(P = N^2+\frac{2}{3}n\) and \(Q = N^2+\frac{1}{3}n\), the difference between \((N^2+n)^{1/2}\) and \(N\sqrt{P/Q}\) is less than \(\frac{5}{36}n^3N^{-1}P^{-1}Q^{-1}\) when \(N\) and \(n\) are positive. Apply the method to express \(\sqrt{53}\) as a decimal, estimating the degree of accuracy obtained.
Prove that, if two infinite series of positive terms \(\sum u_n, \sum v_n\) are such that \(u_n/v_n\) tends to a finite limiting value, not zero, when \(n\) tends to infinity, the series are both convergent or both divergent. Deduce a rule for the convergence or divergence of \(\sum u_n\), when \(u_{n+1}/u_n\) tends to a limit \(k\). Examine for different positive or negative values of \(z\) and \(p\) the convergence or divergence of the series whose \(n\)th terms are \[ \text{(i) } z^n, \quad \text{(ii) } \frac{\sqrt{n+1}-\sqrt{n}}{n^p}. \]
The side \(BC\) of a triangle \(ABC\) is divided at \(D\) so that \(BD:DC=m:n\), where \(m+n=1\). Prove that, if \(R, R_1, R_2\) are the radii of the circles \(ADB, ADC, ABC\), then \[ bR_1 = cR_2 = R(mb^2+nc^2-mna^2)^{1/2}. \] Verify the results obtained in the limiting case when \(m\) tends to zero.
In order to obtain the height \(z\) of an aeroplane above the horizontal plane of a triangle \(ABC\) its angular altitude \(\alpha\) is observed at \(C\), and simultaneously at \(A, B\) are observed its bearings \(\theta, \phi\) measured in the horizontal plane from \(AB\) and \(BA\) respectively towards \(C\). Prove that \[ z^2\cot^2\alpha\sin(\theta+\phi) = b^2\sin\theta\cos\phi+a^2\sin\phi\cos\theta - 2ab\sin C\sin\theta\sin\phi - z^2\sin\theta\sin\phi\cot(\theta+\phi). \]
Find the real linear and quadratic factors of \(z^n-1\) when \(n\) is an odd positive integer. Deduce that \[ \sin\frac{\pi}{n}\sin\frac{2\pi}{n}\dots\sin\frac{(n-1)\pi}{n} = \frac{n}{2^{n-1}}. \]
Draw the graph of the function \(a\csc x + b\sec x\) for values of \(x\) between \(0\) and \(2\pi\), taking \(a,b\) to be positive. Determine the number of real roots between \(0\) and \(2\pi\) of the equation \(a\csc x + b\sec x = 1\), distinguishing the cases \[ a^{2/3}+b^{2/3} > , =, < 1. \]
Arrange the following numbers in order so that as \(x\) increases without limit the ratio of each number to the preceding may tend to infinity: \[ x^2, 2^x, x^x, e^x, x^{\log x}, (\log x)^x, 2^{\log x}. \] Find the limiting values of \[ (\cos x)^{1/x}, \quad (\cos x)^{1/x^2}, \quad (\cos x)^{1/x^3} \] as \(x\) tends to zero through positive or negative values.
Find a formula of reduction for the integral \[ \int\sin^m\theta\cos^n\theta\,d\theta \] when \(n\) is an odd positive integer. Find the indefinite integrals \[ \int \left(1-\frac{a}{x^2}\right)^{1/2}\,dx, \quad \int (x^2+x+1)^{-3/2}\,dx. \]
Draw a rough sketch of the curve \[ (x+2)^2y^2 - x(x+2)y + \frac{1}{4}(2x^2-1) = 0, \] and prove that the area enclosed by the curve is equal to \((2-\sqrt{3})\pi\).