Problems

Evaluate the integrals

1. [(i)] \(\int (x+1)\sqrt{x^2+x+1}\,dx\),
2. [(ii)] \(\int \frac{dx}{x^4+1}\),
3. [(iii)] \(\int x^2\sin^{-1}x\,dx\),
4. [(iv)] \(\int \frac{dx}{\sin(x-a)\sin(x-b)}\).

Solve the equations

1. [(i)] \((x^2+1)^2=4(2x-1)\),
2. [(ii)] \(xyz = a(y^2+z^2)=b(z^2+x^2)=c(x^2+y^2)\).

Along a straight line are placed \(n\) points. The distance between the first two points is one inch; and the distance between the \(r\)th and \((r+1)\)th points exceeds one inch by \(\frac{1}{p}\)th of the distance between the \((r-1)\)th and \(r\)th points, for all values of \(r\) from 2 to \((n-1)\). Find the distance between the first and last points. Shew that, if \(n\) is large, the distance approximates to \(\frac{pn}{p-1} - \frac{p}{(p-1)^2}\) inches; and shew that, if \(n=p=10\), the distance is exactly 9.87654321 inches.

Given that \[ \frac{x}{y+z}=a, \quad \frac{y}{z+x}=b, \quad \frac{z}{x+y}=c, \] find the relation between \(a,b\) and \(c\); and prove that \[ \frac{x^2}{a-abc} = \frac{y^2}{b-abc} = \frac{z^2}{c-abc}. \]

Find an expression for the number of combinations of \(n\) things \(r\) at a time. A pack of cards has been dealt in the usual way to four players. One player has just one ace; prove that the chance that his partner has the other three aces is \(\frac{11}{203}\).

Prove that there are three values of \(c\) for which the equation \[ ax^3+3bx^2+3cx+d=0 \] has equal roots; and that if \(b^3=a^2d\), two of these values coincide and the third is \(-\frac{2}{3}\) of either.

If \[ (x-a)\cos\theta+y\sin\theta = (x-a)\cos\phi+y\sin\phi = a, \] and \[ \tan\frac{\theta}{2}\tan\frac{\phi}{2}=2e, \] and \(\theta, \phi\) are unequal angles less than 360\(^\circ\), prove that \[ y^2=2ax-(1-e^2)x^2. \]

Solve the equation \[ 2\sin x.\sin 3x=1. \] If \[ \tan\beta = \frac{n\sin\alpha.\cos\alpha}{1-n\sin^2\alpha}, \] prove that \[ \tan(\alpha-\beta) = (1-n)\tan\alpha. \]

An observer looking up the line of greatest slope of an inclined plane sees a vertical tower due East of him. He walks \(l\) feet up the plane in a direction \(\alpha\) North of East, and has then reached the level of the foot of the tower, and finds its elevation is \(\beta\). The plane makes an angle \(\gamma\) with the horizontal. Prove that the height of the tower is \[ \frac{l\tan\beta.\cos\gamma}{\sqrt{\cot^2\alpha+\cos^2\gamma}}. \]

If \(\cos(\alpha+i\beta)=\cos\phi+i\sin\phi\), and \(\alpha,\beta,\phi\) are real, prove that \[ \sin\phi=\pm\sin^2\alpha, \] and \[ e^\beta-e^{-\beta}=\pm 2\sin\alpha. \]