Find the sum of the cubes of \(1, 3, 5, \dots, 2n-1\). Prove that the sum of the products of these numbers taken three together is \[ \frac{1}{6}n^2(n-1)(n-2)(n^2-n-1). \]
Find four consecutive numbers which are divisible by 5, 7, 9, 11 respectively.
Solution: \begin{align*} &&n &\equiv 0 \pmod {5} \\ &&n &\equiv 6 \pmod {7} \\ &&n &\equiv 7 \pmod {9} \\ &&n &\equiv 8 \pmod {11} \\ \Rightarrow && 5k &\equiv 6 \pmod{7} \\ \Rightarrow && k & \equiv 18 \equiv 4 \pmod{7} \\ \Rightarrow && n &= 35k+20 \\ \Rightarrow && 35k+20 &\equiv 7 \pmod{9} \\ \Rightarrow && -k &\equiv -13 \pmod{9} \\ \Rightarrow && k &\equiv 4 \pmod{9} \\ \Rightarrow && n &= 35\cdot(9k+4)+20 \\ &&&= 5 \cdot 7 \cdot 9k + 160 \\ \Rightarrow && 5 \cdot 7 \cdot 9k + 160 &\equiv 8 \pmod{11} \\ \Rightarrow && 5 \cdot (-4) \cdot (-2) k+6 &\equiv 8 \pmod{11} \\ \Rightarrow && -15k&\equiv 2 \pmod{11} \\ \Rightarrow && -4k &\equiv 2 \pmod{11} \\ \Rightarrow && 9k &\equiv 1 \pmod{11} \\ && k &\equiv 5 \pmod {11} \\ \Rightarrow && n &= 5 \cdot 7 \cdot 9 \cdot (11k+5) + 160 \\ &&&= 5\cdot 7 \cdot 9 \cdot 11 +1735 \end{align*} The smallest set of \(4\) numbers is \(1735, 1736, 1737, 1738\)
Prove that the equation \[ x^4+4rx+3s=0 \] has no real roots if \(r^4 < s^3\).
A diagonal of a quadrilateral makes angles \(\alpha, \beta\) with the sides at one of its ends, and angles \(\gamma, \delta\) with the sides at the other end, the angles \(\alpha\) and \(\gamma\) being on the same side of the diagonal. Prove that the acute angle between the diagonals is \[ \tan^{-1}\frac{\cot\alpha+\cot\beta+\cot\gamma+\cot\delta}{\cot\alpha\cot\delta \sim \cot\beta\cot\gamma}. \]
Prove that, if \(X\) and \(Y\) are the lengths of the sides of regular polygons of \(n\) sides inscribed in a circle and circumscribing it, the circumference of the circle is approximately \(\frac{n}{3}(2X+Y)\). Shew (without using tables) that when \(n=10\) the error is approximately one of \(\cdot05\) per cent.
The focus of a parabola and one point on it are given. Find the locus of the vertex.
Find the limiting value of \((1-x)^{\log x}\) when \(x \to 0\). The equation of a curve is \[ x^2(x+y)+x-y+1=0. \] Find the asymptotes and trace the curve.
Prove that from a given point on a cubic curve four tangents can be drawn to the cubic in addition to the tangent at the point; and that the points of contact lie on a conic which touches the cubic at the given point; also that the curvature of the conic at this point is half that of the cubic at the point.
Two heavy beads of weights \(P\) and \(Q\) respectively are strung on a light endless string of length \(l\) which passes through two rings \(A\) and \(B\), and the system is in equilibrium as in the diagram. The vertical distance between \(A\) and \(B\) is \(h\), and the horizontal distance is \(k\). Shew that the angles \(\alpha\) and \(\beta\) which \(AP, AQ\) make with the vertical are independent of \(h\), and that, if \(\alpha=30^\circ\) and \(\beta=\sin^{-1}\frac{1}{4}\), then \(8P=7Q\) and \(l=9k\); and, if in addition \(AB\) makes an angle of \(30^\circ\) with the horizontal, then the horizontal distance between \(P\) and \(Q = \frac{3}{4}k\).
A particle can move in a smooth circular tube which revolves about a fixed vertical tangent with uniform velocity \(\omega\). Find the position of relative equilibrium of the particle, and shew that the time of a small oscillation about that position is \[ \frac{2\pi}{\omega}\left\{\frac{\sin\alpha}{1+\sin^2\alpha}\right\}^{\frac{1}{2}}, \] where \(\alpha\) is the angle of inclination to the vertical of the radius to the particle when in relative equilibrium.