Problems

A body is projected from the ground with velocity \(V\) at inclination \(\alpha\) to the horizontal. At the highest point of the trajectory the body is broken into two parts by an internal explosion which creates \(E\) ft. lbs. of energy without altering the direction of motion. Find the distance between the parts when they reach the ground.

A smooth straight tube rotates in a horizontal plane about a point in itself with uniform angular velocity \(\omega\). At time \(t=0\) a particle is placed inside the tube, at rest relative to the tube and distant \(b\) from the point of rotation. Show that at time \(t\) the distance of the particle from the point of rotation is \(b \cosh \omega t\).

Shew that the product \((a^3+b^3+c^3-3abc)(x^3+y^3+z^3-3xyz)\) can be expressed in the form \(A^3+B^3+C^3-3ABC\), all the quantities involved being real. Find \(A, B, C\) in terms of \(a,b,c,x,y,z\). Shew also that if \(n\) is a positive integer of the form \(3m+1\) (\(m\) being any positive integer), then \((y-z)^n + (z-x)^n + (x-y)^n\) has a factor \(\Sigma a^2 - \Sigma yz\).

(i) Find the sum to \(n\) terms of the series: \(1 + 2^2x + 3^2x^2 + \dots\). (ii) Find the sum of the infinite series: \(1+3x+\frac{5x^2}{2!} + \frac{7x^3}{3!} + \dots\).

(i) Solve \[ \frac{x^2-a^2}{(x-a)^3} - \frac{x^2-b^2}{(x-b)^3} + \frac{x^2-c^2}{(x-c)^3} = 0 \] \[ \frac{(x+a)^3}{(x+a)^3} - \frac{(x+b)^3}{(x+b)^3} + \frac{(x+c)^3}{(x+c)^3} = 0 \] for \(x\), where \(a,b,c\) are unequal. [Note: The second equation appears to have typos in the source; it's transcribed as written, but likely intended to be different.] (ii) Shew that if the roots of the equation \(ax^4+bx^3+cx^2+dx+e=0\) are in harmonic progression, then \(d^3=4cde-8be^2\), and \(25ad^2e = (cd-eb)(11eb-cd)\). Verify these conditions in the case of \(40x^4-22x^3-21x^2+2x+1=0\) and solve for \(x\).

Shew that the number of distinct sets of three positive integers (none zero) whose sum is the odd integer \(2n+1\), is given by the least integer containing \(\frac{n^2+n}{3}\).

(a) Differentiate with respect to \(x\): (i) \(x^{x^{\cosh^{-1}x}}\); (ii) \(\tan^{-1}\left[\tan x \frac{1+\cos 2x}{1-\cos 2x}\right]\). (b) Expand \(y=\tan^{-1}x\) as a series in increasing powers of \(x\), stating any theorems you may use, or conditions you may apply to \(x\).

Prove the formula \(\rho=r\frac{dr}{dp}\) for the radius of curvature of a curve at a point \(P\) where the length of the radius vector is \(r\) and the length of the perpendicular from the origin on to the tangent at \(P\) is \(p\). Sketch roughly the curve for which \(\frac{p}{r}\) is a constant and shew that \(\rho\) also is in constant ratio to \(r\) and \(p\). Prove that the radius of curvature can never be less than the corresponding radius vector.

Shew that if \(f(x)\) and \(\phi(x)\) are functions of \(x\) having derivatives \(f'(x), \phi'(x)\) in the range \((a,b)\), then \(\frac{f'(\xi)}{\phi'(\xi)} = \frac{f(b)-f(a)}{\phi(b)-\phi(a)}\) for some value \(\xi\) of \(x\) between \(a\) and \(b\). What restrictions (if any) are to be placed on the behaviour in the interval \((a,b)\) of \(f'(x)\) and \(\phi'(x)\)? Hence or otherwise prove that \(x^\lambda \log_e x \to 0\) as \(x \to 0\), if \(\lambda > 0\).

Find a reduction formula for \(f(m,n) = \int_0^1 x^{n-1}(1-x)^{m-1}dx\) and shew that \[ f(m,n) = \frac{(m-1)!}{n(n+1)\dots(n+m-1)}, \] \(m\) and \(n\) being positive integers. Shew further that if \(I(n) = \int_0^\infty e^{-x} x^{n-1}dx\), then (i) \(I(n+1) = n.I(n)\), (ii) \(f(m,n) = \frac{I(n).I(m)}{I(m+n)}\).