10273 problems found
If \(0 < \theta_1 < \theta_2 < \pi\), prove that the volume swept out in one complete revolution about the line \(\theta=0\) by the plane region bounded by the curve \(r=f(\theta)\) and the lines \(\theta=\theta_1, \theta=\theta_2\) is \[ \frac{2\pi}{3} \int_{\theta_1}^{\theta_2} r^3 \sin\theta \, d\theta. \] Prove that the locus of a point which moves so that the product of its distances from two fixed points at a distance \(2c\) apart is \(c^2\) is a surface of revolution enclosing a volume \[ \pi c^3 \left\{ \log(1+\sqrt{2}) - \frac{\sqrt{2}}{3} \right\}. \]
Find the values of:
A hyperbola may be defined as the locus of a point \(P\) whose distances from two fixed points \(A, B\) have a constant difference; from this definition deduce, without assuming any other property of the hyperbola, the following sequence of results:
The points \(P(x,y)\), \(P'(x',y')\) of a plane are said to correspond when their coordinates are connected by the relations \[ x' : y' : 1 = a(b+h)x + bhy : akx + b(a+k)y : ax+by+ab; \] shew that, if \(P\) describes a line \(\lambda\), \(P'\) describes a line \(\lambda'\). Determine all the points \(P\) which coincide with their corresponding points \(P'\) and all the lines \(\lambda\) which coincide with their corresponding lines \(\lambda'\). Prove that all corresponding pairs of points \(P, P'\) are collinear with a fixed point \(A\) and that all corresponding pairs of lines \(\lambda, \lambda'\) are concurrent with a fixed line \(a\), and shew that, if \(ah+bk+2ab=0\), every pair of corresponding points \(P,P'\) are harmonically separated by \(A\) and \(a\).
If \(\alpha, \beta, \gamma, \delta\) are the roots of the equation \[ x^4 + px^2 + qx + r = 0, \] prove that the equation whose roots are \(\beta+\gamma, \gamma+\alpha, \alpha+\beta\) is \[ y^3 + 2py^2 + (p+2s^2)y - q=0, \text{ (OCR error, likely } p^2 \text{ or similar)} \] and deduce that the equation whose roots are \(\beta+\gamma, \gamma+\alpha, \alpha+\beta, \alpha+\delta, \beta+\delta, \gamma+\delta\) is \[ z^2(z^2+p)^2 - 4rz^2 - q^2=0. \] The roots of the equation \(x^4 + 4ax^3+6bx^2+4cx+d=0\) are \(\lambda_1, \lambda_2, \lambda_3, \lambda_4\). By reducing this equation to the form \(y^4+py^2+qy+r=0\), or otherwise, prove that the equation whose roots are \[ \tfrac{1}{4}(\lambda_1+\lambda_4-\lambda_2-\lambda_3)^2, \quad \tfrac{1}{4}(\lambda_2+\lambda_4-\lambda_3-\lambda_1)^2, \quad \tfrac{1}{4}(\lambda_3+\lambda_4-\lambda_1-\lambda_2)^2 \] is \[ z(z+6b-6a^2)^2 - 4(d-4ac+6a^2b-3a^4)z - (4c-12ab+8a^3)^2 = 0. \] Deduce that, if the roots of the equation \[ x^4+4ax^3+6a^2x^2+4cx+d=0 \] are all real, then they are all equal.
Prove that, if \(a_1, a_2, \dots, a_n\) are all different, the polynomial of degree \(n-1\) which takes the values \(A_1, A_2, \dots, A_n\) when \(x=a_1, a_2, \dots, a_n\) respectively is \[ \sum_{i=1}^n A_i \frac{(x-a_1)\dots(x-a_{i-1})(x-a_{i+1})\dots(x-a_n)}{(a_i-a_1)\dots(a_i-a_{i-1})(a_i-a_{i+1})\dots(a_i-a_n)}. \] Deduce the values of \[ \sum_{i=1}^n \frac{a_i^m}{(a_i-a_1)\dots(a_i-a_{i-1})(a_i-a_{i+1})\dots(a_i-a_n)} \] for \(m=1,2,\dots, n-2, n-1\).
State how to find the differential coefficient with respect to \(x\) of \[ \int_u^v f(x,t)dt, \] where \(u, v\) are functions of \(x\) and \(f(x,t)\) is a function of \(x,t\). If \[ \psi(x) = x^n \int_0^x \psi(t) dt, \] prove that \[ x \frac{d\psi(x)}{dx} = (n+x^{n+1})\psi(x), \] and hence find the form of \(\psi(x)\).
Explain what is meant by \[ \left( x \frac{\partial}{\partial x} + y \frac{\partial}{\partial y} \right)^2 z, \] where \(z\) is a function of \(x\) and \(y\), and prove that it is not in general equal to \[ x^2 \frac{\partial^2 z}{\partial x^2} + 2xy \frac{\partial^2 z}{\partial x \partial y} + y^2 \frac{\partial^2 z}{\partial y^2}. \] If \(z=x^n f(y/x)\), prove that \[ x \frac{\partial z}{\partial x} + y \frac{\partial z}{\partial y} = nz, \] and that \[ x^2 \frac{\partial^2 z}{\partial x^2} + 2xy \frac{\partial^2 z}{\partial x \partial y} + y^2 \frac{\partial^2 z}{\partial y^2} = n(n-1)z. \] Find the value of \[ c_0 x^p \frac{\partial^p z}{\partial x^p} + c_1 x^{p-1} y \frac{\partial^p z}{\partial x^{p-1} \partial y} + \dots + c_k x^{p-k} y^k \frac{\partial^p z}{\partial x^{p-k} \partial y^k} + \dots + c_p y^p \frac{\partial^p z}{\partial y^p}, \] where \(c_0, c_1, \dots, c_k, \dots, c_p\) are the coefficients in order in the binomial expansion of \((1-\xi)^p\).
A light rectangular rigid table, which has a leg at each corner of the top, has a particle of weight \(W\) placed on it at a point whose coordinates are \((x,y)\), referred to rectangular axes; the edges of the table are given by the equations \(x \pm a=0, y \pm b=0\). The floor is slightly elastic and gives under each leg by a small amount proportional to the pressure of the leg on the floor. Shew that the forces acting in the legs are \(\frac{1}{4}W(1\pm \frac{x}{a} \pm \frac{y}{b})\), provided that \(x\) and \(y\) are such that all these four numbers are positive.
A trolley consists of a uniform rectangular platform of length \(2c\) with two pairs of wheels of radius \(a\). The wheels are mounted on axles in the plane of the platform. The axles are both parallel to the front edge of the platform and are at a distance \(d\) in front of and behind the centre of the platform. The trolley is drawn by a light handle attached to the middle point of the front edge of the platform, and the handle slopes at an angle \(\alpha\) to the horizontal. The weight of the platform is \(W\) and the weight of the wheels is neglected. The friction at the axle is such that the couple needed to turn each pair is \(\lambda\) times the pressure on it. Find