If \(e^{x-y^2} = x-y\), prove that \(y^2\dfrac{\partial z}{\partial x} + x\dfrac{\partial z}{\partial y} = x^2-y^2\), and also that \[ y^2\frac{\partial^2 z}{\partial x^2} + x^2\frac{\partial^2 z}{\partial y^2} + 2xy\frac{\partial^2 z}{\partial x \partial y} + x\frac{\partial z}{\partial x} + y\frac{\partial z}{\partial y} = 0. \] % There seems to be a typo in the question, z is not defined. Assuming z = x-y.
Find the equations to the tangent and normal to the curve \(y=f(x)\) at any point. A circle is described passing through the origin and touching the curve \[ ax^3 = (x^2+y^2)y^2 \] at the point \((x,y)\). Shew that the circle cuts the axis of \(x\) also at the point whose abscissa is \(3x^2a/y^2\).
Prove the formula for the radius of curvature \(\rho=r\dfrac{dr}{dp}\). At any point of a rectangular hyperbola prove that \(3\rho\dfrac{d^2 p}{ds^2} - 2\left(\dfrac{dp}{ds}\right)^2\) is constant.
If \[ E(m) = 1+m+\frac{m^2}{2!} + \dots + \frac{m^r}{r!} + \dots, \] prove that \[ E(m) \times E(n) = E(m+n). \] Hence shew that \(E(x) = \{E(1)\}^x\) for all real values of \(x\).
Prove that the geometric mean between two quantities is also the geometric mean between their arithmetic and harmonic means. Sum the series \[ a+(a+b)r+(a+2b)r^2+(a+3b)r^3+\dots+(a+\overline{n-1}b)r^{n-1}. \]
Prove that, if \(p\) is a prime number, and \(x\) is any number less than \(p\) except \(1\) and \(p-1\), then another such number \(y\) can be found so that \(xy \equiv 1 \pmod p\). Hence prove that \((p-1)!+1 \equiv 0 \pmod p\).
Prove that the value of a determinant is unaltered by adding to each element of one column the same multiple of the corresponding element of another column. Evaluate \[ \begin{vmatrix} 12 & 34 & 56 \\ 14 & 26 & 35 \\ 16 & 18 & 14 \end{vmatrix}. \]
Prove that if \(A\) and \(B\) are acute angles while \(A+B\) is obtuse, \[ \cos(A+B) = \cos A\cos B - \sin A\sin B. \] Prove that \[ \sin(\alpha+\theta)\cos^3(\alpha-\theta) - \sin(\alpha-\theta)\cos^3(\alpha+\theta) = \sin 2\theta\{1+\cos 2\alpha\sin(\alpha+\theta)\sin(\alpha-\theta)\}. \]
If \(a, b, c\) are the sides and \(A, B, C\) the angles of a triangle prove, ab initio,
Find an expression for all the angles which have the same cosine as a given angle. Prove \[ \cos n\phi - \cos n\theta = 2^{n-1} \prod_0^{n-1} \left\{\cos\phi - \cos\left(\theta+\frac{2r\pi}{n}\right)\right\}. \] Deduce \[ \cos\alpha = 2^{n-1} \prod_0^{n-1} \sin \frac{2\alpha+(2r+1)\pi}{2n}. \]