Find the conditions that \(ax^2+2bx+c\) should be positive for all real values of \(x\). Find the greatest value of \[ (a-x)(x+\sqrt{b^2+x^2}). \]
If \[ (1+x)^n = c_0+c_1x+c_2x^2+\dots \] find \[ c_0 - c_1 + c_2 - \dots + (-1)^n c_n. \] Prove that the coefficient of \(x^n\) in the expansion in powers of \(x\) of \[ \frac{1}{(1-x)(1-x^2)(1-x^5)} \] is \((n+1)^2\). (Note: The original text had a typo in the denominator, \(x^5\) vs \(x^3\). Transcribed as written. Also, the claim regarding the coefficient seems incorrect for the given function.)
Prove Fermat's theorem that \(a^n-x\) is divisible by \(n\) if \(n\) is a prime and \(x\) any positive integer. Prove that \(x^{45}-x\) is divisible by 69.
Define the differential coefficient of a function of \(x\). If \(f(x)\) is positive shew that \(f(x)\) is increasing. Prove that \(x\log x > x-1\) if \(x\) is positive. Differentiate \(x^x, \tan^{-1}\left(\frac{x\sin\alpha}{1-x\cos\alpha}\right)\).
If \(x=r\cos\theta, y=r\sin\theta\), find the values of \(\frac{\partial r}{\partial x}\) and \(\frac{\partial r}{\partial y}\). Transform the variables from \(x, y\) to \(r, \theta\) in \[ x\frac{\partial u}{\partial y} - y\frac{\partial u}{\partial x}. \]
Prove that if \(p\) is the perpendicular from the origin on the tangent to a curve \(r=f(\theta)\), \[ \frac{1}{p^2} = \frac{1}{r^2} + \frac{1}{r^4}\left(\frac{dr}{d\theta}\right)^2. \] Prove that the feet of the perpendiculars from the origin on the normals to the curve \(r^2=a^2\cos 2\theta\) lie on the curve \[ 4r^2/a^2 = \cos 2\theta + \cos(\frac{1}{3}\pi+\theta). \] (Note: The constant in the cosine term is transcribed as it appears.)
Trace the curve \(x^4 - x^2y+y^3=0\).
Evaluate the integrals \[ \int \frac{dx}{(2+x)\sqrt{1+x}}, \quad \int \cos x \cos 3x \,dx, \quad \int_0^\pi \frac{d\theta}{a+b\cos\theta} \quad (a>b). \]
Explain how the area of a plane curve may be obtained. Find the area contained between the parabolas \(y^2=4ax\) and \(x^2=4ay\).
Solve the equations \begin{align*} x^2+2yz &= -11, \\ y^2+2zx &= -2, \\ z^2+2xy &= 13. \end{align*} Given that \(x=a\) is one root of the equation \[ (x^2+x+1)^3 = M(x^2+x)^2, \] prove that the other roots are \[ \frac{1}{a}, \quad -(a+1), \quad -\frac{1}{a+1}, \quad -\frac{a}{a+1}, \quad -\frac{a+1}{a}. \]