Applications of Derivatives: IITJEE Maths tutorial problem set: III

Slightly difficult questions, I hope, but will certainly re-inforce core concepts:

  1. Prove that the segment of the tangent to the curve y=c/x which is contained between the co-ordinate axes, is bisected at the point of tangency.
  2. Find all tangents to the curve y=\cos{(x+y)} for -\pi \leq x \leq \pi that are parallel to the line x+2y=0.
  3. Prove that the curves y=f(x), where f(x)>0 and y=f(x)\sin(x), where f(x) is a differentiable function, have common tangents at common points.
  4. Find the condition that the lines x\cos{\alpha} + y \sin{\alpha}=p may touch the curve (\frac{x}{a})^{m} + (\frac{y}{b})^{m}=1.
  5. If p_{1} and p_{2} are lengths of the perpendiculars from origin on the tangent and normal to the curve x^{2/3} + y^{2/3}=a^{2/3} respectively, prove that 4p_{1}^{2}+p_{2}^{2}=a^{2}.
  6. Show that the curve x=1-3t^{2}, y=t-3t^{3} is symmetrical about x-axis and has no real points for x>1. If the tangent at the point t is inclined at an angle \psi to OX, prove that 3t = \tan{\psi} + \sec{\psi}. If the tangent at P(-2,2) meets the curve again at Q, prove that the tangents at P and Q are at right angles.
  7. A tangent at a point P_{1} other than (0,0) on the curve y=x^{3} meets the curve again at P_{2}. The tangent at P_{2} meets the curve at P_{3} and so on. Show that the abscissae of P_{1}, P_{2}, \ldots, P_{n} form a GP. Also, find the ratio of area \frac{\Delta P_{1}P_{2}P_{3}}{area \hspace{0.1in} P_{2}P_{3}P_{4}}.
  8. Show that the square roots of two successive natural numbers greater than N^{2} differ by less than \frac{1}{2N}.
  9. Show that the derivative of the function f(x) = x \sin {(\frac{\pi}{x})}, when x>0, and f(x)=0 when x=0 vanishes on an infinite set of points of the interval (0,1).
  10. Prove that \frac{x}{(1+x)} < \log {(1+x)} < x for x>0.

More later, cheers,

Nalin Pithwa.

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