## Applications of Derivatives: IITJEE Maths tutorial problem set: III

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

- Prove that the segment of the tangent to the curve which is contained between the co-ordinate axes, is bisected at the point of tangency.
- Find all tangents to the curve for that are parallel to the line .
- Prove that the curves , where and , where is a differentiable function, have common tangents at common points.
- Find the condition that the lines may touch the curve .
- If and are lengths of the perpendiculars from origin on the tangent and normal to the curve respectively, prove that .
- Show that the curve , is symmetrical about x-axis and has no real points for . If the tangent at the point t is inclined at an angle to OX, prove that . If the tangent at meets the curve again at Q, prove that the tangents at P and Q are at right angles.
- A tangent at a point other than on the curve meets the curve again at . The tangent at meets the curve at and so on. Show that the abscissae of form a GP. Also, find the ratio of area .
- Show that the square roots of two successive natural numbers greater than differ by less than .
- Show that the derivative of the function , when , and when vanishes on an infinite set of points of the interval .
- Prove that for .

More later, cheers,

Nalin Pithwa.

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