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.