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 .
I have a bright kid working towards his IITJEE Foundation math. Some days back I had suggested to him to solve some hard word problems based on simultaneous equations from a very old classic text. He started on his own, almost with some guidance from me. Until he attacked very well — those “age” kind of problems. Father’s age vs. son’s age, etc. But, in this case, he suddenly sprang to his feet: He almost yelled, “Sir! Please check my solution! I am getting the answer as husband’s age is 48 and wife’s age is 23!” Actually, I too was a bit shocked; but, I checked his calculations in detail; they were mathematically correct. It suddenly flashed in my head:
“Do you know Vedant? Mathematics is not human! It has no emotions, no feelings, whatsoever! I told him the following quote of Bertrand Russell: ” Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty, cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.””
And, of course, equations//numbers don’t lie! 🙂 🙂 🙂