Mathematics Hothouse

Limits Part 2: IITJEE Mains maths: Mathematics Hothouse

January 9, 2019 – 1:36 am

Limits part 1: video lecture: IITJEE Mains maths: Mathematics Hothouse

January 9, 2019 – 1:14 am

By Nalin Pithwa | Posted in calculus, IITJEE Mains | Tagged Calculus for IITJEE | Comments (0)

IIT JEE Foundation video lecture: Plane Geometry: From Mathematics Hothouse

January 8, 2019 – 9:37 pm

By Nalin Pithwa | Posted in IITJEE Foundation mathematics | Comments (0)

On the value of time…essence of time management

November 27, 2018 – 10:40 pm

Time is Life.

Money lost can come back, but time lost can never come back. Time is more valuable than money. (Bertrand Russell).

By Nalin Pithwa | Posted in time management | Comments (0)

Mathematics versus Physics

November 27, 2018 – 10:36 pm

The object of pure Physics is the unfolding of the laws of the intelligible world; the object of pure Mathematic that of unfolding the laws of human intelligence. — J. J. Sylvester.

In my opinion, for example, Boole’s Laws of (Human) Thought.

By Nalin Pithwa | Posted in applications of maths, careers in mathematics, mathematicians, miscellaneous, motivational stuff | Comments (0)

An observation about Sir Isaac Newton

November 27, 2018 – 10:07 pm

Newton’s patience was limitless. Truth, he said much later, was the offspring of silence and meditation. And, he said, I keep the subject constantly before me and wait till the first dawnings open slowly, by little and little into a full and clear light.

By Nalin Pithwa | Posted in mathematicians, motivational stuff | Comments (0)

Applications of Derivatives IITJEE Maths tutorial: practice problems part IV

November 13, 2018 – 6:26 am

Question 1.

If the point on $y = x \tan {\alpha} - \frac{ax^{2}}{2u^{2}\cos^{2}{\alpha}}$ , where $\alpha>0$ , where the tangent is parallel to $y=x$ has an ordinate $\frac{u^{2}}{4a}$ , then what is the value of $\alpha$ ?

Question 2:

Prove that the segment of the tangent to the curve $y=c/x$ , which is contained between the coordinate axes is bisected at the point of tangency.

Question 3:

Find all the tangents to the curve $y = \cos{(x+y)}$ for $-\pi \leq x \leq \pi$ that are parallel to the line $x+2y=0$ .

Question 4:

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.

Question 5:

Find the condition that the lines $x \cos{\alpha} + y \sin{\alpha} = p$ may touch the curve $(\frac{x}{a})^{m} + (\frac{y}{b})^{m}=1$ .

Question 6:

Find the equation of a straight line which is tangent to one point and normal to the point on the curve $y=8t^{3}-1$ , and $x=4t^{2}+3$ .

Question 7:

Three normals are drawn from the point $(c,0)$ to the curve $y^{2}=x$ . Show that c must be greater than 1/2. One normal is always the x-axis. Find c for which the two other normals are perpendicular to each other.

Question 8:

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 $4p_{1}^{2} + p_{2}^{2}=a^{2}$ .

Question 9:

Show that the curve $x=1-3t^{2}$ , and $y=t-3t^{3}$ 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.

Question 10:

Find the condition that the curves $ax^{2}+by^{2}=1$ and $a^{'}x^{2} + b^{'}y^{2}=1$ intersect orthogonality and hence show that the curves $\frac{x^{2}}{(a^{2}+b_{1})} + \frac{y^{2}}{(b^{2}+b_{1})} = 1$ and $\frac{x^{2}}{a^{2}+b_{2}} + \frac{y^{2}}{(b^{2}+b_{2})} =1$ also intersect orthogonally.

More later,

Nalin Pithwa.

By Nalin Pithwa | Posted in applications of maths, calculus, IITJEE Mains, ISI Kolkatta Entrance Exam, KVPY | Comments (0)

Applications of Derivatives: IITJEE Maths tutorial problem set: III

October 23, 2018 – 9:43 pm

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

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.
Find all tangents to the curve $y=\cos{(x+y)}$ for $-\pi \leq x \leq \pi$ that are parallel to the line $x+2y=0$ .
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.
Find the condition that the lines $x\cos{\alpha} + y \sin{\alpha}=p$ may touch the curve $(\frac{x}{a})^{m} + (\frac{y}{b})^{m}=1$ .
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 $4p_{1}^{2}+p_{2}^{2}=a^{2}$ .
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.
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 $\frac{\Delta P_{1}P_{2}P_{3}}{area \hspace{0.1in} P_{2}P_{3}P_{4}}$ .
Show that the square roots of two successive natural numbers greater than $N^{2}$ differ by less than $\frac{1}{2N}$ .
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 $(0,1)$ .
Prove that $\frac{x}{(1+x)} < \log {(1+x)} < x$ for $x>0$ .

More later, cheers,

Nalin Pithwa.

By Nalin Pithwa | Posted in IITJEE Foundation Math IITJEE Main and Advanced Math and RMO/INMO of (TIFR and Homibhabha) | Comments (0)

Applications of Derivatives: Tutorial: IITJEE Maths: Part II

October 23, 2018 – 8:32 pm

Another set of “easy to moderately difficult” questions:

The function $y = \frac{}x{1+x^{2}}$ decreases in the interval (a) $(-1,1)$ (b) $[1, \infty)$ (c) $(-\infty, -1]$ (d) $(-\infty, \infty)$ . There are more than one correct choices. Which are those?
The function $f(x) = \arctan (x) - x$ decreases in the interval (a) $(1,\infty)$ (b) $(-1, \infty)$ (c) $(-\infty, -\infty)$ (d) $(0, \infty)$ . There is more than one correct choice. Which are those?
For $x>1$ , $y = \log(x)$ satisfies the inequality: (a) $x-1>y$ (b) $x^{2}-1>y$ (c) $y>x-1$ (d) $\frac{x-1}{x}<y$ . There is more than one correct choice. Which are those?
Suppose $f^{'}(x)$ exists for each x and $h(x) = f(x) - (f(x))^{2} + (f(x))^{3}$ for every real number x. Then, (a) h is increasing whenever f is increasing (b) h is increasing whenever f is decreasing (c) h is decreasing whenever f is decreasing (d) nothing can be said in general. Find the correct choice(s).
If $f(x)=3x^{2}+12x-1$ , when $-1 \leq x \leq 2$ , and $f(x)=37-x$ , when $2<x\leq 3$ . Then, (a) $f(x)$ is increasing on $[-1,2]$ (b) $f(x)$ is continuous on $[-1,3]$ (c) $f^{'}(2)$ doesn’t exist (d) $f(x)$ has the maximum value at $x=2$ . Find all the correct choice(s).
In which interval does the function $y=\frac{x}{\log(x)}$ increase?
Which is the larger of the functions $\sin(x) + \tan(x)$ and $f(x)=2x$ in the interval $(0<x<\pi/2)$ ?
Find the set of all x for which $\log {(1+x)} \leq x$ .
Let $f(x) = |x-1| + a$ , if $x \leq 1$ ; and, $f(x)=2x+3$ , if $x>1$ . If $f(x)$ has local minimum at $x=1$ , then $a \leq$ ?
There are exactly two distinct linear functions (find them), such that they map $[-1,1]$ and $[0,2]$ .

more later, cheers,

Nalin Pithwa.

By Nalin Pithwa | Posted in applications of maths, calculus, IITJEE Mains | Comments (0)

Applications of Derivatives: Tutorial Set 1: IITJEE Mains Maths

October 19, 2018 – 6:33 pm

“Easy” questions:

Question 1:

Find the slope of the tangent to the curve represented by the curve $x=t^{2}+3t-8$ and $y=2t^{2}-2t-5$ at the point $(2,-1)$ .

Question 2:

Find the co-ordinates of the point P on the curve $y^{2}=2x^{3}$ , the tangent at which is perpendicular to the line $4x-3y+2=0$ .

Question 3:

Find the co-ordinates of the point $P(x,y)$ lying in the first quadrant on the ellipse $x^{2}/8 + y^{2}/18=1$ so that the area of the triangle formed by the tangent at P and the co-ordinate axes is the smallest.