Limits Part 2: IITJEE Mains maths: Mathematics Hothouse

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

 

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

On the value of time…essence of time management

Time is Life.

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

Mathematics versus Physics

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. 

An observation about Sir Isaac Newton

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.

Applications of Derivatives IITJEE Maths tutorial: practice problems part IV

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.

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.

Applications of Derivatives: Tutorial: IITJEE Maths: Part II

Another set of “easy to moderately difficult” questions:

  1. 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?
  2. 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?
  3. 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?
  4. 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).
  5. 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).
  6. In which interval does the function y=\frac{x}{\log(x)} increase?
  7. Which is the larger of the functions \sin(x) + \tan(x) and f(x)=2x in the interval (0<x<\pi/2)?
  8. Find the set of all x for which \log {(1+x)} \leq x.
  9. 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 ?
  10. There are exactly two distinct linear functions (find them), such that they map [-1,1] and [0,2].

more later, cheers,

Nalin Pithwa.

Applications of Derivatives: Tutorial Set 1: IITJEE Mains Maths

“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.

Question 4:

The function f(x) = \frac{\log (\pi+x)}{\log (e+x)}, where x \geq 0 is

(a) increasing on (-\infty, \infty)

(b) decreasing on [0, \infty)

(c) increasing on [0, \pi/e) and decreasing on [\pi/e, \infty)

(d) decreasing on [0, \pi/e) and increasing on [\pi/e, \infty).

Fill in the correct multiple choice. Only one of the choices is correct.

Question 5:

Find the length of a longest interval in which the function 3\sin(x) -4\sin^{3}(x) is increasing.

Question 6:

Let f(x)=x e^{x(1-x)}, then f(x) is

(a) increasing on [-1/2, 1]

(b) decreasing on \Re

(c) increasing on \Re

(d) decreasing on [-1/2, 1].

Fill in the correct choice above. Only one choice holds true.

Question 7:

Consider the following statements S and R:

S: Both \sin(x) and \cos (x) are decreasing functions in the interval (\pi/2, \pi).

R: If a differentiable function decreases in the interval (a,b), then its derivative also decreases in (a,b).

Which of the following is true?

(i) Both S and R are wrong.

(ii) Both S and R are correct, but R is not the correct explanation for S.

(iii) S is correct and R is the correct explanation for S.

(iv) S is correct and R is wrong.

Indicate the correct choice. Only one choice is correct.

Question 8:

For which of the following functions on [0,1], the Lagrange’s Mean Value theorem is not applicable:

(i) f(x) = 1/2 -x, when x<1/2; and f(x) = (1/2-x)^{2}, when x \geq 1/2.

(ii) f(x) = \frac{\sin(x)}{x}, when x \neq 0; and f(x)=1, when x=0.

(iii) f(x)=x |x|

(iv) f(x)=|x|.

Only one choice is correct. Which one?

Question 9:

How many real roots does the equation e^{x-1}+x-2=0 have?

Question 10:

What is the difference between the greatest and least values of the function f(x) = \cos(x) + \frac{1}{2}\cos(2x) -\frac{1}{3}\cos(3x)?

More later,

Nalin Pithwa.