Monthly Archives: October 2018

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.