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.

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