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- Date Submitted: 01/04/2010 9:57 PM
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Workbook for Class XII CHAPTER

5

CONTINUITY AND DIFFERENTIABILITY

INSTRUCTIONS

HOTS

WARNINGS

33

(i) This chapter contains 36 questions divided among 1 mark, 4 marks and 6 marks. (ii) The marks of each question are indicated in the bracket immediately after the question. (iii) Blank space has been provided against each question. Try to restrict your answer in the provided space. Use extra sheets if required. (iv) The solutions to all the questions have been provided at the end of the book.

(i) Do not see the solutions before attempting these questions. (ii) Do not attempt these questions if you are not prepared for the chapter.

Q.1

If n ∈ N and [x] denotes the greatest integer less than or equal to x, then is lim

logxn − [x] exists? [x] x →0 (1 mark)

Q.2

Find the number of points at which the function f ( x) =

1 is discontinuous. log | x |

(1 mark)

Q.3

Find the number of points at which the function f (x) =|x – 0.5| + | x – 1| + tan x does not have a derivative in the interval (0, 2). (1 mark)

34

Workbook for Class XII

Q.4

Find lim

a n + bn − bn

n→ ∞ n a

, where a > b > 1.

(1 mark)

Q.5

sin(1/x),x ≠ 0 Find the value of k which makes f (x) = k , x = 0 continuous at x = 0.

{

(1 mark)

Q.6

Discuss the continuity of the function [cos x] at x = π/2, where [ ⋅ ] denotes the greatest integer function. (1 mark)

Q.7

dy −1 If y = sin x − ax − a − ax , then find the value of . dx

(1 mark)

Q.8

Explain the difference between the average rate of change of y = f (x) as x changes from a to b and the instantaneous rate of change of y at x = a. (4 marks)

Workbook for Class XII

35

Q.9

Find all points of discontinuity for the functions whose graph is shown below:

f(x)

(4 marks)

3 3 –2 –2 2 x

Q.10 Find all points of discontinuity for the functions whose graph is shown below:

y

(4 marks)

x –3 –1 1 3

36

Workbook for Class XII

Q.11...