## CONTINUITY AND DIFFERENTIABILITY-12

As per analysis for previous years, it has been observed that students preparing for JEE MAINS find Mathematics out of all the sections to be complex to handle and the majority of them are not able to comprehend the reason behind it. This problem arises especially because these aspirants appearing for the examination are more inclined to have a keen interest in Mathematics due to their ENGINEERING background. Furthermore, sections such as Mathematics are dominantly based on theories, laws, numerical in comparison to a section of Engineering which is more of fact-based, Physics, and includes substantial explanations. By using the table given below, you easily and directly access to the topics and respective links of MCQs. Moreover, to make learning smooth and efficient, all the questions come with their supportive solutions to make utilization of time even more productive. Students will be covered for all their studies as the topics are available from basics to even the most advanced. .

Q1. The function defined by
is continuous from right at the point x=2, then k is equal to
•  0
•  1/4
•  -1/2
•  None of these
Solution

Q2. The number of points of discontinuity of the function f(x)=1/log⁡|x|
•  4
•  3
•  2
•  1
Solution
Clearly, log⁡|x| is discontinuous at x=0
f(x)=1/log⁡|x| is not defined at x=±1
Hence, f(x) is discontinuous s at x=0,1,-1

Q3. Let f(x)=(x+|x|)|x|. The, for all x
•  f and f' are continuous
•  f is differentiable for some x
•  f' is not continuous
•  f'' is continuous
Solution
we have,

As is evident from the graph of f(x) that it is continuous and differentiable for all x
Also, we have

Clearly, f''(x) is continuous for all x but it is not differentiable at x=0

Q4.
then derivative of f(x) at x=0
•  Is equal to 1
•  Is equal to 0
•  Is equal to -1
•  Does not exist
Solution
We have,

Hence, f'(x) at x=0 does not exist

Q5. Function f(x)=|x-1|+|x-2|,x∈R is
•  Differentiable everywhere in R
•  Except x=1 and x=2 differentiable everywhere in R
•  Not continuous at x=1 and x=2
•  Increasing in R
Solution

Hence, except x=1 and x=2,f(x) is differentiable everywhere in R

Q6. The function f(x) is defined as f(x)=(2x-sin-1x)/(2x+tan-1⁡x), if x≠0. The value of f to be assigned at x=0 so that the function is continuous there, is
•  -1/3
•  1
•  2/3
•  1/3
Solution

Q7.
Then, f' (1) is equal to
•  -1
•  1
•  0
•  None of these
Solution

Q8. The function

•  Is discontinuous at finitely many points
•  Is continuous everywhere
•  Is discontinuous only at x=±1/n,n∈Z-{0} and x=0
•  None of these
Solution
The function f is clearly continuous for |x|>1 We observe that

Q9. The set of points where the function f(x)=√(1-e-x2) is differentiable is
•  (-∞,∞)
•  (-∞,0)∪(0,∞)
•  (-1,∞)
•  None of these
Solution

Q10. if f(x)=|x|3, then f' (0) equals
•  0
•  1/2
•  -1
• -1/2
Solution

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