## CONTINUITY AND DIFFERENTIABILITY-16

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 value of f at x=0 so that function f(x)=(2x-2(-x))/x,x≠0 is continuous at x=0, is
•  0
•  log⁡2
•  4
•  log⁡4
Solution

Q2. If f(x)={((1-cos⁡x)/x,x≠0) (k , x=0) is continuous at x=0, then the value of k is
•  0
•  1/2
•  1/4
•  -1/2
Solution

Q3.
(loge⁡a)n,a>0,a≠0, then at x=0,f(x) is
•   Everywhere continuous but not differentiable
•  Everywhere differentiable
•  Nowhere continuous
•  None of these
Solution

Q4. For the function f(x)=(e(1/x)-1)/(e(1/x)+1),x=0, which of the following is correct?
•  lim(x→0)⁡ f(x) does not exist
•  lim(x→0)⁡ f(x)=1
•  lim(x→0)⁡ f(x) exists but f(x) is not continuous at x=0
•  f(x) is continuous at x=0
Solution

Q5. The function f(x)=|cos⁡x | is
•  Everywhere continuous and differentiable
•  Everywhere continuous and but not differentiable at (2n+1) Ï€/2,n∈Z
•  Neither continuous nor differentiable at (2n+1) Ï€/2,n∈Z
•  None of these
Solution
It can be easily seen from the graph of f(x)=|cos⁡x | that it is everywhere continuous but not differentiable at odd multiples of Ï€/2

Q6.
is continuous function at x=0, then the value of a is equal to
•  3
•  1
•  2
•  0
Solution

Q7. Let f(x)=(ex-1)2/sin⁡ (x/a) log⁡(1+x/4) for x≠0 andf(0)=12. If f is continuous at x=0, then the value of a is equal to
•  1
•  -1
•  2
•  3
Solution

Q8.
then at x=0, the derivative f'(x) is
•  1
•  0
•  Infinite
•  Does not exist
Solution
We have,
Lf' (0)=0 and Rf' (0)=0+cos⁡ 0°=1
∴Lf'(0)≠Rf'(0)
Hence, f'(x) does not exist at x=0

Q9.

•  f and f' are continuous at x=0
•  f is derivable at x=0 and f' is continuous at x=0
•  f is derivable at x=0 and f' is not continuous at x=0
•  f' is derivable at x=0
Solution

Q10. f(x)=|[x]+x| in -1 < x < 2 is
•  Continuous at x=0
•  Discontinuous at x=1
•  Not differentiable at x=2,0
•  All the above
Solution

It is evident from the graph of this function that it is continuous but not differentiable at x=0. Also, it is discontinuous at x=1 and non-differentiable at x=2

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