## CONTINUITY AND DIFFERENTIABILITY-19

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. f(x)={ (2x, x < 0) (2x + 1, x ≥ 0) then,
•  f(x) is continuous at x=0
•  f(|x|) is continuous at x=0
•  f(x) is discontinuous at x=0
•  None of the above
Solution

Q2. The function f(x)=max⁡ [(1-x),(1+x),2], x∈(-∞,∞) is
•  Continuous at all points
•  Differentiable at all points
•  Differentiable at all points except at x=1 and x=-1
•  None of the above
Solution

It is clear from the graph that f(x) is continuous everywhere and also differentiable everywhere except {-1,1} due to sharp edge

Q3. If f(x)={ ((x-1)/(2x2-7x+5), for x≠1) (-1/3, for x=1), then f' (1) is equal to
•   -1/9
•  -2/9
•   -1/3
•  1/3
Solution

Q4. The function f(x)=e(-|x|) is
•  Continuous everywhere but not differentiable at x=0
•  Continuous and differentiable everywhere
•  Not continuous at x=0
•  None of the above
Solution
It is clear from the figure that f(x) is continuous everywhere and not differentiable at x=0 due to sharp edge

Q5. Let [ ] denotes the greatest integer function and f(x)=[tan2⁡x]. Then,
•  lim(x→0)⁡ f(x) does not exist
•  f(x) is continuous at x=0
•  f(x) is not differentiable at x=0
•  f(x)=1
Solution

Q6. The function f(x)=|x|+|x-1|, is
•  Continuous at x=1, but not differentiable
•  Both continuous and differentiable at x=1
•  Not continuous at x=1
•  None of these
Solution

Q7. If f(x)=loge⁡ (1+x2 tan⁡x)/sin⁡ x3 ,x≠0, is to be continuous at x=0, then f(0) must be defined as
•  1
•  0
•  1/2
•  -1
Solution
nishhh

Q8 .Let f(x+y)=f(x)f(y) and f(x)=1+(sin⁡ 2 x)g(x) where g(x) is continuous. Then, f'(x) equals
•  f(x)g(0)
•  2f(x)g(0)
•  2g(0)
•  None of the above
Solution

Q9. Let f(x+y)=f(x)f(y)and f(x)=1+xg(x)G(x), where lim(x→0)⁡ g(x)=a and lim(x→0)⁡ G(x)=b. Then f'(x) is equal to
•  1+ab
•  ab
•  a/b
•  None of these
Solution

Q10. Let f(x)={ (sin⁡ x,for x≥0) (1-cos⁡ x, for x≤0 ) and g(x)=ex. Then, (gof)'(0) is
•  1
•  -1
•  0
•  None of these
Solution

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