## CONTINUITY AND DIFFERENTIABILITY-20

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. Let f(x)=[x] and g(x)={ (0,x∈Z) (x2,x∈R-Z) Then, which one of the following is incorrect?
•  lim(x→1)⁡ g(x) exists, but g(x) is not continuous at x=1
•  lim(x→1)⁡ f(x) does not exist and f(x) is not continuous at x=1
•  gof is continuous for all x
•  fog is continuous for all x
Solution

Q2.

is continuous at x=0, then k equals
•  16√2 log⁡2 log⁡3
•  16√2 ln⁡6
•  16√2 ln⁡2 ln⁡3
•  None of these
Solution

Q3. Let f(x) be a function such that f(x+y)=f(x)+f(y) and f(x)=sin⁡x g(x) for all x,y∈R. If g(x) is a continuous function such that g(0)=k, then f'(x) is equal to
•   k
•  kx
•  kg(x)
•  None of these
Solution

Q4. If f' (a)=2 and f(a)=4, then lim(x→a)⁡ (xf(a)-af(x))/(x-a) equals
•  2a-4
•  4-2a
•  2a+4
•  None of these
Solution

Q5.

is differentiable at x=1, then
•  a= 1/2,b=-1/2
•  a=- 1/2,b=-3/2
•  a=b= 1/2
•  a=b=- 1/2
Solution

Q6.

Then, f(x) is continuous at x=4, when
•  a=0,b=0
•  a=1,b=1
•  a=-1,b=1
•  a=1,b=-1
Solution

Q7.

then f' (2+) is equal to
•  0
•  2
•  3
•  4
Solution

Q8.

the value of a so that f(x) is continuous at x=Ï€/4 is
•  2
•  4
•  3
•  1
Solution

Q9. Let f(x)=[x3-x], where [x]the greatest integer function is. Then the number of points in the interval (1, 2), where function is discontinuous is
•  4
•  5
•  6
•  7
Solution
Given, f(x)=[x3-3] Let g(x)=x3-x it is in increasing function ∴ g(1)=1-3=-2 and g(2)=8-3=5 Here, f(x) is discontinuous at six points

Q10. The value of f(0), so that the function

Becomes continuous for all x, is given by

•  a3/2
•  a1/2
•  -a1/2
•  -a3/2
Solution

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Quiz-20 CONTINUITY AND DIFFERENTIABILITY