## CONTINUITY AND DIFFERENTIABILITY-17

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)=sin⁡ 4 Ï€ [x]/(1+[x]2), where [x] is the greatest integer less than or equal to x, then
•  f(x) is not differentiable at some points
•  f'(x) exists but is different from zero
•  f' (x)=0 for all x
•  f' (x)=0 but f is not a constant function
Solution
We have,
f(x)=sin⁡ 4 Ï€[x]/(1+[x]2)=0 for all x [∵4Ï€[x]is a multiple of Ï€]
⇒f' (x)=0 for all x

Q2.
is continuous t x=Ï€/2, then
•  m=1,n=0
•  m=nÏ€/2+1
•  n=m Ï€/2
•  m=n=Ï€/2
Solution

Q3.
is continuous at x=0, then the value of k is
•   1
•  -2
•  2
•  1/2
Solution

Q4. Let g(x) be the inverse of an invertible function f(x) which is differentiable at x=c, then g' (f(c) ) equals
•  f'(c)
•  1/(f'(c))
•  f(c)
•  None of these
Solution

Q5. Let f(x)={ (sin⁡Ï€x/5x, x≠0) (k, x=0), if f(x) is continuous at x=0, then k is equal to
•  Ï€/5
•  5/Ï€
•  1
•  0
Solution

Q6. The value of k which makes f(x)={ (sin⁡ (1/k),x≠0) (k,x=0) continuous at x=0 is
•  8
•  1
• -1
•  None of these
Solution
We have,
lim(x→0)⁡ f(x) =lim(x→0)⁡sin⁡1/x
⇒lim(x→0)⁡ f(x) = An oscillating number which oscillates between -1 and 1
Hence, lim(x→0)⁡ f(x) does not exist
Consequently, f(x) cannot be continuous at x=0 for any value of k

Q7. If lim(x→c)⁡ (f(x)-f(c))/(x-c) exists finitely, then
•  lim(x→c)⁡ f(x)=f(c)
•  lim(x→c)⁡ f'(x)=f'(c)
•  lim(x→c)⁡ f(x) does not exist
•  lim(x→c)⁡ f(x) may or may not exist
Solution
It is a fact.

Q8. Let f(x) be a function satisfying f(x+y)=f(x)+f(y) and f(x)=x g (x) for all x,y∈R, where g(x) is continuous. Then,
•  f' (x)=g'(x)
•  f' (x)=g(x)
•  f' (x)=g(0)
•  None of these
Solution

Q9. Let f and g be differentiable functions satisfying g' (a)=2,g(a)=b and fog=I (identify function). Then, f'(b) is equal to
•  1/2
•  2
•  2/3
•  None of these
Solution
We have,
fog=I
⇒fog(x)=x for all x
⇒f' (g(x) ) g' (x)=1 for all x
⇒f' (g(a) )=1/(g'(a))=1/2
⇒f' (b)=1/2 [∵f(a)=b]

Q10. The function f:R/{0}→R given by f(x)=1/x-2/(e2x-1) Can be made continuous at x=0 by defining f(0) as function
•  2
•  -1
•  0
•  1
Solution

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