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CONTINUITY AND DIFFERENTIABILITY-11

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. If f:R→R is defined by
,then f is continuous on the set
•  R
•  R-{-2}
•  R-{-1}
•  R-(-1,-2)
Solution
Since, f(x) is continuous for every value of R except {-1, -2}. Now, we have to check that points
At x=-2

⇒ LHL=RHL≠f(-1) ∴ It is not continuous at x=-1 The required function is continuous in R-{-1}

Q2. Let f(x+y)=f(x)f(y) for all x,y∈R. Suppose that f(3)=3 and f^' (0)=11 then, f'(3) is equal to
•  22
•  44
•  28
•  None of these
Solution
It is a fact.

Q3.
is continuous at x=Ï€/2, then k=
•   -1/16
•  -1/32
•  -1/64
•  -1/28
Solution
For f(x) to be continuous at x=Ï€/2, we must have

Q4. The set of points where the function f(x)=x|x| is differentiable is
•  (-∞,∞)
•  (-∞,0)∪(0,∞)
•  (0,∞)
•  [0,∞)
Solution
Clearly, f(x) is differentiable for all x<0.So, we check the differentiable at x=0

∴ (LHD at x=0)=(RHD at x=0)
So, f(x) is differentiable for all x ie, the set of all points where f(x) is differentiable is (-∞,∞)

Q5. The value of f(0) so that (-ex+2x)/x may be continuous at x=0 is
•  log⁡(1/2)
•  0
•  4
•  -1+log⁡2
Solution
Since, f(x) is continuous at x=0

Q6. For the function f(x)
if f(x) is continuous at x=a, then b is equal to
•  a2
•  2a2
•  3a2
•  4a2
Solution

Q7. f(x)=x+|x| is continuous for
•  x∈(-∞,∞)
•  x∈(-∞,∞)-{0}
•  Only x>0
•  No value of x
Solution
Given, f(x)=x+|x|

It is clear from the graph of f(x) is continuous for every value of x Alternate
Since, x and |x| is continuous for every value of x, so their sum is also continous for every value of x

Q8.
then at x=1,f(x) is
•  Continuous and differentiable
•  Differentiable but not continuous
•  Continuous but not differentiable
•  Neither continuous nor differentiable
Solution
We have,

So, f(x) is not continuous at x=1 and hence it is not differentiable at x=1

Q9. If f:R→R given by
Function on R, then (a,b) is equal to
•  (1/2, 1/2)
•  (0, -1)
•  (0, 2)
•  (1, 0)
Solution

Since, function is continuous.
∴ RHL=LHL ⇒ a=b
From the given options only (a) ie,(1/2,1/2) satisfies this condition

Q10. If f(x)=[x sin⁡Ï€ x], then which of the following is incorrect?
•  f(x) is continuous at x=0
•  f(x) is continuous in (-1,0)
•  f(x) is differentiable at x=1
• f(x) is differentiable in (-1,1)
Solution

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