## CONTINUITY AND DIFFERENTIABILITY-13

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 function f(x)={(x,if x is rational)(1-x,if x is irrational) then the number of points at which f(x) is continuous, is
•
•  1
•  0
•  None of these
Solution
At no point, function is continuous

Q2.
is continuous everywhere, then k is equal to
•  1/2 loge⁡2
•  loge⁡4
•  loge⁡8
•  loge⁡2
Solution

Q3. If f(x)=sin-1⁡(2x/(1+x2)) , then f(x) is differentiable on
•   [-1,1]
•  R-{-1,1}
•  R-(-1,1)
•  None of these
Solution

Q4. Let g(x) be the inverse of the function f(x)and f' (x)=1/(1+x3). Then, g'(x) is equal to
•  1/(1+(g(x))3)
•  1/(1+(f(x))3)
•  1+(g(x))3
•  1+(f(x))3
Solution

Q5. If f(x)=ae|x|+b|x|2; a,b ∈R and f(x) is differentiable at x=0. Then a and b are
•  a=0,b∈R
•  a=1,b=2
•  b=0,a∈R
•  a=4,b=5
Solution
Given, f(x)=ae|x| +b |x|2 We know e|x| is not differentiable at x=0 and |x|2 is differentiable at x=0
∴ f(x) is differentiable at x=0, if a=0 and b∈R

Q6. The set of points of differentiability of the function

•  R
•  [0,∞]
• (-∞,0)
•  R-{0}
Solution

Q7. Let f(x)=||x|-1|, then points where f(x) is not differentiable, is/(are)
•  0,±1
•  ±1
•  0
•  1
Solution
y=|(|x|-1)| we have 3 sharp edges at x=-1,0,1
∴f(x) is not differentiable at {0,±1}

Q8. If the function
is continuous at x=0, then the value of k is
•  1
•  0
•  1/2
•  -1
Solution

Q9.
is continuous at x=0, then
•  a=- 3/2,b=0,c=1/2
•  a=- 3/2,b=1,c=-1/2
•  a=- 3/2,b∈R-{0},c=1/2
•  None of these
Solution

Q10. If f(x)=(x+1)cot⁡x be continuous at =0, then f(0) is equal to
•  0
•  -e
•  e
•  None of these
Solution

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