Topic: Continuity and Differentiability (Test 1)



Topic: Continuity and Differentiability
Q.1
limx01cosxx2 is equal to
A. 1/2
B. -1
C. 0
D. 1
Answer : Option A
Explaination / Solution:

 (UsingL’hospital Rule ).

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Q.2
 is equal to
A. 0
B. none of these
C. 2/3
D. -2/3
Answer : Option D
Explaination / Solution:

  
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Q.3
is equal to
A. -1/48
B. none of these
C. 1/24
D. 1/48
Answer : Option A
Explaination / Solution:


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Q.4
If k be an integer, then  (x –[x])
A. is equal to – 1
B. does not exists.
C. 1
D. is equal to 0
Answer : Option C
Explaination / Solution:

 = k - (k - 1) = 1 for all 
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Q.5

 is equal to


A. none of these.
B. 0
C. 1
D. 1/2
Answer : Option D
Explaination / Solution:

  ( using L’Hospital Rule)

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Q.6
 is equal to
A. 1/2
B. 8
C. 0
D. none of these.
Answer : Option B
Explaination / Solution:



( using L’Hospital Rule)


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Q.7

 is equal to


A. 0
B. 1
C. does not exist
D. none of these
Answer : Option B
Explaination / Solution:

limx0tanxlog(1+x)=limx0tanxxxlog(1+x)=limx0tanxx.limx011xlog(1+x)=1.11=1
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Q.8
The function f (x) = 1 + | sin x l is
A. differentiable nowhere
B. differentiable everywhere.
C. continuous nowhere
D. continuous everywhere
Answer : Option D
Explaination / Solution:

  is not derivable at those x for which  is continuous everywhere (being the sum of two continuous functions)
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Q.9
The function, f (x) = (x – a) sin for x a and f (a) = 0 is
A. Derivable at x = a
B. Not continuous at x = a
C. None of these
D. Continuous but not derivable at x = a
Answer : Option D
Explaination / Solution:

) 
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Q.10
Let f (x) = [x], then f (x) is
A. differentiable for all x∈(R−I).
B. continuous for all x∈R
C. continuous nowhere
D. differentiable for all x∈R
Answer : Option A
Explaination / Solution:

f(x) = [x] is derivable at all x except at integral points i.e. on R – I .

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