Exercise 5.1 Part-2
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NCERT Solutions class 12 Continuity & Differentiability
7. Find the relationship between 
and 
so that the function 
defined by
is continuous at 
Ans. Given: 
Continuity at 
Also
NCERT Solutions class 12 Maths Exercise 5.1 Part-2
18. For what value of 
is the function defined by 
continuous at 
What about continuity at 
Ans. Since 
is continuous at 
And 
Here, therefore should be L.H.L. = R.H.L.
0 = 1, which is not possible.
Again Since is continuous at 
And 
Here, therefore should be L.H.L. = R.H.L.
NCERT Solutions class 12 Maths Exercise 5.1 Part-2
19. Show that the function defined by 
is discontinuous at all integral points. Here 
denotes the greatest integer less than or equal to 
Ans. For any real number 
we use the symbol to denote the fractional part or decimal part of For example,
[3.45] = 0.45
[–7.25] = 0.25
[3] = 0
[–7] = 0
The function 
: R 
R defined by 
is called the fractional part function. It is observed that the domain of the fractional part function is the set R of all real numbers and the range of the set [0, 1).
Hence given function is discontinuous function.
NCERT Solutions class 12 Maths Exercise 5.1 Part-2
20. Is the function 
continuous at 
?
Ans. Given:
L.H.L. = 
R.H.L. = 
And 
Since L.H.L. = R.H.L. = 
Therefore, is continuous at
NCERT Solutions class 12 Maths Exercise 5.1 Part-2
21. Discuss the continuity of the following functions:
(a) 
(b) 
(c) 
Ans. (a) Let be an arbitrary real number then 
= 
= 
= 
Similarly, we have 
Therefore, is continuous at 
Since, is an arbitrary real number, therefore, is continuous.
(b) Let be an arbitrary real number then
= 
= 
= 
Similarly, we have
Therefore, is continuous at
Since, is an arbitrary real number, therefore, is continuous.
(c) Let be an arbitrary real number then
= 
= 
= 
Similarly, we have
Therefore, is continuous at
Since, is an arbitrary real number, therefore, is continuous.
NCERT Solutions class 12 Maths Exercise 5.1 Part-2
22. Discuss the continuity of cosine, cosecant, secant and cotangent functions.
Ans. (a) Let be an arbitrary real number then 
= 
= 
= 
for all 
R
Therefore, is continuous at
Since, is an arbitrary real number, therefore, 
is continuous.
(b)
and domain 
I
= 
= 
= 
= 
Therefore, is continuous at
Since, is an arbitrary real number, therefore, 
is continuous.
(c) 
and domain 
I
= 
= 
= 
= 
Therefore, is continuous at
Since, is an arbitrary real number, therefore, 
is continuous.
(d)
and domain I
= 
= 
= 
= 
Therefore, is continuous at
Since, is an arbitrary real number, therefore, 
is continuous.
23. Find all points of discontinuity of 
where 
.
Ans. Given: 
At 
L.H.L. = 
R.H.L. = 
is continuous at
When 
and are continuous, then 
is also continuous.
When 
is a polynomial, then is continuous.
Therefore, is continuous at any point.
24. Determine if defined by 
is a continuous function.
Ans. Here, 
= 0 x a finite quantity = 0
Also 
Since, 
therefore, the function is continuous at
NCERT Solutions class 12 Maths Exercise 5.1 Part-2
25. Examine the continuity of where is defined by 
.
Ans. At L.H.L. = 
= 
R.H.L. = 
= 
And
Therefore, is discontinuous at
NCERT Solutions class 12 Maths Exercise 5.1 Part-2
Find the values of 
so that the function is continuous at the indicated point in Exercise 26 to 29.
26. 
at 
Ans. Here, 
Putting 
where 
= 
= 
= 
= 
= 
……….(i)
And 
……….(ii)
when 
[Given]
Because 
is continuous at
From eq. (i) and (ii),
27. 
at 
Ans. Here, 
and 
Now, when , we have
Therefore, is continuous at 
when .
28. 
at 
Ans. Here, 
And 
Also 
Since the given function is continuous at 
29. 
at 
Ans. When 
we have 
which being a polynomial is continuous at each point 
And, when 
we have 
which being a polynomial is continuous at each point 
Now 
……….(i)
= 
…….(i)
Since function is continuous, therefore, eq. (i) = eq. (ii)
NCERT Solutions class 12 Maths Exercise 5.1 Part-2
30. Find the values of 
and 
such that the function defined by 
is a continuous function.
Ans. For 
function is 
constant, therefore it is continuous.
For 
function 
polynomial, therefore, it is continuous.
For 
function is 
constant, therefore it is continuous.
For continuity at 
……….(i)
For continuity at 
……….(ii)
Solving eq. (i) and eq. (ii), we get
and 
NCERT Solutions class 12 Maths Exercise 5.1 Part-2
31. Show that the function defined by 
is a continuous function.
Ans. Let 
and 
, then
Now 
and 
being continuous it follows that their composite 
is continuous.
Hence 
is continuous function.
32. Show that the function defined by 
is a continuous function.
Ans. Given: ….(i)
has a real and finite value for all 
R.
Domain of is R.
Let and 
Since 
and 
being cosine function and modulus function are continuous for all real 
Now, 
being the composite function of two continuous functions is continuous, but not equal to
Again, 
[Using eq. (i)]
Therefore, 
being the composite function of two continuous functions is continuous.
33. Examine that 
is a continuous function.
Ans. Let 
and 
, then
Now, and being continuous, it follows that their composite, is continuous.
Therefore, is continuous.
34. Find all points of discontinuity of defined by 
Ans. Given: 
When 
= 
When 
When 
At 
L.H.L. = 
R.H.L. = 
And 
Therefore, at is continuous.
At 
L.H.L. = 
R.H.L. = 
And 
Therefore, at is continuous.
Hence, there is no point of discontinuity.