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NCERT Solutions for Class 12 Maths Chapter 5 are provided here. These NCERT solutions are created bu expert team at careers360 considering the latest syllabus of CBSE 2023-24. Questions based on the topics like continuity, differentiability, and relations between them are covered in the NCERT solutions for class 12 maths chapter 5. In NCERT Class 12 maths book, there are 48 solved examples to understand the concepts of continuity and differentiability class 12. If you are finding difficulties in solving them, you can take help from NCERT maths chapter 5 class 12 solutions.
In NCERT class 11 Maths solutions, you have already learned the differentiation of certain functions like polynomial functions and trigonometric functions. In this chapter, you will get NCERT solutions for class 12 maths chapter 5 continuity and differentiability. If you are interested in the chapter 5 class 12 maths NCERT solutions then you can check NCERT solutions for class 12 other subjects.
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>> Continuity: A function f(x) is continuous at a point x = a if:
f(a) exists (finite, definite, and real).
lim(x → a) f(x) exists.
lim(x → a) f(x) = f(a).
>> Discontinuity: f(x) is discontinuous in an interval if it is discontinuous at any point in that interval.
Algebra of Continuous Functions:
Sum, difference, product, and quotient of continuous functions are continuous.
Differentiation:
The derivative of f(x) at x = a, denoted as f'(a), represents the slope of the tangent line to the graph.
Chain Rule:
If f = v o u, where t = u(x), and if both dt/dx and dv/dx exist, then: df/dx = dv/dt * dt/dx.
Derivatives of Some Standard Functions:
d/dx(xn) = nxn-1
d/dx(sin x) = cos x
d/dx(cos x) = -sin x
d/dx(tan x) = sec2 x
d/dx(cot x) = -csc2 x
d/dx(sec x) = sec x * tan x
d/dx(csc x) = -csc x * cot x
d/dx(ax) = ax * ln(a)
d/dx(ex) = ex
d/dx(ln x) = 1/x
As per latest 2024 syllabus. Physics formulas, equations, & laws of class 11 & 12th chapters
Mean Value Theorem:
Mean Value Theorem states that if f(x) is continuous on [a, b] and differentiable on (a, b), then there exists some c in (a, b) such that: f'(c) = (f(b) - f(a)) / (b - a).
Rolle's Theorem:
Rolle's Theorem states that if f(x) is continuous on [a, b], differentiable on (a, b), and f(a) = f(b), then there exists some c in (a, b) such that f'(c) = 0.
Lagrange's Mean Value Theorem:
Lagrange's Mean Value Theorem states that if f(x) is continuous on [a, b] and differentiable on (a, b), then there exists some c in (a, b) such that:
f'(c) = (f(b) - f(a)) / (b - a).
Free download Continuity And Differentiability Class 12 NCERT Solutions for CBSE Exam.
NCERT Continuity And Differentiability Class 12 Solutions : Excercise: 5.1
Question:1 . Prove that the function is continuous at and at
Answer:
Given function is
Hence, function is continous at x = 0
Hence, function is continous at x = -3
Hence, function is continuous at x = 5
Question:2 . Examine the continuity of the function
Answer:
Given function is
at x = 3
Hence, function is continous at x = 3
Question:3 Examine the following functions for continuity.
Answer:
Given function is
Our function is defined for every real number say k
and value at x = k ,
and also,
Hence, the function is continuous at every real number
Question:3 b) Examine the following functions for continuity.
Answer:
Given function is
For every real number k ,
We get,
Hence, function continuous for every real value of x,
Question:3 c) Examine the following functions for continuity.
Answer:
Given function is
For every real number k ,
We gwt,
Hence, function continuous for every real value of x ,
Question:3 d) Examine the following functions for continuity.
Answer:
Given function is
for x > 5 , f(x) = x - 5
for x < 5 , f(x) = 5 - x
SO, different cases are their
case(i) x > 5
for every real number k > 5 , f(x) = x - 5 is defined
Hence, function f(x) = x - 5 is continous for x > 5
case (ii) x < 5
for every real number k < 5 , f(x) = 5 - x is defined
Hence, function f(x) = 5 - x is continous for x < 5
case(iii) x = 5
for x = 5 , f(x) = x - 5 is defined
Hence, function f(x) = x - 5 is continous for x = 5
Hence, the function is continuous for each and every real number
Question:4 . Prove that the function is continuous at x = n, where n is a positive integer
Answer:
GIven function is
the function is defined for all positive integer, n
Hence, the function is continuous at x = n, where n is a positive integer
Question:5. Is the function f defined by
continuous at x = 0? At x = 1? At x = 2?
Answer:
Given function is
function is defined at x = 0 and its value is 0
Hence , given function is continous at x = 0
given function is defined for x = 1
Now, for x = 1 Right-hand limit and left-hand limit are not equal
R.H.L L.H.L.
Therefore, given function is not continous at x =1
Given function is defined for x = 2 and its value at x = 2 is 5
Hence, given function is continous at x = 2
Question:6. Find all points of discontinuity of f, where f is defined by
Answer:
Given function is
given function is defined for every real number k
There are different cases for the given function
case(i) k > 2
Hence, given function is continuous for each value of k > 2
case(ii) k < 2
Hence, given function is continuous for each value of k < 2
case(iii) x = 2
Right hand limit at x= 2 Left hand limit at x = 2
Therefore, x = 2 is the point of discontinuity
Question:7. Find all points of discontinuity of f, where f is defined by
Answer:
Given function is
GIven function is defined for every real number k
Different cases are their
case (i) k < -3
Hence, given function is continuous for every value of k < -3
case(ii) k = -3
Hence, given function is continous for x = -3
case(iii) -3 < k < 3
Hence, for every value of k in -3 < k < 3 given function is continous
case(iv) k = 3
Hence . x = 3 is the point of discontinuity
case(v) k > 3
Hence, given function is continuous for each and every value of k > 3
Question:8. Find all points of discontinuity of f, where f is defined by
Answer:
Given function is
if x > 0 ,
if x < 0 ,
given function is defined for every real number k
Now,
case(i) k < 0
Hence, given function is continuous for every value of k < 0
case(ii) k > 0
Hence, given function is continuous for every value of k > 0
case(iii) x = 0
Hence, 0 is the only point of discontinuity
Question:9. Find all points of discontinuity of f, where f is defined by
Answer:
Given function is
if x < 0 ,
Now, for any value of x, the value of our function is -1
Therefore, the given function is continuous for each and every value of x
Hence, no point of discontinuity
Question:10. Find all points of discontinuity of f, where f is defined by
Answer:
Given function is
given function is defined for every real number k
There are different cases for the given function
case(i) k > 1
Hence, given function is continuous for each value of k > 1
case(ii) k < 1
Hence, given function is continuous for each value of k < 1
case(iii) x = 1
Hence, at x = 2 given function is continuous
Therefore, no point of discontinuity
Question:11. Find all points of discontinuity of f, where f is defined by
Answer:
Given function is
given function is defined for every real number k
There are different cases for the given function
case(i) k > 2
Hence, given function is continuous for each value of k > 2
case(ii) k < 2
Hence, given function is continuous for each value of k < 2
case(iii) x = 2
Hence, given function is continuous at x = 2
There, no point of discontinuity
Question:12. Find all points of discontinuity of f, where f is defined by
Answer:
Given function is
given function is defined for every real number k
There are different cases for the given function
case(i) k > 1
Hence, given function is continuous for each value of k > 1
case(ii) k < 1
Hence, given function is continuous for each value of k < 1
case(iii) x = 1
Hence, x = 1 is the point of discontinuity
Question:13. Is the function defined by
Answer:
Given function is
given function is defined for every real number k
There are different cases for the given function
case(i) k > 1
Hence, given function is continuous for each value of k > 1
case(ii) k < 1
Hence, given function is continuous for each value of k < 1
case(iii) x = 1
Hence, x = 1 is the point of discontinuity
Question:14. Discuss the continuity of the function f, where f is defined by
Answer:
Given function is
GIven function is defined for every real number k
Different cases are their
case (i) k < 1
Hence, given function is continous for every value of k < 1
case(ii) k = 1
Hence, given function is discontinous at x = 1
Therefore, x = 1 is he point od discontinuity
case(iii) 1 < k < 3
Hence, for every value of k in 1 < k < 3 given function is continous
case(iv) k = 3
Hence. x = 3 is the point of discontinuity
case(v) k > 3
Hence, given function is continous for each and every value of k > 3
case(vi) when k < 3
Hence, for every value of k in k < 3 given function is continous
Question:15 Discuss the continuity of the function f, where f is defined by
Answer:
Given function is
Given function is satisfies for the all real values of x
case (i) k < 0
Hence, function is continuous for all values of x < 0
case (ii) x = 0
L.H.L at x= 0
R.H.L. at x = 0
L.H.L. = R.H.L. = f(0)
Hence, function is continuous at x = 0
case (iii) k > 0
Hence , function is continuous for all values of x > 0
case (iv) k < 1
Hence , function is continuous for all values of x < 1
case (v) k > 1
Hence , function is continuous for all values of x > 1
case (vi) x = 1
Hence, function is not continuous at x = 1
Question:16. Discuss the continuity of the function f, where f is defined by
Answer:
Given function is
GIven function is defined for every real number k
Different cases are their
case (i) k < -1
Hence, given function is continuous for every value of k < -1
case(ii) k = -1
Hence, given function is continous at x = -1
case(iii) k > -1
Hence, given function is continous for all values of x > -1
case(vi) -1 < k < 1
Hence, for every value of k in -1 < k < 1 given function is continous
case(v) k = 1
Hence.at x =1 function is continous
case(vi) k > 1
Hence, given function is continous for each and every value of k > 1
case(vii) when k < 1
Hence, for every value of k in k < 1 given function is continuous
Therefore, continuous at all points
Question:17. Find the relationship between a and b so that the function f defined by
is continuous at x = 3.
Answer:
Given function is
For the function to be continuous at x = 3 , R.H.L. must be equal to L.H.L.
For the function to be continuous
Question:18. For what value of l is the function defined by
continuous at x = 0? What about continuity at x = 1?
Answer:
Given function is
For the function to be continuous at x = 0 , R.H.L. must be equal to L.H.L.
For the function to be continuous
Hence, for no value of function is continuous at x = 0
For x = 1
Hence, given function is continuous at x =1
Answer:
Given function is
Given is defined for all real numbers k
Hence, by this, we can say that the function defined by is discontinuous at all integral points
Question:20. Is the function defined by continuous at x = ?
Answer:
Given function is
Clearly, Given function is defined at x =
Hence, the function defined by continuous at x =
Question:21. Discuss the continuity of the following functions:
a)
Answer:
Given function is
Given function is defined for all real number
We, know that if two function g(x) and h(x) are continuous then g(x)+h(x) , g(x)-h(x) , g(x).h(x) allare continuous
Lets take g(x) = sin x and h(x) = cos x
Let suppose x = c + h
if
Hence, function is a continuous function
Now,
h(x) = cos x
Let suppose x = c + h
if
Hence, function is a continuous function
We proved independently that sin x and cos x is continous function
So, we can say that
f(x) = g(x) + h(x) = sin x + cos x is also a continuous function
Question:21. b) Discuss the continuity of the following functions:
Answer:
Given function is
Given function is defined for all real number
We, know that if two function g(x) and h(x) are continuous then g(x)+h(x) , g(x)-h(x) , g(x).h(x) allare continuous
Lets take g(x) = sin x and h(x) = cos x
Let suppose x = c + h
if
Hence, function is a continuous function
Now,
h(x) = cos x
Let suppose x = c + h
if
Hence, function is a continuous function
We proved independently that sin x and cos x is continous function
So, we can say that
f(x) = g(x) - h(x) = sin x - cos x is also a continuous function
Question:21 c) Discuss the continuity of the following functions:
Answer:
Given function is
Given function is defined for all real number
We, know that if two function g(x) and h(x) are continuous then g(x)+h(x) , g(x)-h(x) , g(x).h(x) allare continuous
Lets take g(x) = sin x and h(x) = cos x
Let suppose x = c + h
if
Hence, function is a continuous function
Now,
h(x) = cos x
Let suppose x = c + h
if
Hence, function is a continuous function
We proved independently that sin x and cos x is continous function
So, we can say that
f(x) = g(x).h(x) = sin x .cos x is also a continuous function
Question:22. Discuss the continuity of the cosine, cosecant, secant and cotangent functions.
Answer:
We, know that if two function g(x) and h(x) are continuous then
Lets take g(x) = sin x and h(x) = cos x
Let suppose x = c + h
if
Hence, function is a continuous function
Now,
h(x) = cos x
Let suppose x = c + h
if
Hence, the function is a continuous function
We proved independently that sin x and cos x is a continous function
So, we can say that
cosec x = is also continuous except at
sec x = is also continuous except at
cot x = is also continuous except at
Question:23. Find all points of discontinuity of f, where
Answer:
Given function is
Hence, the function is continuous
Therefore, no point of discontinuity
Question:24. Determine if f defined by
is a continuous function?
Answer:
Given function is
Given function is defined for all real numbers k
when x = 0
Hence, function is continuous at x = 0
when
Hence, the given function is continuous for all points
Question:25 . Examine the continuity of f, where f is defined by
Answer:
Given function is
Given function is defined for all real number
We, know that if two function g(x) and h(x) are continuous then g(x)+h(x) , g(x)-h(x) , g(x).h(x) allare continuous
Lets take g(x) = sin x and h(x) = cos x
Let suppose x = c + h
if
Hence, function is a continuous function
Now,
h(x) = cos x
Let suppose x = c + h
if
Hence, function is a continuous function
We proved independently that sin x and cos x is continous function
So, we can say that
f(x) = g(x) - h(x) = sin x - cos x is also a continuous function
When x = 0
Hence, function is also continuous at x = 0
Question:26. Find the values of k so that the function f is continuous at the indicated point in Exercises
Answer:
Given function is
When
For the function to be continuous
Therefore, the values of k so that the function f is continuous is 6
Question:27 . Find the values of k so that the function f is continuous at the indicated point in Exercises
Answer:
Given function is
When x = 2
For the function to be continuous
f(2) = R.H.L. = LH.L.
Hence, the values of k so that the function f is continuous at x= 2 is
Question:28 . Find the values of k so that the function f is continuous at the indicated point in Exercises
Answer:
Given function is
When x =
For the function to be continuous
f( ) = R.H.L. = LH.L.
Hence, the values of k so that the function f is continuous at x= is
Question:29 Find the values of k so that the function f is continuous at the indicated point in Exercises
Answer:
Given function is
When x = 5
For the function to be continuous
f(5) = R.H.L. = LH.L.
Hence, the values of k so that the function f is continuous at x= 5 is
Question:30 Find the values of a and b such that the function defined by
is a continuous function.
Answer:
Given continuous function is
The function is continuous so
By solving equation (i) and (ii)
a = 2 and b = 1
Hence, values of a and b such that the function defined by is a continuous function is 2 and 1 respectively
Question:31. Show that the function defined by is a continuous function.
Answer:
Given function is
given function is defined for all real values of x
Let x = k + h
if
Hence, the function is a continuous function
Question:32. Show that the function defined by is a continuous function.
Answer:
Given function is
given function is defined for all values of x
f = g o h , g(x) = |x| and h(x) = cos x
Now,
g(x) is defined for all real numbers k
case(i) k < 0
Hence, g(x) is continuous when k < 0
case (ii) k > 0
Hence, g(x) is continuous when k > 0
case (iii) k = 0
Hence, g(x) is continuous when k = 0
Therefore, g(x) = |x| is continuous for all real values of x
Now,
h(x) = cos x
Let suppose x = c + h
if
Hence, function is a continuous function
g(x) is continuous , h(x) is continuous
Therefore, f(x) = g o h is also continuous
Question:33 . Examine that sin | x| is a continuous function.
Answer:
Given function is
f(x) = sin |x|
f(x) = h o g , h(x) = sin x and g(x) = |x|
Now,
g(x) is defined for all real numbers k
case(i) k < 0
Hence, g(x) is continuous when k < 0
case (ii) k > 0
Hence, g(x) is continuous when k > 0
case (iii) k = 0
Hence, g(x) is continuous when k = 0
Therefore, g(x) = |x| is continuous for all real values of x
Now,
h(x) = sin x
Let suppose x = c + h
if
Hence, function is a continuous function
g(x) is continuous , h(x) is continuous
Therefore, f(x) = h o g is also continuous
Question:34. Find all the points of discontinuity of f defined by
Answer:
Given function is
Let g(x) = |x| and h(x) = |x+1|
Now,
g(x) is defined for all real numbers k
case(i) k < 0
Hence, g(x) is continuous when k < 0
case (ii) k > 0
Hence, g(x) is continuous when k > 0
case (iii) k = 0
Hence, g(x) is continuous when k = 0
Therefore, g(x) = |x| is continuous for all real values of x
Now,
g(x) is defined for all real numbers k
case(i) k < -1
Hence, h(x) is continuous when k < -1
case (ii) k > -1
Hence, h(x) is continuous when k > -1
case (iii) k = -1
Hence, h(x) is continuous when k = -1
Therefore, h(x) = |x+1| is continuous for all real values of x
g(x) is continuous and h(x) is continuous
Therefore, f(x) = g(x) - h(x) = |x| - |x+1| is also continuous
NCERT class 12 maths chapter 5 question answer: Excercise: 5.2
Question:1. Differentiate the functions with respect to x in
Answer:
Given function is
when we differentiate it w.r.t. x.
Lets take . then,
(By chain rule)
Now,
Therefore, the answer is
Question:2. Differentiate the functions with respect to x in
Answer:
Given function is
Lets take then,
( By chain rule)
Now,
Therefore, the answer is
Question:3. Differentiate the functions with respect to x in
Answer:
Given function is
when we differentiate it w.r.t. x.
Lets take . then,
(By chain rule)
Now,
Therefore, the answer is
Question:4 . Differentiate the functions with respect to x in
Answer:
Given function is
when we differentiate it w.r.t. x.
Lets take . then,
take . then,
(By chain rule)
Now,
Therefore, the answer is
Question:5. Differentiate the functions with respect to x in
Answer:
Given function is
We know that,
and
Lets take
Then,
(By chain rule)
-(i)
Similarly,
-(ii)
Now, put (i) and (ii) in
Therefore, the answer is
Question:6. Differentiate the functions with respect to x in
Answer:
Given function is
Differentitation w.r.t. x is
Lets take
Our functions become,
and
Now,
( By chain rule)
-(i)
Similarly,
-(ii)
Put (i) and (ii) in
Therefore, the answer is
Question:7. Differentiate the functions with respect to x in
Answer:
Give function is
Let's take
Now, take
Differentiation w.r.t. x
-(By chain rule)
So,
( Multiply and divide by and multiply and divide by )
There, the answer is
Question:8 Differentiate the functions with respect to x in
Answer:
Let us assume :
Differentiating y with respect to x, we get :
or
or
Question:9 . Prove that the function f given by is not differentiable at x = 1.
Answer:
Given function is
We know that any function is differentiable when both
and are finite and equal
Required condition for function to be differential at x = 1 is
Now, Left-hand limit of a function at x = 1 is
Right-hand limit of a function at x = 1 is
Now, it is clear that
R.H.L. at x= 1 L.H.L. at x= 1
Therefore, function is not differentiable at x = 1
Question:10. Prove that the greatest integer function defined by is not differentiable at
Answer:
Given function is
We know that any function is differentiable when both
and are finite and equal
Required condition for function to be differential at x = 1 is
Now, Left-hand limit of the function at x = 1 is
Right-hand limit of the function at x = 1 is
Now, it is clear that
R.H.L. at x= 1 L.H.L. at x= 1 and L.H.L. is not finite as well
Therefore, function is not differentiable at x = 1
Similary, for x = 2
Required condition for function to be differential at x = 2 is
Now, Left-hand limit of the function at x = 2 is
Right-hand limit of the function at x = 1 is
Now, it is clear that
R.H.L. at x= 2 L.H.L. at x= 2 and L.H.L. is not finite as well
Therefore, function is not differentiable at x = 2
NCERT class 12 maths chapter 5 question answer: Exercise: 5.3
Question:1. Find dy/dx in the following:
Answer:
Given function is
We can rewrite it as
Now, differentiation w.r.t. x is
Therefore, the answer is
Question:2. Find dy/dx in the following:
Answer:
Given function is
We can rewrite it as
Now, differentiation w.r.t. x is
Therefore, the answer is
Question:3. Find dy/dx in the following:
Answer:
Given function is
We can rewrite it as
Now, differentiation w.r.t. x is
Therefore, the answer is
Question:4. Find dy/dx in the following:
Answer:
Given function is
We can rewrite it as
Now, differentiation w.r.t. x is
Therefore, the answer is
Question:5. Find dy/dx in the following:
Answer:
Given function is
We can rewrite it as
Now, differentiation w.r.t. x is
Therefore, the answer is
Question:6 Find dy/dx in the following:
Answer:
Given function is
We can rewrite it as
Now, differentiation w.r.t. x is
Therefore, the answer is
Question:7 . Find dy/dx in the following:
Answer:
Given function is
Now, differentiation w.r.t. x is
Therefore, the answer is
Question:8. Find dy/dx in the following:
Answer:
Given function is
We can rewrite it as
Now, differentiation w.r.t. x is
Therefore, the answer is
Question:9 Find dy/dx in the following:
Answer:
Given function is
Lets consider
Then,
Now,
Our equation reduces to
Now, differentiation w.r.t. x is
Therefore, the answer is
Question:10. Find dy/dx in the following:
Answer:
Given function is
Lets consider
Then,
Now,
Our equation reduces to
Now, differentiation w.r.t. x is
Therefore, the answer is
Question:11. Find dy/dx in the following:
Answer:
Given function is
Let's consider
Then,
Now,
Our equation reduces to
Now, differentiation w.r.t. x is
Therefore, the answer is
Question:12 . Find dy/dx in the following:
Answer:
Given function is
We can rewrite it as
Let's consider
Then,
Now,
Our equation reduces to
Now, differentiation w.r.t. x is
Therefore, the answer is
Question:13. Find dy/dx in the following:
Answer:
Given function is
We can rewrite it as
Let's consider
Then,
Now,
Our equation reduces to
Now, differentiation w.r.t. x is
Therefore, the answer is
Question:14 . Find dy/dx in the following:
Answer:
Given function is
Lets take
Then,
And
Now, our equation reduces to
Now, differentiation w.r.t. x
Therefore, the answer is
Question:15 . Find dy/dx in the following:
Answer:
Given function is
Let's take
Then,
And
Now, our equation reduces to
Now, differentiation w.r.t. x
Therefore, the answer is
NCERT class 12 maths chapter 5 question answer: Exercise 5.4
Question:1. Differentiate the following w.r.t. x:
Answer:
Given function is
We differentiate with the help of Quotient rule
Therefore, the answer is
Question:2 . Differentiate the following w.r.t. x:
Answer:
Given function is
Let
Then,
Now, differentiation w.r.t. x
-(i)
Put this value in our equation (i)
Question:3 . Differentiate the following w.r.t. x:
Answer:
Given function is
Let
Then,
Now, differentiation w.r.t. x
-(i)
Put this value in our equation (i)
Therefore, the answer is
Question:4. Differentiate the following w.r.t. x:
Answer:
Given function is
Let's take
Now, our function reduces to
Now,
-(i)
And
Put this value in our equation (i)
Therefore, the answer is
Question:5 . Differentiate the following w.r.t. x:
Answer:
Given function is
Let's take
Now, our function reduces to
Now,
-(i)
And
Put this value in our equation (i)
Therefore, the answer is
Question:6 . Differentiate the following w.r.t. x:
Answer:
Given function is
Now, differentiation w.r.t. x is
Therefore, answer is
Question:7 . Differentiate the following w.r.t. x:
Answer:
Given function is
Lets take
Now, our function reduces to
Now,
-(i)
And
Put this value in our equation (i)
Therefore, the answer is
Question:8 Differentiate the following w.r.t. x:
Answer:
Given function is
Lets take
Now, our function reduces to
Now,
-(i)
And
Put this value in our equation (i)
Therefore, the answer is
Question:9. Differentiate the following w.r.t. x:
Answer:
Given function is
We differentiate with the help of Quotient rule
Therefore, the answer is
Question:10. Differentiate the following w.r.t. x:
Answer:
Given function is
Lets take
Then , our function reduces to
Now, differentiation w.r.t. x is
-(i)
And
Put this value in our equation (i)
Therefore, the answer is
Class 12 Maths Chapter 5 NCERT solutions: Exercise: 5.5
Question:1 Differentiate the functions w.r.t. x.
Answer:
Given function is
Now, take log on both sides
Now, differentiation w.r.t. x
There, the answer is
Question:2. Differentiate the functions w.r.t. x.
Answer:
Given function is
Take log on both the sides
Now, differentiation w.r.t. x is
Therefore, the answer is
Question:3 Differentiate the functions w.r.t. x.
Answer:
Given function is
take log on both the sides
Now, differentiation w.r.t x is
Therefore, the answer is
Question:4 Differentiate the functions w.r.t. x.
Answer:
Given function is
Let's take
take log on both the sides
Now, differentiation w.r.t x is
Similarly, take
Now, take log on both sides and differentiate w.r.t. x
Now,
Therefore, the answer is
Question:5 Differentiate the functions w.r.t. x.
Answer:
Given function is
Take log on both sides
Now, differentiate w.r.t. x we get,
Therefore, the answer is
Question:6 Differentiate the functions w.r.t. x.
Answer:
Given function is
Let's take
Now, take log on both sides
Now, differentiate w.r.t. x
we get,
Similarly, take
Now, take log on both sides
Now, differentiate w.r.t. x
We get,
Now,
Therefore, the answer is
Question:7 Differentiate the functions w.r.t. x.
Answer:
Given function is
Let's take
Now, take log on both the sides
Now, differentiate w.r.t. x
we get,
Similarly, take
Now, take log on both sides
Now, differentiate w.r.t. x
We get,
Now,
Therefore, the answer is
Question:8 Differentiate the functions w.r.t. x.
Answer:
Given function is
Lets take
Now, take log on both the sides
Now, differentiate w.r.t. x
we get,
Similarly, take
Now, differentiate w.r.t. x
We get,
Now,
Therefore, the answer is
Question:9 Differentiate the functions w.r.t. x
Answer:
Given function is
Now, take
Now, take log on both sides
Now, differentiate it w.r.t. x
we get,
Similarly, take
Now, take log on both the sides
Now, differentiate it w.r.t. x
we get,
Now,
Therefore, the answer is
Question:10 Differentiate the functions w.r.t. x.
Answer:
Given function is
Take
Take log on both the sides
Now, differentiate w.r.t. x
we get,
Similarly,
take
Now. differentiate it w.r.t. x
we get,
Now,
Therefore, the answer is
Question:11 Differentiate the functions w.r.t. x.
Answer:
Given function is
Let's take
Now, take log on both sides
Now, differentiate w.r.t. x
we get,
Similarly, take
Now, take log on both the sides
Now, differentiate w.r.t. x
we get,
Now,
Therefore, the answer is
Question:12 Find dy/dx of the functions given in Exercises 12 to 15
Answer:
Given function is
Now, take
take log on both sides
Now, differentiate w.r.t x
we get,
Similarly, take
Now, take log on both sides
Now, differentiate w.r.t. x
we get,
Now,
Therefore, the answer is
Question:13 Find dy/dx of the functions given in Exercises 12 to 15.
Answer:
Given function is
Now, take
take log on both sides
Now, differentiate w.r.t x
we get,
Similarly, take
Now, take log on both sides
Now, differentiate w.r.t. x
we get,
Now,
Therefore, the answer is
Question:14 Find dy/dx of the functions given in Exercises 12 to 15.
Answer:
Given function is
Now, take log on both the sides
Now, differentiate w.r.t x
By taking similar terms on the same side
We get,
Therefore, the answer is
Question:15 Find dy/dx of the functions given in Exercises 12 to 15.
Answer:
Given function is
Now, take take log on both the sides
Now, differentiate w.r.t x
By taking similar terms on same side
We get,
Therefore, the answer is
Question:16 Find the derivative of the function given by and hence find
Answer:
Given function is
Take log on both sides
NOW, differentiate w.r.t. x
Therefore,
Now, the vale of is
Question:17 (1) Differentiate in three ways mentioned below:
(i) by using product rule
Answer:
Given function is
Now, we need to differentiate using the product rule
Therefore, the answer is
Question:17 (2) Differentiate in three ways mentioned below:
(ii) by expanding the product to obtain a single polynomial.
Answer:
Given function is
Multiply both to obtain a single higher degree polynomial
Now, differentiate w.r.t. x
we get,
Therefore, the answer is
Question:17 (3) Differentiate in three ways mentioned below:
(iii) by logarithmic differentiation.
Do they all give the same answer?
Answer:
Given function is
Now, take log on both the sides
Now, differentiate w.r.t. x
we get,
Therefore, the answer is
And yes they all give the same answer
Question:18 If u, v and w are functions of x, then show that in two ways - first by repeated application of product rule, second by logarithmic differentiation.
Answer:
It is given that u, v and w are the functions of x
Let
Now, we differentiate using product rule w.r.t x
First, take
Now,
-(i)
Now, again by the product rule
Put this in equation (i)
we get,
Hence, by product rule we proved it
Now, by taking the log
Again take
Now, take log on both sides
Now, differentiate w.r.t. x
we get,
Hence, we proved it by taking the log
Class 12 Maths Chapter 5 NCERT solutions: Exercise:5.6
Answer:
Given equations are
Now, differentiate both w.r.t t
We get,
Similarly,
Now,
Therefore, the answer is
Answer:
Given equations are
Now, differentiate both w.r.t
We get,
Similarly,
Now,
Therefore, answer is
Answer:
Given equations are
Now, differentiate both w.r.t t
We get,
Similarly,
Now,
Therefore, the answer is
Answer:
Given equations are
Now, differentiate both w.r.t t
We get,
Similarly,
Now,
Therefore, the answer is
Answer:
Given equations are
Now, differentiate both w.r.t
We get,
Similarly,
Now,
Therefore, answer is
Answer:
Given equations are
Now, differentiate both w.r.t
We get,
Similarly,
Now,
Therefore, the answer is
Answer:
Given equations are
Now, differentiate both w.r.t t
We get,
Similarly,
Now,
Therefore, the answer is
Answer:
Given equations are
Now, differentiate both w.r.t t
We get,
Similarly,
Now,
Therefore, the answer is
Answer:
Given equations are
Now, differentiate both w.r.t
We get,
Similarly,
Now,
Therefore, the answer is
Answer:
Given equations are
Now, differentiate both w.r.t
We get,
Similarly,
Now,
Therefore, the answer is
Question:11 If , show that
Answer:
Given equations are
differentiating with respect to x
Class 12 Maths Chapter 5 NCERT solutions: Exercise: 5.7
Question:1 Find the second order derivatives of the functions given in Exercises 1 to 10.
Answer:
Given function is
Now, differentiation w.r.t. x
Now, second order derivative
Therefore, the second order derivative is
Question:2 Find the second order derivatives of the functions given in Exercises 1 to 10.
Answer:
Given function is
Now, differentiation w.r.t. x
Now, the second-order derivative is
Therefore, second-order derivative is
Question:3 Find the second order derivatives of the functions given in Exercises 1 to 10.
Answer:
Given function is
Now, differentiation w.r.t. x
Now, the second-order derivative is
Therefore, the second-order derivative is
Question:4 Find the second order derivatives of the functions given in Exercises 1 to 10.
Answer:
Given function is
Now, differentiation w.r.t. x
Now, second order derivative is
Therefore, second order derivative is
Question:5 Find the second order derivatives of the functions given in Exercises 1 to 10.
Answer:
Given function is
Now, differentiation w.r.t. x
Now, the second-order derivative is
Therefore, the second-order derivative is
Question:6 Find the second order derivatives of the functions given in Exercises 1 to 10.
Answer:
Given function is
Now, differentiation w.r.t. x
Now, second order derivative is
Therefore, second order derivative is
Question:7 Find the second order derivatives of the functions given in Exercises 1 to 10.
Answer:
Given function is
Now, differentiation w.r.t. x
Now, second order derivative is
Therefore, second order derivative is
Question:8 Find the second order derivatives of the functions given in Exercises 1 to 10.
Answer:
Given function is
Now, differentiation w.r.t. x
Now, second order derivative is
Therefore, second order derivative is
Question:9 Find the second order derivatives of the functions given in Exercises 1 to 10.
Answer:
Given function is
Now, differentiation w.r.t. x
Now, second order derivative is
Therefore, second order derivative is
Question:10 Find the second order derivatives of the functions given in Exercises 1 to 10.
Answer:
Given function is
Now, differentiation w.r.t. x
Now, second order derivative is
Using Quotient rule
Therefore, second order derivative is
Question:11 If prove that
Answer:
Given function is
Now, differentiation w.r.t. x
Now, the second-order derivative is
Now,
Hence proved
Question:12 If Find in terms of y alone.
Answer:
Given function is
Now, differentiation w.r.t. x
Now, second order derivative is
-(i)
Now, we want in terms of y
Now, put the value of x in (i)
Therefore, answer is
Question:13 If , show that
Answer:
Given function is
Now, differentiation w.r.t. x
-(i)
Now, second order derivative is
By using the Quotient rule
-(ii)
Now, from equation (i) and (ii) we will get
Now, we need to show
Put the value of from equation (i) and (ii)
Hence proved
Question:14 If , show that
Answer:
Given function is
Now, differentiation w.r.t. x
-(i)
Now, second order derivative is
-(ii)
Now, we need to show
Put the value of from equation (i) and (ii)
Hence proved
Question:15 If , show that
Answer:
Given function is
Now, differentiation w.r.t. x
-(i)
Now, second order derivative is
-(ii)
Now, we need to show
Put the value of from equation (ii)
Hence, L.H.S. = R.H.S.
Hence proved
Question:16 If show that
Answer:
Given function is
We can rewrite it as
Now, differentiation w.r.t. x
-(i)
Now, second order derivative is
-(ii)
Now, we need to show
Put value of from equation (i) and (ii)
Hence, L.H.S. = R.H.S.
Hence proved
Question:17 If show that
Answer:
Given function is
Now, differentiation w.r.t. x
-(i)
Now, the second-order derivative is
By using the quotient rule
-(ii)
Now, we need to show
Put the value from equation (i) and (ii)
Hence, L.H.S. = R.H.S.
Hence proved
Class 12 Maths Chapter 5 NCERT solutions: Excercise: 5.8
Question:1 Verify Rolle’s theorem for the function
Answer:
According to Rolle's theorem function must be
a ) continuous in given closed interval say [x,y]
b ) differentiable in given open interval say (x,y)
c ) f(x) = f(y)
Then their exist a such that
If all these conditions are satisfies then we can verify Rolle's theorem
Given function is
Now, being a polynomial function, is both continuous in [-4,2] and differentiable in (-4,2)
Now,
Similalrly,
Therefore, value of and value of f(x) at -4 and 2 are equal
Now,
According to roll's theorem their is point c , such that
Now,
And
Hence, Rolle's theorem is verified for the given function
Answer:
According to Rolle's theorem function must be
a ) continuous in given closed interval say [x,y]
b ) differentiable in given open interval say (x,y)
c ) f(x) = f(y)
Then their exist a such that
If all these conditions are satisfied then we can verify Rolle's theorem
Given function is
It is clear that Given function is not continuous for each and every point in [5,9]
Now, lets check differentiability of f(x)
L.H.L. at x = n ,
Now,
R.H.L. at x = n ,
We can clearly see that R.H.L. is not equal to L.H.L.
Therefore, the function is not differential in (5,9)
Hence, Rolle's theorem is not applicable for given function ,
Answer:
According to Rolle's theorem function must be
a ) continuous in given closed interval say [x,y]
b ) differentiable in given open interval say (x,y)
c ) f(x) = f(y)
Then their exist a such that
If all these conditions are satisfies then we can verify Rolle's theorem
Given function is
It is clear that Given function is not continuous for each and every point in [-2,2]
Now, lets check differentiability of f(x)
L.H.L. at x = n ,
Now,
R.H.L. at x = n ,
We can clearly see that R.H.L. is not equal to L.H.L.
Therefore, function is not differential in (-2,2)
Hence, Rolle's theorem is not applicable for given function ,
Answer:
According to Rolle's theorem function must be
a ) continuous in given closed interval say [x,y]
b ) differentiable in given open interval say (x,y)
c ) f(x) = f(y)
Then there exist a such that
If all these conditions are satisfied then we can verify Rolle's theorem
Given function is
Now, being a polynomial , function is continuous in [1,2] and differentiable in(1,2)
Now,
And
Therefore,
Therefore, All conditions are not satisfied
Hence, Rolle's theorem is not applicable for given function ,
Question:3 If is a differentiable function and if does not vanish
anywhere, then prove that
Answer:
It is given that
is a differentiable function
Now, f is a differential function. So, f is also a continuous function
We obtain the following results
a ) f is continuous in [-5,5]
b ) f is differentiable in (-5,5)
Then, by Mean value theorem we can say that there exist a c in (-5,5) such that
Now, it is given that does not vanish anywhere
Therefore,
Hence proved
Question:4 Verify Mean Value Theorem, if in the interval [a, b], where
a = 1 and b = 4.
Answer:
Condition for M.V.T.
If
a ) f is continuous in [a,b]
b ) f is differentiable in (a,b)
Then, there exist a c in (a,b) such that
It is given that
and interval is [1,4]
Now, f is a polynomial function , is continuous in[1,4] and differentiable in (1,4)
And
and
Then, by Mean value theorem we can say that their exist a c in (1,4) such that
Now,
And
Hence, mean value theorem is verified for the function
Answer:
Condition for M.V.T.
If
a ) f is continuous in [a,b]
b ) f is differentiable in (a,b)
Then, their exist a c in (a,b) such that
It is given that
and interval is [1,3]
Now, f being a polynomial function , is continuous in[1,3] and differentiable in (1,3)
And
and
Then, by Mean value theorem we can say that their exist a c in (1,4) such that
Now,
And
Hence, mean value theorem is varified for following function and is the only point where f '(c) = 0
Answer:
According to Mean value theorem function
must be
a ) continuous in given closed interval say [a,b]
b ) differentiable in given open interval say (a,b)
Then their exist a such that
If all these conditions are satisfies then we can verify mean value theorem
Given function is
It is clear that Given function is not continuous for each and every point in [5,9]
Now, lets check differentiability of f(x)
L.H.L. at x = n ,
Now,
R.H.L. at x = n ,
We can clearly see that R.H.L. is not equal to L.H.L.
Therefore, function is not differential in (5,9)
Hence, Mean value theorem is not applicable for given function ,
Similaly,
Given function is
It is clear that Given function is not continuous for each and every point in [-2,2]
Now, lets check differentiability of f(x)
L.H.L. at x = n ,
Now,
R.H.L. at x = n ,
We can clearly see that R.H.L. is not equal to L.H.L.
Therefore, function is not differential in (-2,2)
Hence, Mean value theorem is not applicable for given function ,
Similarly,
Given function is
Now, being a polynomial , function is continuous in [1,2] and differentiable in(1,2)
Now,
And
Now,
Now,
And
Therefore, mean value theorem is applicable for the function
NCERT class 12 continuity and differentiability ncert solutions Miscellaneous Excercise
Question:1 Differentiate w.r.t. x the function in Exercises 1 to 11.
Answer:
Given function is
Now, differentiation w.r.t. x is
Therefore, differentiation w.r.t. x is
Question:2 Differentiate w.r.t. x the function in Exercises 1 to 11.
Answer:
Given function is
Now, differentiation w.r.t. x is
Therefore, differentiation w.r.t. x is
Question:3 Differentiate w.r.t. x the function in Exercises 1 to 11.
Answer:
Given function is
Take, log on both the sides
Now, differentiation w.r.t. x is
By using product rule
Therefore, differentiation w.r.t. x is
Question:4 Differentiate w.r.t. x the function in Exercises 1 to 11.
Answer:
Given function is
Now, differentiation w.r.t. x is
Therefore, differentiation w.r.t. x is
Question:5 Differentiate w.r.t. x the function in Exercises 1 to 11.
Answer:
Given function is
Now, differentiation w.r.t. x is
By using the Quotient rule
Therefore, differentiation w.r.t. x is
Question:6 Differentiate w.r.t. x the function in Exercises 1 to 11.
Answer:
Given function is
Now, rationalize the [] part
Given function reduces to
Now, differentiation w.r.t. x is
Therefore, differentiation w.r.t. x is
Question:7 Differentiate w.r.t. x the function in Exercises 1 to 11.
Answer:
Given function is
Take log on both sides
Now, differentiate w.r.t.
Therefore, differentiation w.r.t x is
Question:8 , for some constant a and b.
Answer:
Given function is
Now, differentiation w.r.t x
Therefore, differentiation w.r.t x
Question: 9
Answer:
Given function is
Take log on both the sides
Now, differentiate w.r.t. x
Therefore, differentiation w.r.t x is
Question:10 , for some fixed a > 0 and x > 0
Answer:
Given function is
Lets take
Now, take log on both sides
Now, differentiate w.r.t x
-(i)
Similarly, take
take log on both the sides
Now, differentiate w.r.t x
-(ii)
Similarly, take
take log on both the sides
Now, differentiate w.r.t x
-(iii)
Similarly, take
take log on both the sides
Now, differentiate w.r.t x
-(iv)
Now,
Put values from equation (i) , (ii) ,(iii) and (iv)
Therefore, differentiation w.r.t. x is
Question: 11
Answer:
Given function is
take
Now, take log on both the sides
Now, differentiate w.r.t x
-(i)
Similarly,
take
Now, take log on both the sides
Now, differentiate w.r.t x
-(ii)
Now
Put the value from equation (i) and (ii)
Therefore, differentiation w.r.t x is
Question:12 Find dy/dx if
Answer:
Given equations are
Now, differentiate both y and x w.r.t t independently
And
Now
Therefore, differentiation w.r.t x is
Question:13 Find dy/dx if
Answer:
Given function is
Now, differentiatiate w.r.t. x
Therefore, differentiatiate w.r.t. x is 0
Question:14 If
Answer:
Given function is
Now, squaring both sides
Now, differentiate w.r.t. x is
Hence proved
Question:15 If , for some c > 0, prove that is a constant independent of a and b.
Answer:
Given function is
- (i)
Now, differentiate w.r.t. x
-(ii)
Now, the second derivative
Now, put values from equation (i) and (ii)
Now,
Which is independent of a and b
Hence proved
Question:16 If , with , prove that
Answer:
Given function is
Now, Differentiate w.r.t x
Hence proved
Question:17 If and find
Answer:
Given functions are
and
Now, differentiate both the functions w.r.t. t independently
We get
Similarly,
Now,
Now, the second derivative
Therefore,
Question:18 If , show that f ''(x) exists for all real x and find it.
Answer:
Given function is
Now, differentiate in both the cases
And
In both, the cases f ''(x) exist
Hence, we can say that f ''(x) exists for all real x
and values are
Question:19 Using mathematical induction prove that for all positive integers n.
Answer:
Given equation is
We need to show that for all positive integers n
Now,
For ( n = 1)
Hence, true for n = 1
For (n = k)
Hence, true for n = k
For ( n = k+1)
Hence, (n = k+1) is true whenever (n = k) is true
Therefore, by the principle of mathematical induction we can say that is true for all positive integers n
Question:20 Using the fact that and the differentiation,
obtain the sum formula for cosines.
Answer:
Given function is
Now, differentiate w.r.t. x
Hence, we get the formula by differentiation of sin(A + B)
Answer:
Consider f(x) = |x| + |x +1|
We know that modulus functions are continuous everywhere and sum of two continuous function is also a continuous function
Therefore, our function f(x) is continuous
Now,
If Lets differentiability of our function at x = 0 and x= -1
L.H.D. at x = 0
R.H.L. at x = 0
R.H.L. is not equal to L.H.L.
Hence.at x = 0 is the function is not differentiable
Now, Similarly
R.H.L. at x = -1
L.H.L. at x = -1
L.H.L. is not equal to R.H.L, so not differentiable at x=-1
Hence, exactly two points where it is not differentiable
Question:22 If , prove that
Answer:
Given that
We can rewrite it as
Now, differentiate w.r.t x
we will get
Hence proved
Question:23 If , show that
Answer:
Given function is
Now, differentiate w.r.t x we will get
-(i)
Now, again differentiate w.r.t x
-(ii)
Now, we need to show that
Put the values from equation (i) and (ii)
Hence proved
If you are looking for continuity and differentiability class 12 NCERT solutions of exercises then these are listed below.
5.1 Introduction
5.2 Continuity
5.2.1 Algebra of continuous functions
5.3. Differentiability
5.3.1 Derivatives of composite functions
5.3.2 Derivatives of implicit functions
5.3.3 Derivatives of inverse trigonometric functions
5.4 Exponential and Logarithmic Functions
5.5. Logarithmic Differentiation
5.6 Derivatives of Functions in Parametric Form
The mathematical definition of Continuity and Differentiability -
Let f be a real function and c be a point in the domain of f. Then f is continuous at c if . A function f is differentiable at point c in its domain if it is continuous at point c. A function is said to be differentiable in an interval [a, b] if it is differentiable at every point of [a, b].
'Continuity and differentiability' is one of the very important and time-consuming chapters of the NCERT Class 12 maths syllabus. It contains 8 exercises with 121 questions and also 23 questions in the miscellaneous exercise. In this article, you will find all NCERT solutions for class 12 maths chapter 5 continuity and differentiability including miscellaneous exercises.
Also read,
NCERT exemplar solutions class 12 maths chapter 5
The main topics covered in chapter 5 maths class 12 are:
Continuity
A function is continuous at a given point if the left-hand limit, right-hand limit and value of function exist and are equal. In this class 12 NCERT topics elaborate concepts related to continuity, point of discontinuity, algebra of continuous function. Continuity and Differentiability class 12 solutions include a comprehensive module of quality questions.
Differentiability
This ch 5 maths class 12 discuss differentiability concepts of different functions including derivatives of composite functions, derivatives of implicit functions, derivatives of inverse trigonometric functions. To get command on these concepts you can refer to NCERT solutions for class 12 maths chapter 5.
Exponential and Logarithmic Functions
This ch 5 maths class 12 also includes concepts of exponential and logarithmic functions including natural log and their graphical representation. maths class 12 chapter 5 also contains fundamental properties of the logarithmic function. You can refer to class 12 NCERT solutions for questions about these concepts.
Logarithmic Differentiation
this class 12 ncert chapter discusses a special technique of differentiation known as logarithmic differentiation. to get command of these concepts you can go through the NCERT solution for class 12 maths chapter 5.
Derivatives of Functions in Parametric Forms
concepts to differentiate a function which is not implicit and explicit but given in the parametric form are explained in this chapter. Continuity and Differentiability class 12 solutions include problems to understand the concepts.
ch 5 maths class 12 also discuss in detail the concepts of second-order derivative, mean value theorem, Rolle's theorem. for questions on these concepts, you can browse NCERT solutions for class 12 chapter 5.
Topics enumerated in class 12 NCERT are very important and students are suggested to go through all the concepts discussed in the topics. Questions related to all the above topics are covered in the NCERT solutions for class 12 maths chapter 5
Also read,
Chapter 1 | NCERT Solutions for Class 12 Maths Chapter 1 Relations and Functions |
Chapter 2 | NCERT solutions for class 12 maths chapter 2 Inverse Trigonometric Functions |
Chapter 3 | NCERT solutions for class 12 maths chapter 3 Matrices |
Chapter 4 | NCERT solutions for class 12 maths chapter 4 Determinants |
Chapter 5 | NCERT solutions for class 12 maths chapter 5 Continuity and Differentiability |
Chapter 6 | NCERT solutions for class 12 maths chapter 6 Application of Derivatives |
Chapter 7 | NCERT solutions for class 12 maths chapter 7 Integrals |
Chapter 8 | NCERT solutions for class 12 maths chapter 8 Application of Integrals |
Chapter 9 | NCERT solutions for class 12 maths chapter 9 Differential Equations |
Chapter 10 | NCERT solutions for class 12 maths chapter 10 Vector Algebra |
Chapter 11 | NCERT solutions for class 12 maths chapter 11 Three Dimensional Geometry |
Chapter 12 | NCERT solutions for class 12 maths chapter 12 Linear Programming |
Chapter 13 | NCERT solutions for class 12 maths chapter 13 Probability |
NCERT solutions for class 12 maths chapter 5 continuity and differentiability are very helpful in the preparation of this chapter. But here are some tips to get command on this chapter.
Also check,
Happy learning!!!
Basic concepts of continuity and differentiability, derivatives of composite functions, derivatives of implicit functions, derivatives of inverse trigonometric functions, exponential and logarithmic functions, logarithmic differentiation, derivatives of functions in parametric form are the important topics in this chapter. Practice these class 12 maths ch 5 question answer to command the concepts.
The maths chapter 5 class 12 NCERT solutions created by the experts at Careers360 offer numerous advantages to students preparing for their board exams. These solutions provide comprehensive explanations of each topic, which help students achieve high scores. Additionally, the solutions are based on the latest CBSE syllabus for the 2022-23 academic year. Furthermore, these solutions also assist students in preparing for other competitive exams such as JEE Main and JEE Advanced. For ease, Students can study continuity and differentiability pdf both online and offline
NCERT is the best book for CBSE class 12 maths. Most of the questions in CBSE board exam are directly asked from NCERT textbook. All you need to do is rigorous practice of all the problems given in the NCERT textbook.
According to the given information, there are 8 exercises in NCERT Solutions for maths chapter 5 class 12 . The following is the number of questions in each exercise:
Exercise 5.1: 34 questions
Exercise 5.2: 10 questions
Exercise 5.3: 15 questions
Exercise 5.4: 10 questions
Exercise 5.5: 18 questions
Exercise 5.6: 11 questions
Exercise 5.7: 17 questions
Exercise 5.8: 6 questions
Additionally, there is a Miscellaneous Exercise with 23 questions.
Generally, Continuity and differntiability has 9% weightage in the 12th board final examination. if you want to obtain meritious marks or full marks then you should have good command on concepts that can be developed by practice therefore you should practice NCERT solutions and NCERT exercise solutions.
hello,
Yes you can appear for the compartment paper again since CBSE gives three chances to a candidate to clear his/her exams so you still have two more attempts. However, you can appear for your improvement paper for all subjects but you cannot appear for the ones in which you have failed.
I hope this was helpful!
Good Luck
Hello dear,
If you was not able to clear 1st compartment and now you giving second compartment so YES, you can go for your improvement exam next year but if a student receives an improvement, they are given the opportunity to retake the boards as a private candidate the following year, but there are some requirements. First, the student must pass all of their subjects; if they received a compartment in any subject, they must then pass the compartment exam before being eligible for the improvement.
As you can registered yourself as private candidate for giving your improvement exam of 12 standard CBSE(Central Board of Secondary Education).For that you have to wait for a whole year which is bit difficult for you.
Positive side of waiting for whole year is you have a whole year to preparing yourself for your examination. You have no distraction or something which may causes your failure in the exams. In whole year you have to stay focused on your 12 standard examination for doing well in it. By this you get a highest marks as a comparison of others.
Believe in Yourself! You can make anything happen
All the very best.
Hello Student,
I appreciate your Interest in education. See the improvement is not restricted to one subject or multiple subjects and we cannot say if improvement in one subject in one year leads to improvement in more subjects in coming year.
You just need to have a revision of all subjects what you have completed in the school. have a revision and practice of subjects and concepts helps you better.
All the best.
If you'll do hard work then by hard work of 6 months you can achieve your goal but you have to start studying for it dont waste your time its a very important year so please dont waste it otherwise you'll regret.
Yes, you can take admission in class 12th privately there are many colleges in which you can give 12th privately.
The field of biomedical engineering opens up a universe of expert chances. An Individual in the biomedical engineering career path work in the field of engineering as well as medicine, in order to find out solutions to common problems of the two fields. The biomedical engineering job opportunities are to collaborate with doctors and researchers to develop medical systems, equipment, or devices that can solve clinical problems. Here we will be discussing jobs after biomedical engineering, how to get a job in biomedical engineering, biomedical engineering scope, and salary.
Database professionals use software to store and organise data such as financial information, and customer shipping records. Individuals who opt for a career as data administrators ensure that data is available for users and secured from unauthorised sales. DB administrators may work in various types of industries. It may involve computer systems design, service firms, insurance companies, banks and hospitals.
A career as ethical hacker involves various challenges and provides lucrative opportunities in the digital era where every giant business and startup owns its cyberspace on the world wide web. Individuals in the ethical hacker career path try to find the vulnerabilities in the cyber system to get its authority. If he or she succeeds in it then he or she gets its illegal authority. Individuals in the ethical hacker career path then steal information or delete the file that could affect the business, functioning, or services of the organization.
The invention of the database has given fresh breath to the people involved in the data analytics career path. Analysis refers to splitting up a whole into its individual components for individual analysis. Data analysis is a method through which raw data are processed and transformed into information that would be beneficial for user strategic thinking.
Data are collected and examined to respond to questions, evaluate hypotheses or contradict theories. It is a tool for analyzing, transforming, modeling, and arranging data with useful knowledge, to assist in decision-making and methods, encompassing various strategies, and is used in different fields of business, research, and social science.
Individuals who opt for a career as geothermal engineers are the professionals involved in the processing of geothermal energy. The responsibilities of geothermal engineers may vary depending on the workplace location. Those who work in fields design facilities to process and distribute geothermal energy. They oversee the functioning of machinery used in the field.
Individuals who opt for a career as a remote sensing technician possess unique personalities. Remote sensing analysts seem to be rational human beings, they are strong, independent, persistent, sincere, realistic and resourceful. Some of them are analytical as well, which means they are intelligent, introspective and inquisitive.
Remote sensing scientists use remote sensing technology to support scientists in fields such as community planning, flight planning or the management of natural resources. Analysing data collected from aircraft, satellites or ground-based platforms using statistical analysis software, image analysis software or Geographic Information Systems (GIS) is a significant part of their work. Do you want to learn how to become remote sensing technician? There's no need to be concerned; we've devised a simple remote sensing technician career path for you. Scroll through the pages and read.
The role of geotechnical engineer starts with reviewing the projects needed to define the required material properties. The work responsibilities are followed by a site investigation of rock, soil, fault distribution and bedrock properties on and below an area of interest. The investigation is aimed to improve the ground engineering design and determine their engineering properties that include how they will interact with, on or in a proposed construction.
The role of geotechnical engineer in mining includes designing and determining the type of foundations, earthworks, and or pavement subgrades required for the intended man-made structures to be made. Geotechnical engineering jobs are involved in earthen and concrete dam construction projects, working under a range of normal and extreme loading conditions.
How fascinating it is to represent the whole world on just a piece of paper or a sphere. With the help of maps, we are able to represent the real world on a much smaller scale. Individuals who opt for a career as a cartographer are those who make maps. But, cartography is not just limited to maps, it is about a mixture of art, science, and technology. As a cartographer, not only you will create maps but use various geodetic surveys and remote sensing systems to measure, analyse, and create different maps for political, cultural or educational purposes.
Budget analysis, in a nutshell, entails thoroughly analyzing the details of a financial budget. The budget analysis aims to better understand and manage revenue. Budget analysts assist in the achievement of financial targets, the preservation of profitability, and the pursuit of long-term growth for a business. Budget analysts generally have a bachelor's degree in accounting, finance, economics, or a closely related field. Knowledge of Financial Management is of prime importance in this career.
The invention of the database has given fresh breath to the people involved in the data analytics career path. Analysis refers to splitting up a whole into its individual components for individual analysis. Data analysis is a method through which raw data are processed and transformed into information that would be beneficial for user strategic thinking.
Data are collected and examined to respond to questions, evaluate hypotheses or contradict theories. It is a tool for analyzing, transforming, modeling, and arranging data with useful knowledge, to assist in decision-making and methods, encompassing various strategies, and is used in different fields of business, research, and social science.
A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.
An Investment Banking career involves the invention and generation of capital for other organizations, governments, and other entities. Individuals who opt for a career as Investment Bankers are the head of a team dedicated to raising capital by issuing bonds. Investment bankers are termed as the experts who have their fingers on the pulse of the current financial and investing climate. Students can pursue various Investment Banker courses, such as Banking and Insurance, and Economics to opt for an Investment Banking career path.
An underwriter is a person who assesses and evaluates the risk of insurance in his or her field like mortgage, loan, health policy, investment, and so on and so forth. The underwriter career path does involve risks as analysing the risks means finding out if there is a way for the insurance underwriter jobs to recover the money from its clients. If the risk turns out to be too much for the company then in the future it is an underwriter who will be held accountable for it. Therefore, one must carry out his or her job with a lot of attention and diligence.
Individuals in the operations manager jobs are responsible for ensuring the efficiency of each department to acquire its optimal goal. They plan the use of resources and distribution of materials. The operations manager's job description includes managing budgets, negotiating contracts, and performing administrative tasks.
Welding Engineer Job Description: A Welding Engineer work involves managing welding projects and supervising welding teams. He or she is responsible for reviewing welding procedures, processes and documentation. A career as Welding Engineer involves conducting failure analyses and causes on welding issues.
A career as Transportation Planner requires technical application of science and technology in engineering, particularly the concepts, equipment and technologies involved in the production of products and services. In fields like land use, infrastructure review, ecological standards and street design, he or she considers issues of health, environment and performance. A Transportation Planner assigns resources for implementing and designing programmes. He or she is responsible for assessing needs, preparing plans and forecasts and compliance with regulations.
An expert in plumbing is aware of building regulations and safety standards and works to make sure these standards are upheld. Testing pipes for leakage using air pressure and other gauges, and also the ability to construct new pipe systems by cutting, fitting, measuring and threading pipes are some of the other more involved aspects of plumbing. Individuals in the plumber career path are self-employed or work for a small business employing less than ten people, though some might find working for larger entities or the government more desirable.
Individuals who opt for a career as construction managers have a senior-level management role offered in construction firms. Responsibilities in the construction management career path are assigning tasks to workers, inspecting their work, and coordinating with other professionals including architects, subcontractors, and building services engineers.
Urban Planning careers revolve around the idea of developing a plan to use the land optimally, without affecting the environment. Urban planning jobs are offered to those candidates who are skilled in making the right use of land to distribute the growing population, to create various communities.
Urban planning careers come with the opportunity to make changes to the existing cities and towns. They identify various community needs and make short and long-term plans accordingly.
Highway Engineer Job Description: A Highway Engineer is a civil engineer who specialises in planning and building thousands of miles of roads that support connectivity and allow transportation across the country. He or she ensures that traffic management schemes are effectively planned concerning economic sustainability and successful implementation.
Individuals who opt for a career as an environmental engineer are construction professionals who utilise the skills and knowledge of biology, soil science, chemistry and the concept of engineering to design and develop projects that serve as solutions to various environmental problems.
A Naval Architect is a professional who designs, produces and repairs safe and sea-worthy surfaces or underwater structures. A Naval Architect stays involved in creating and designing ships, ferries, submarines and yachts with implementation of various principles such as gravity, ideal hull form, buoyancy and stability.
Orthotists and Prosthetists are professionals who provide aid to patients with disabilities. They fix them to artificial limbs (prosthetics) and help them to regain stability. There are times when people lose their limbs in an accident. In some other occasions, they are born without a limb or orthopaedic impairment. Orthotists and prosthetists play a crucial role in their lives with fixing them to assistive devices and provide mobility.
A career in pathology in India is filled with several responsibilities as it is a medical branch and affects human lives. The demand for pathologists has been increasing over the past few years as people are getting more aware of different diseases. Not only that, but an increase in population and lifestyle changes have also contributed to the increase in a pathologist’s demand. The pathology careers provide an extremely huge number of opportunities and if you want to be a part of the medical field you can consider being a pathologist. If you want to know more about a career in pathology in India then continue reading this article.
Gynaecology can be defined as the study of the female body. The job outlook for gynaecology is excellent since there is evergreen demand for one because of their responsibility of dealing with not only women’s health but also fertility and pregnancy issues. Although most women prefer to have a women obstetrician gynaecologist as their doctor, men also explore a career as a gynaecologist and there are ample amounts of male doctors in the field who are gynaecologists and aid women during delivery and childbirth.
An oncologist is a specialised doctor responsible for providing medical care to patients diagnosed with cancer. He or she uses several therapies to control the cancer and its effect on the human body such as chemotherapy, immunotherapy, radiation therapy and biopsy. An oncologist designs a treatment plan based on a pathology report after diagnosing the type of cancer and where it is spreading inside the body.
The audiologist career involves audiology professionals who are responsible to treat hearing loss and proactively preventing the relevant damage. Individuals who opt for a career as an audiologist use various testing strategies with the aim to determine if someone has a normal sensitivity to sounds or not. After the identification of hearing loss, a hearing doctor is required to determine which sections of the hearing are affected, to what extent they are affected, and where the wound causing the hearing loss is found. As soon as the hearing loss is identified, the patients are provided with recommendations for interventions and rehabilitation such as hearing aids, cochlear implants, and appropriate medical referrals. While audiology is a branch of science that studies and researches hearing, balance, and related disorders.
The hospital Administrator is in charge of organising and supervising the daily operations of medical services and facilities. This organising includes managing of organisation’s staff and its members in service, budgets, service reports, departmental reporting and taking reminders of patient care and services.
For an individual who opts for a career as an actor, the primary responsibility is to completely speak to the character he or she is playing and to persuade the crowd that the character is genuine by connecting with them and bringing them into the story. This applies to significant roles and littler parts, as all roles join to make an effective creation. Here in this article, we will discuss how to become an actor in India, actor exams, actor salary in India, and actor jobs.
Individuals who opt for a career as acrobats create and direct original routines for themselves, in addition to developing interpretations of existing routines. The work of circus acrobats can be seen in a variety of performance settings, including circus, reality shows, sports events like the Olympics, movies and commercials. Individuals who opt for a career as acrobats must be prepared to face rejections and intermittent periods of work. The creativity of acrobats may extend to other aspects of the performance. For example, acrobats in the circus may work with gym trainers, celebrities or collaborate with other professionals to enhance such performance elements as costume and or maybe at the teaching end of the career.
Career as a video game designer is filled with excitement as well as responsibilities. A video game designer is someone who is involved in the process of creating a game from day one. He or she is responsible for fulfilling duties like designing the character of the game, the several levels involved, plot, art and similar other elements. Individuals who opt for a career as a video game designer may also write the codes for the game using different programming languages.
Depending on the video game designer job description and experience they may also have to lead a team and do the early testing of the game in order to suggest changes and find loopholes.
Radio Jockey is an exciting, promising career and a great challenge for music lovers. If you are really interested in a career as radio jockey, then it is very important for an RJ to have an automatic, fun, and friendly personality. If you want to get a job done in this field, a strong command of the language and a good voice are always good things. Apart from this, in order to be a good radio jockey, you will also listen to good radio jockeys so that you can understand their style and later make your own by practicing.
A career as radio jockey has a lot to offer to deserving candidates. If you want to know more about a career as radio jockey, and how to become a radio jockey then continue reading the article.
The word “choreography" actually comes from Greek words that mean “dance writing." Individuals who opt for a career as a choreographer create and direct original dances, in addition to developing interpretations of existing dances. A Choreographer dances and utilises his or her creativity in other aspects of dance performance. For example, he or she may work with the music director to select music or collaborate with other famous choreographers to enhance such performance elements as lighting, costume and set design.
A multimedia specialist is a media professional who creates, audio, videos, graphic image files, computer animations for multimedia applications. He or she is responsible for planning, producing, and maintaining websites and applications.
A career as social media manager involves implementing the company’s or brand’s marketing plan across all social media channels. Social media managers help in building or improving a brand’s or a company’s website traffic, build brand awareness, create and implement marketing and brand strategy. Social media managers are key to important social communication as well.
In a career as a copywriter, one has to consult with the client and understand the brief well. A career as a copywriter has a lot to offer to deserving candidates. Several new mediums of advertising are opening therefore making it a lucrative career choice. Students can pursue various copywriter courses such as Journalism, Advertising, Marketing Management. Here, we have discussed how to become a freelance copywriter, copywriter career path, how to become a copywriter in India, and copywriting career outlook.
Careers in journalism are filled with excitement as well as responsibilities. One cannot afford to miss out on the details. As it is the small details that provide insights into a story. Depending on those insights a journalist goes about writing a news article. A journalism career can be stressful at times but if you are someone who is passionate about it then it is the right choice for you. If you want to know more about the media field and journalist career then continue reading this article.
For publishing books, newspapers, magazines and digital material, editorial and commercial strategies are set by publishers. Individuals in publishing career paths make choices about the markets their businesses will reach and the type of content that their audience will be served. Individuals in book publisher careers collaborate with editorial staff, designers, authors, and freelance contributors who develop and manage the creation of content.
In a career as a vlogger, one generally works for himself or herself. However, once an individual has gained viewership there are several brands and companies that approach them for paid collaboration. It is one of those fields where an individual can earn well while following his or her passion.
Ever since internet costs got reduced the viewership for these types of content has increased on a large scale. Therefore, a career as a vlogger has a lot to offer. If you want to know more about the Vlogger eligibility, roles and responsibilities then continue reading the article.
Individuals in the editor career path is an unsung hero of the news industry who polishes the language of the news stories provided by stringers, reporters, copywriters and content writers and also news agencies. Individuals who opt for a career as an editor make it more persuasive, concise and clear for readers. In this article, we will discuss the details of the editor's career path such as how to become an editor in India, editor salary in India and editor skills and qualities.
Linguistic meaning is related to language or Linguistics which is the study of languages. A career as a linguistic meaning, a profession that is based on the scientific study of language, and it's a very broad field with many specialities. Famous linguists work in academia, researching and teaching different areas of language, such as phonetics (sounds), syntax (word order) and semantics (meaning).
Other researchers focus on specialities like computational linguistics, which seeks to better match human and computer language capacities, or applied linguistics, which is concerned with improving language education. Still, others work as language experts for the government, advertising companies, dictionary publishers and various other private enterprises. Some might work from home as freelance linguists. Philologist, phonologist, and dialectician are some of Linguist synonym. Linguists can study French, German, Italian.
The career of a travel journalist is full of passion, excitement and responsibility. Journalism as a career could be challenging at times, but if you're someone who has been genuinely enthusiastic about all this, then it is the best decision for you. Travel journalism jobs are all about insightful, artfully written, informative narratives designed to cover the travel industry. Travel Journalist is someone who explores, gathers and presents information as a news article.
Welding Engineer Job Description: A Welding Engineer work involves managing welding projects and supervising welding teams. He or she is responsible for reviewing welding procedures, processes and documentation. A career as Welding Engineer involves conducting failure analyses and causes on welding issues.
A quality controller plays a crucial role in an organisation. He or she is responsible for performing quality checks on manufactured products. He or she identifies the defects in a product and rejects the product.
A quality controller records detailed information about products with defects and sends it to the supervisor or plant manager to take necessary actions to improve the production process.
A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.
A QA Lead is in charge of the QA Team. The role of QA Lead comes with the responsibility of assessing services and products in order to determine that he or she meets the quality standards. He or she develops, implements and manages test plans.
A metallurgical engineer is a professional who studies and produces materials that bring power to our world. He or she extracts metals from ores and rocks and transforms them into alloys, high-purity metals and other materials used in developing infrastructure, transportation and healthcare equipment.
An Azure Administrator is a professional responsible for implementing, monitoring, and maintaining Azure Solutions. He or she manages cloud infrastructure service instances and various cloud servers as well as sets up public and private cloud systems.
An AWS Solution Architect is someone who specializes in developing and implementing cloud computing systems. He or she has a good understanding of the various aspects of cloud computing and can confidently deploy and manage their systems. He or she troubleshoots the issues and evaluates the risk from the third party.
Careers in computer programming primarily refer to the systematic act of writing code and moreover include wider computer science areas. The word 'programmer' or 'coder' has entered into practice with the growing number of newly self-taught tech enthusiasts. Computer programming careers involve the use of designs created by software developers and engineers and transforming them into commands that can be implemented by computers. These commands result in regular usage of social media sites, word-processing applications and browsers.
A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.
Individuals in the information security manager career path involves in overseeing and controlling all aspects of computer security. The IT security manager job description includes planning and carrying out security measures to protect the business data and information from corruption, theft, unauthorised access, and deliberate attack
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