{"id":536627,"date":"2021-09-16T05:41:34","date_gmt":"2021-09-16T05:41:34","guid":{"rendered":"https:\/\/www.indcareer.com\/schools\/?p=536627"},"modified":"2023-09-21T07:22:09","modified_gmt":"2023-09-21T07:22:09","slug":"ncert-solutions-for-12th-class-maths-chapter-5-continuity-and-differentiability","status":"publish","type":"post","link":"https:\/\/www.indcareer.com\/schools\/ncert-solutions-for-12th-class-maths-chapter-5-continuity-and-differentiability\/","title":{"rendered":"NCERT Solutions for 12th Class Maths: Chapter 5-Continuity and Differentiability"},"content":{"rendered":"\n

Class 12: Maths Chapter 5 solutions. Complete Class 12 Maths Chapter 5 Notes.<\/p>\n\n\n\n

NCERT Solutions for 12th Class Maths: Chapter 5-Continuity and Differentiability<\/h2>\n\n\n\n

Class 12: Maths Chapter 5 solutions. Complete Class 12 Maths Chapter 5 Notes.<\/p>\n\n\n\n

Ex 5.1 Class 12 Maths Question 1.<\/strong>
Prove that the function f (x) = 5x \u2013 3 is continuous at x = 0, at x = \u2013 3 and at x = 5.
Solution:<\/strong>
(i) At x = 0. limx\u2013>0<\/sub> f (x) = limx\u2013>0<\/sub> (5x \u2013 3) = \u2013 3 and
f(0) = \u2013 3
\u2234f is continuous at x = 0
(ii) At x = \u2013 3, limx\u2013>3<\/sub> f(x)= limx\u2013>-3<\/sub> (5x \u2013 3) = \u2013 18
and f( \u2013 3) = \u2013 18
\u2234 f is continuous at x = \u2013 3
(iii) At x = 5, limx\u2013>5<\/sub> f(x) = limx\u2013>5<\/sub> (5x \u2013 3) = 22 and
f(5) = 22
\u2234 f is continuous at x = 5<\/p>\n\n\n\n

Ex 5.1 Class 12 Maths Question 6.<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


Solution:<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

 at x\u22602
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.1 Class 12 Maths Question 7.<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


Solution:<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n



Ex 5.1 Class 12 Maths Question 8.<\/strong>
Test the continuity of the function f (x) at x = 0
<\/p>\n\n\n\n


Solution:<\/strong>
We have;
(LHL at x=0)
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.1 Class 12 Maths Question 9.<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


Solution:<\/strong>
<\/p>\n\n\n\n\n\n


<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.1 Class 12 Maths Question 10.<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


Solution:<\/strong>
<\/p>\n\n\n\n\n\n


<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.1 Class 12 Maths Question 11.<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


Solution:<\/strong>
<\/p>\n\n\n\n\n\n


At x = 2, L.H.L. limx\u2013>2-<\/sub> (x\u00b3 \u2013 3) = 8 \u2013 3 = 5
R.H.L. = limx\u2013>2+<\/sub> (x\u00b2 + 1) = 4 + 1 = 5
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.1 Class 12 Maths Question 12.<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


Solution:<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.1 Class 12 Maths Question 13.<\/strong>
Is the function defined by <\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

 a continuous function?
Solution:<\/strong>
At x = 1,L.H.L.= limx\u2013>1-<\/sub> f(x) = limx\u2013>1-<\/sub> (x + 5) = 6,
R.HL. = limx\u2013>1+<\/sub> f(x) = limx\u2013>1+<\/sub> (x \u2013 5) = \u2013 4
f(1) = 1 + 5 = 6,
f(1) = L.H.L. \u2260 R.H.L.
=> f is not continuous at x = 1
At x = c < 1, limx\u2013>c<\/sub> (x + 5) = c + 5 = f(c)
At x = c > 1, limx\u2013>c<\/sub> (x \u2013 5) = c \u2013 5 = f(c)
\u2234 f is continuous at all points x \u2208 R except x = 1.<\/p>\n\n\n\n

Discuss the continuity of the function f, where f is defined by<\/strong><\/p>\n\n\n\n

Ex 5.1 Class 12 Maths Question 14.<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


Solution:<\/strong>
<\/p>\n\n\n\n\n\n


In the interval 0 \u2264 x \u2264 1,f(x) = 3; f is continuous in this interval.
At x = 1,L.H.L. = lim f(x) = 3,
R.H.L. = limx\u2013>1+<\/sub> f(x) = 4 => f is discontinuous at
x = 1
At x = 3, L.H.L. = limx\u2013>3-<\/sub> f(x)=4,
R.H.L. = limx\u2013>3+<\/sub> f(x) = 5 => f is discontinuous at
x = 3
=> f is not continuous at x = 1 and x = 3.<\/p>\n\n\n\n

Ex 5.1 Class 12 Maths Question 15.<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


Solution:<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


At x = 0, L.H.L. = limx\u2013>0-<\/sub> 2x = 0 ,
R.H.L. = limx\u2013>0+<\/sub> (0)= 0 , f(0) = 0
=> f is continuous at x = 0
At x = 1, L.H.L. = limx\u2013>1-<\/sub> (0) = 0,
R.H.L. = limx\u2013>1+<\/sub> 4x = 4
f(1) = 0, f(1) = L.H.L.\u2260R.H.L.
\u2234 f is not continuous at x = 1
when x < 0 f (x) = 2x, being a polynomial, it is
continuous at all points x < 0. when x > 1. f (x) = 4x being a polynomial, it is
continuous at all points x > 1.
when 0 \u2264 x \u2264 1, f (x) = 0 is a continuous function
the point of discontinuity is x = 1.<\/p>\n\n\n\n

Ex 5.1 Class 12 Maths Question 16.<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


Solution:<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


At x = \u2013 1,L.H.L. = limx\u2013>1-<\/sub> f(x) = \u2013 2, f(-1) = \u2013 2,
R.H.L. = limx\u2013>1+<\/sub> f(x) = \u2013 2
=> f is continuous at x = \u2013 1
At x= 1, L.H.L. = limx\u2013>1-<\/sub> f(x) = 2,f(1) = 2
\u2234 f is continuous at x = 1,
R.H.L. = limx\u2013>1+<\/sub> f(x) = 2
Hence, f is continuous function.<\/p>\n\n\n\n

Ex 5.1 Class 12 Maths Question 17.<\/strong>
Find the relationship between a and b so that the function f defined by
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


is continuous at x = 3
Solution:<\/strong>
At x = 3, L.H.L. = limx\u2013>3-<\/sub> (ax+1) = 3a+1 ,
f(3) = 3a + 1, R.H.L. = limx\u2013>3+<\/sub> (bx+3) = 3b+3
f is continuous ifL.H.L. = R.H.L. = f(3)
3a + 1 = 3b + 3 or 3(a \u2013 b) = 2
a \u2013 b = \"\\\\ or a = b + \"\\\\, for any arbitrary value of b.
Therefore the value of a corresponding to the value of b.<\/p>\n\n\n\n

Ex 5.1 Class 12 Maths Question 18.<\/strong>
For what value of \u03bb is the function defined by
\"f(x)=\\begin{cases}0 \\end{cases} “>
continuous at x = 0? What about continuity at x = 1?
Solution:<\/strong>
At x = 0, L.H.L. = limx\u2013>0-<\/sub> \u03bb (x\u00b2 \u2013 2x) = 0 ,
R.H.L. = limx\u2013>0+<\/sub> (4x+ 1) = 1, f(0)=0
f (0) = L.H.L. \u2260 R.H.L.
=> f is not continuous at x = 0,
whatever value of \u03bb \u2208 R may be
At x = 1, limx\u2013>1<\/sub> f(x) = limx\u2013>1<\/sub> (4x + l) = f(1)
=> f is not continuous at x = 0 for any value of \u03bb but f is continuous at x = 1 for all values of \u03bb.<\/p>\n\n\n\n

Ex 5.1 Class 12 Maths Question 19.<\/strong>
Show that the function defined by g (x) = x \u2013 [x] is discontinuous at all integral points. Here [x] denotes the greatest integer less than or equal to x.
Solution:<\/strong>
Let c be an integer, [c \u2013 h] = c \u2013 1, [c + h] = c, [c] = c, g(x) = x \u2013 [x].
At x = c, limx\u2013>c-<\/sub> (x \u2013 [x]) = limh\u2013>0<\/sub> [(c \u2013 h) \u2013 (c \u2013 1)]
= limh\u2013>0<\/sub> (c \u2013 h \u2013 (c \u2013 1)) = 1[\u2235 [c \u2013 h] = c \u2013 1]
R.H.L. = limx\u2013>c+<\/sub> (x \u2013 [x])= limh\u2013>0<\/sub> (c + h \u2013 [c + h])
= limh\u2013>0<\/sub> [c + h \u2013 c] = 0
f(c) = c \u2013 [c] = 0,
Thus L.H.L. \u2260 R.H.L. = f (c) => f is not continuous at integral points.<\/p>\n\n\n\n

Ex 5.1 Class 12 Maths Question 20.<\/strong>
Is the function defined by f (x) = x\u00b2 \u2013 sin x + 5 continuous at x = \u03c0?
Solution:<\/strong>
Let f(x) = x\u00b2 \u2013 sinx + 5,
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.1 Class 12 Maths Question 21.<\/strong>
Discuss the continuity of the following functions:
(a) f (x) = sin x + cos x
(b) f (x) = sin x \u2013 cos x
(c) f (x) = sin x \u00b7 cos x
Solution:<\/strong>
(a) f(x) = sinx + cosx
<\/p>\n\n\n\n\n\n

<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n


Ex 5.1 Class 12 Maths Question 22.<\/strong>
Discuss the continuity of the cosine, cosecant, secant and cotangent functions.
Solution:<\/strong>
(a) Let f(x) = cosx
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n


Ex 5.1 Class 12 Maths Question 23.<\/strong>
Find all points of discontinuity of f, where
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


Solution:<\/strong>
At x = 0
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.1 Class 12 Maths Question 24.<\/strong>
Determine if f defined by <\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

 is a continuous function?
Solution:<\/strong>
At x = 0
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.1 Class 12 Maths Question 25.<\/strong>
Examine the continuity of f, where f is defined by <\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


Solution:<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Find the values of k so that the function is continuous at the indicated point in Questions 26 to 29.<\/strong><\/p>\n\n\n\n

Ex 5.1 Class 12 Maths Question 26.<\/strong>
<\/p>\n\n\n\n\n\n


Solution:<\/strong>
At x = \"\\frac
L.H.L = <\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.1 Class 12 Maths Question 27.<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


Solution:<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n


<\/p>\n\n\n\n

Ex 5.1 Class 12 Maths Question 28.<\/strong>
<\/p>\n\n\n\n\n\n


Solution:<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.1 Class 12 Maths Question 29.<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


Solution:<\/strong>
<\/p>\n\n\n\n\n\n


<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.1 Class 12 Maths Question 30.<\/strong>
Find the values of a and b such that the function defined by
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


to is a continuous function.
Solution:<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.1 Class 12 Maths Question 31.<\/strong>
Show that the function defined by f(x)=cos (x\u00b2) is a continuous function.
Solution:<\/strong>
Now, f (x) = cosx\u00b2, let g (x)=cosx and h (x) x\u00b2
\u2234 goh(x) = g (h (x)) = cos x\u00b2
Now g and h both are continuous \u2200 x \u2208 R.
f (x) = goh (x) = cos x\u00b2 is also continuous at all x \u2208 R.<\/p>\n\n\n\n

Ex 5.1 Class 12 Maths Question 32.<\/strong>
Show that the function defined by f (x) = |cos x| is a continuous function.
Solution:<\/strong>
Let g(x) =|x|and h (x) = cos x, f(x) = goh(x) = g (h (x)) = g (cosx) = |cos x |
Now g (x) = |x| and h (x) = cos x both are continuous for all values of x \u2208 R.
\u2234 (goh) (x) is also continuous.
Hence, f (x) = goh (x) = |cos x| is continuous for all values of x \u2208 R.<\/p>\n\n\n\n

Ex 5.1 Class 12 Maths Question 33.<\/strong>
Examine that sin |x| is a continuous function.
Solution:<\/strong>
Let g (x) = sin x, h (x) = |x|, goh (x) = g (h(x))
= g(|x|) = sin|x| = f(x)
Now g (x) = sin x and h (x) = |x| both are continuous for all x \u2208 R.
\u2234f (x) = goh (x) = sin |x| is continuous at all x \u2208 R.<\/p>\n\n\n\n

Ex 5.1 Class 12 Maths Question 34.<\/strong>
Find all the points of discontinuity of f defined by f(x) = |x|-|x+1|.
Solution:<\/strong>
f(x) = |x|-|x+1|, when x< \u2013 1,
f(x) = -x-[-(x+1)] = \u2013 x + x + 1 = 1
when -1 \u2264 x < 0, f(x) = \u2013 x \u2013 (x + 1) = \u2013 2x \u2013 1,
when x \u2265 0, f(x) = x \u2013 (x + 1) = \u2013 1
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.2 Class 12 Maths Question 1.<\/strong>
sin(x\u00b2 + 5)
Solution:<\/strong>
Let y = sin(x2 + 5),
put x\u00b2 + 5 = t
y = sint
t = x\u00b2+5
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


= cos (x\u00b2 + 5) \u00d7 2x
= 2x cos (x\u00b2 + 5)<\/p>\n\n\n\n

Ex 5.2 Class 12 Maths Question 2.<\/strong>
cos (sin x)
Solution:<\/strong>
let y = cos (sin x)
put sinx = t
\u2234 y = cost,
t = sinx
\u2234<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


Putting the value of t, <\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.2 Class 12 Maths Question 3.<\/strong>
sin(ax+b)
Solution:<\/strong>
let = sin(ax+b)
put ax+bx = t
\u2234 y = sint
t = ax+b
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n\n\n


<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


<\/p>\n\n\n\n

Ex 5.2 Class 12 Maths Question 4.<\/strong>
sec(tan(\u221ax))
Solution:<\/strong>
let y = sec(tan(\u221ax))
by chain rule
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Ex 5.2 Class 12 Maths Question 5.<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


Solution:<\/strong>
y = <\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

 = <\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


u = sin(ax+b)
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.2 Class 12 Maths Question 6.<\/strong>
cos x\u00b3 . sin\u00b2(x5<\/sup>) = y(say)
Solution:<\/strong>
Let u = cos x\u00b3 and v = sin\u00b2 x5<\/sup>,
put x\u00b3 = t
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.2 Class 12 Maths Question 7.<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


Solution:<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.2 Class 12 Maths Question 8.<\/strong>
cos(\u221ax) = y(say)
Solution:<\/strong>
cos(\u221ax) = y(say)
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

Ex 5.2 Class 12 Maths Question 9.<\/strong>
Prove that the function f given by f (x) = |x \u2013 1|,x \u2208 R is not differential at x = 1.
Solution:<\/strong>
The given function may be written as
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.2 Class 12 Maths Question 10.<\/strong>
Prove that the greatest integer function defined by f (x)=[x], 0 < x < 3 is not differential at x = 1 and x = 2.
Solution:<\/strong>
(i) At x = 1
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.3 Class 12 Maths Question 1.<\/strong>
2x + 3y = sinx
Solution:<\/strong>
2x + 3y = sinx
Differentiating w.r.t x,
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


=><\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.3 Class 12 Maths Question 2.<\/strong>
2x + 3y = siny
Solution:<\/strong>
2x + 3y = siny
Differentiating w.r.t x,
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


=><\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.3 Class 12 Maths Question 3.<\/strong>
ax + by\u00b2 = cosy
Solution:<\/strong>
ax + by\u00b2 = cosy
Differentiate w.r.t. x,
<\/p>\n\n\n\n\n\n


=><\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.3 Class 12 Maths Question 4.<\/strong>
xy + y\u00b2 = tan x + y
Solution:<\/strong>
xy + y\u00b2 = tanx + y
Differentiating w.r.t. x,
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.3 Class 12 Maths Question 5.<\/strong>
x\u00b2 + xy + y\u00b2 = 100
Solution:<\/strong>
x\u00b2 + xy + xy = 100
Differentiating w.r.t. x,
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.3 Class 12 Maths Question 6.<\/strong>
x\u00b3 + x\u00b2y + xy\u00b2 + y\u00b3 = 81
Solution:<\/strong>
Given that
x\u00b3 + x\u00b2y + xy\u00b2 + y\u00b3 = 81
Differentiating both sides we get
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.3 Class 12 Maths Question 7.<\/strong>
sin\u00b2 y + cos xy = \u03c0
Solution:<\/strong>
Given that
sin\u00b2 y + cos xy = \u03c0
Differentiating both sides we get
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.3 Class 12 Maths Question 8.<\/strong>
sin\u00b2x + cos\u00b2y = 1
Solution:<\/strong>
Given that
sin\u00b2x + cos\u00b2y = 1
Differentiating both sides, we get
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.3 Class 12 Maths Question 9.<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


Solution:<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


put x = tan\u03b8
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.3 Class 12 Maths Question 10.<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


Solution:<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


put x = tan\u03b8
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.3 Class 12 Maths Question 11.<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


Solution:<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


put x = tan\u03b8
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.3 Class 12 Maths Question 12.<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


Solution:<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


put x = tan\u03b8
we get
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.3 Class 12 Maths Question 13.<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


Solution:<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


put x = tan\u03b8
we get
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.3 Class 12 Maths Question 14.<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


Solution:<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


put x = tan\u03b8
we get
<\/p>\n\n\n\n\n\n

\"\"<\/figure>\n\n\n\n



<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.3 Class 12 Maths Question 15.<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


Solution:<\/strong>
<\/p>\n\n\n\n\n\n


put x = tan\u03b8
we get
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.4 Class 12 Maths Question 1.<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


Solution:<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n


<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n



<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.4 Class 12 Maths Question 2.<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


Solution:<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n


<\/p>\n\n\n\n


x=sint
<\/p>\n\n\n\n\n\n


<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.4 Class 12 Maths Question 3.<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


Solution:<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n



<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.4 Class 12 Maths Question 4.<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


Solution:<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.4 Class 12 Maths Question 5.<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


Solution:<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.4 Class 12 Maths Question 6.<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

\u2026<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


Solution:<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


\"{\u2026\"+{
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.4 Class 12 Maths Question 7.<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


Solution:<\/strong>
y = <\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n



<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.4 Class 12 Maths Question 8.<\/strong>
log(log x),x>1
Solution:<\/strong>
y = log(log x),
put y = log t, t = log x,
differentiating
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.4 Class 12 Maths Question 9.<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


Solution:<\/strong>
let <\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.4 Class 12 Maths Question 10.<\/strong>
cos(log x+ex<\/sup>),x>0
Solution:<\/strong>
y = cos(log x+ex<\/sup>),x>0
put y = cos t,t = log x+ex<\/sup>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Differentiate the functions given in Questions 1 to 11 w.r.to x<\/strong><\/p>\n\n\n\n

Ex 5.5 Class 12 Maths Question 1.<\/strong>
cos x. cos 2x. cos 3x
Solution:<\/strong>
Let y = cos x. cos 2x . cos 3x,
Taking log on both sides,
log y = log (cos x. cos 2x. cos 3x)
log y = log cos x + log cos 2x + log cos 3x,
Differentiating w.r.t. x, we get
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.5 Class 12 Maths Question 2.<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


Solution:<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


taking log on both sides
log y = log <\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.5 Class 12 Maths Question 3.<\/strong>
(log x)cosx<\/sup>
Solution:<\/strong>
let y = (log x)cosx<\/sup>
Taking log on both sides,
log y = log (log x)cosx<\/sup>
log y = cos x log (log x),
Differentiating w.r.t. x,
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.5 Class 12 Maths Question 4.<\/strong>
x \u2013 2sinx<\/sup>
Solution:<\/strong>
let y = x \u2013 2sinx<\/sup>,
y = u \u2013 v
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.5 Class 12 Maths Question 5.<\/strong>
(x+3)2<\/sup>.(x + 4)3<\/sup>.(x + 5)4<\/sup>
Solution:<\/strong>
let y = (x + 3)2<\/sup>.(x + 4)3<\/sup>.(x + 5)4<\/sup>
Taking log on both side,
logy = log [(x + 3)2<\/sup> \u2022 (x + 4)3<\/sup> \u2022 (x + 5)4<\/sup>]
= log (x + 3)2<\/sup> + log (x + 4)3<\/sup> + log (x + 5)4<\/sup>
log y = 2 log (x + 3) + 3 log (x + 4) + 4 log (x + 5)
Differentiating w.r.t. x, we get
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.5 Class 12 Maths Question 6.<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


Solution:<\/strong>
let <\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


let <\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.5 Class 12 Maths Question 7.<\/strong>
(log x)x<\/sup> + xlogx<\/sup>
Solution:<\/strong>
let y = (log x)x<\/sup> + xlogx<\/sup> = u+v
where u = (log x)x<\/sup>
\u2234 log u = x log(log x)
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.5 Class 12 Maths Question 8.<\/strong>
(sin x)x<\/sup>+sin-1<\/sup> \u221ax
Solution:<\/strong>
Let y = (sin x)x <\/sup>+ sin-1 <\/sup>\u221ax
let u = (sin x)x, v = sin-1<\/sup> \u221ax
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.5 Class 12 Maths Question 9.<\/strong>
xsinx<\/sup> + (sin x)cosx<\/sup>
Solution:<\/strong>
let y = xsinx<\/sup> + (sin x)cosx<\/sup> = u+v
where u = xsinx<\/sup>
log u = sin x log x
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.5 Class 12 Maths Question 10.<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


Solution:<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


y = u + v
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.5 Class 12 Maths Question 11.<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


Solution:<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


Let u = (x cosx)x<\/sup>
logu = x log(x cosx)
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Find  of the functions given in Questions 12 to 15.<\/strong><\/p>\n\n\n\n

Ex 5.5 Class 12 Maths Question 12.<\/strong>
xy<\/sup> + yx<\/sup> = 1
Solution:<\/strong>
xy<\/sup> + yx<\/sup> = 1
let u = xy<\/sup> and v = yx<\/sup>
\u2234 u + v = 1,
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


Now u = x
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.5 Class 12 Maths Question 13.<\/strong>
yx <\/sup>= xy<\/sup>
Solution:<\/strong>
y = x
x logy = y logx
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.5 Class 12 Maths Question 14.<\/strong>
(cos x)y<\/sup> = (cos y)x<\/sup>
Solution:<\/strong>
We have
(cos x)y<\/sup> = (cos y)x<\/sup>
=> y log (cosx) = x log (cosy)
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.5 Class 12 Maths Question 15.<\/strong>
xy = e(x-y)<\/sup>
Solution:<\/strong>
log(xy) = log e(x-y)<\/sup>
=> log(xy) = x \u2013 y
=> logx + logy = x \u2013 y
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.5 Class 12 Maths Question 16.<\/strong>
Find the derivative of the function given by f (x) = (1 + x) (1 + x2<\/sup>) (1 + x4<\/sup>) (1 + x8<\/sup>) and hence find f'(1).
Solution:<\/strong>
Let f(x) = y = (1 + x)(1 + x2<\/sup>)(1 + x4<\/sup>)(1 + x8<\/sup>)
Taking log both sides, we get
logy = log [(1 + x)(1 + x2<\/sup>)(1 + x4<\/sup>)(1 + x8<\/sup>)]
logy = log(1 + x) + log (1 + x2<\/sup>) + log(1 + x4<\/sup>) + log(1 + x8<\/sup>)
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.5 Class 12 Maths Question 17.<\/strong>
Differentiate (x2<\/sup> \u2013 5x + 8) (x3<\/sup> + 7x + 9) in three ways mentioned below:
(i) by using product rule
(ii) by expanding the product to obtain a single polynomial.
(iii) by logarithmic differentiation.
Do they all give the same answer?
Solution:<\/strong>
(i) By using product rule
f\u2019 = (x2<\/sup> \u2013 5x + 8) (3x2<\/sup> + 7) + (x3<\/sup> + 7x + 9) (2x \u2013 5)
f = 5x4<\/sup> \u2013 20x3<\/sup> + 45x2<\/sup> \u2013 52x + 11.
(ii) By expanding the product to obtain a single polynomial, we get
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.5 Class 12 Maths Question 18.<\/strong>
If u, v and w are functions of w then show that
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


in two ways-first by repeated application of product rule, second by logarithmic differentiation.
Solution:<\/strong>
Let y = u.v.w
=> y = u. (vw)
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

If x and y are connected parametrically by the equations given in Questions 1 to 10, without eliminating the parameter. Find .<\/strong><\/p>\n\n\n\n

Ex 5.6 Class 12 Maths Question 1.<\/strong>
x = 2at\u00b2, y = at4<\/sup>
Solution:<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.6 Class 12 Maths Question 2.<\/strong>
x = a cos\u03b8,y = b cos\u03b8
Solution:<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.6 Class 12 Maths Question 3.<\/strong>
x = sin t, y = cos 2t
Solution:<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.6 Class 12 Maths Question 4.<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


Solution:<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.6 Class 12 Maths Question 5.<\/strong>
x = cos \u03b8 \u2013 cos 2\u03b8, y = sin \u03b8 \u2013 sin 2\u03b8
Solution:<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.6 Class 12 Maths Question 6.<\/strong>
x = a(\u03b8 \u2013 sin\u03b8), y = a(1 + cos\u03b8)
Solution:<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.6 Class 12 Maths Question 7.<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


Solution:<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.6 Class 12 Maths Question 8.<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


Solution:<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.6 Class 12 Maths Question 9.<\/strong>
x = a sec \u03b8,y = b tan \u03b8
Solution:<\/strong>
x = a sec \u03b8,y = b tan \u03b8
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.6 Class 12 Maths Question 10.<\/strong>
x = a(cos\u03b8+\u03b8sin\u03b8), y = a(sin\u03b8-\u03b8cos\u03b8)
Solution:<\/strong>
x = a(cos\u03b8+\u03b8sin\u03b8), y = a(sin\u03b8-\u03b8cos\u03b8)
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.6 Class 12 Maths Question 11.<\/strong>
If <\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

 show that <\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


Solution:<\/strong>
Given that
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.7 Class 12 Maths Question 1.<\/strong>
x\u00b2 + 3x + 2 = y(say)
Solution:<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.7 Class 12 Maths Question 2.<\/strong>
x20<\/sup> = y(say)
Solution:<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.7 Class 12 Maths Question 3.<\/strong>
x.cos x = y(say)
Solution:<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.7 Class 12 Maths Question 4.<\/strong>
log x = y (say)
Solution:<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.7 Class 12 Maths Question 5.<\/strong>
x3<\/sup> log x = y (say)
Solution:<\/strong>
x3<\/sup> log x = y
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.7 Class 12 Maths Question 6.<\/strong>
ex<\/sup> sin5x = y
Solution:<\/strong>
ex<\/sup> sin5x = y
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.7 Class 12 Maths Question 7.<\/strong>
e6x<\/sup> cos3x = y
Solution:<\/strong>
e6x<\/sup> cos3x = y
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.7 Class 12 Maths Question 8.<\/strong>
tan-1<\/sup> x = y
Solution:<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.7 Class 12 Maths Question 9.<\/strong>
log(logx) = y
Solution:<\/strong>
log(logx) = y
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.7 Class 12 Maths Question 10.<\/strong>
sin(log x) = y
Solution:<\/strong>
sin(log x) = y
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.7 Class 12 Maths Question 11.<\/strong>
If y = 5 cosx \u2013 3 sin x, prove that <\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


Solution:<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


Hence proved<\/p>\n\n\n\n

Ex 5.7 Class 12 Maths Question 12.<\/strong>
If y = cos-1<\/sup> x, Find <\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

 in terms of y alone.
Solution:<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.7 Class 12 Maths Question 13.<\/strong>
If y = 3 cos (log x) + 4 sin (log x), show that
\"{
Solution:<\/strong>
Given that
y = 3 cos (log x) + 4 sin (log x)
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.7 Class 12 Maths Question 14.<\/strong>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


Solution:<\/strong>
Given that
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.7 Class 12 Maths Question 15.<\/strong>
If y = 500e7x<\/sup> + 600e-7x<\/sup>, show that <\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

.
Solution:<\/strong>
we have
y = 500e7x<\/sup> + 600e-7x<\/sup>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.7 Class 12 Maths Question 16.<\/strong>
If ey<\/sup>(x+1) = 1,show that <\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n


Solution:<\/strong>
\"{{ e }^{ y }=\\frac { 1 }{ x+1 } “>
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.7 Class 12 Maths Question 17.<\/strong>
If y=(tan-1<\/sup> x)\u00b2 show that (x\u00b2+1)\u00b2y2<\/sub>+2x(x\u00b2+1)y1<\/sub>=2
Solution:<\/strong>
we have
y=(tan-1<\/sup> x)\u00b2
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.8 Class 12 Maths Question 5.<\/strong>
Verify Mean Value Theorem, if f (x)=x3<\/sup> \u2013 5x2<\/sup> \u2013 3x in the interval [a, b], where a = 1 and b = 3. Find all c \u2208 (1,3) for which f\u2019 (c) = 0.
Solution:<\/strong>
f (x)=x3<\/sup> \u2013 5x2<\/sup> \u2013 3x,
It is a polynomial. Therefore it is continuous in the interval [1,3] and derivable in the interval (1,3)
Also, f'(x)=3x\u00b2-10x-3
<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Ex 5.8 Class 12 Maths Question 6.<\/strong>
Examine the applicability of Mean Value theroem for all three functions given in the above Question 2.
Solution:<\/strong>
(i) F (x)= [x] for x \u2208 [5,9], f (x) = [x] in the interval [5, 9] is neither continuous, nor differentiable.
(ii) f (x) = [x], for x \u2208 [-2,2],
Again f (x) = [x] in the interval [-2,2] is neither continuous, nor differentiable.
(iii) f(x) = x\u00b2-1 for x \u2208 [1,2], It is a polynomial. Therefore it is continuous in the interval [1,2] and differentiable in the interval (1,2)
f (x) = 2x, f(1) = 1 \u2013 1 = 0 ,
f(2) = 4 \u2013 1 = 3, f'(c) = 2c
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NCERT Solutions for 12th Class Maths: Chapter 5: Download PDF<\/strong><\/h2>\n\n\n\n

NCERT Solutions for 12th Class Maths: Chapter 5-Continuity and Differentiability<\/p>\n\n\n\n

Download PDF<\/strong>: NCERT Solutions for 12th Class Maths: Chapter 5-Continuity and Differentiability PDF<\/a><\/p>\n\n\n\n

Chapterwise NCERT Solutions for Class 12 Maths :<\/strong><\/h2>\n\n\n\n