NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability — Exercise 5.1

Question 1:Prove that the function f (x) = 5x – 3 is continuous at x = 0, at x = – 3 and at x = 5.

Question 2:Examine the continuity of the function f (x) = 2x2 – 1 at x = 3.

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Question 3:Examine the following functions for continuity. (a) f (x) = x – 5 (b) f (x) = 1 x − 5 , x ¹ 5 (c) f (x) = 2 25 5 x x − + , x ¹ –5 (d) f (x) = | x – 5|

Question 4:Prove that the function f (x) = xn is continuous at x = n, where n is a positive integer.

Question 5:Is the function f defined by , if 1 ( ) 5, if >1 x x f x x
≤ = continuous at x = 0? At x = 1? At x = 2? Find all points of discontinuity of f, where f is defined by

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Question 6:2 3, if 2 ( ) 2 3, if > 2 x x f x x x
+ ≤ =   −

Question 7:| | 3, if 3 ( ) 2 , if 3 < 3 6 2, if 3 x x f x x x x x + ≤ −
 = − − <  + ³ 

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Question 8:| | , if 0 ( ) 0, if 0 x x f x x x
 ¹ =  =

Question 9:, if 0 ( ) | | 1, if 0 x x f x x x
 < = − ³

Question 10:2 1, if 1 ( ) 1, if 1 x x f x x x + ³
 =  + < 

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Question 11:3 2 3, if 2 ( ) 1, if 2 x x f x x x 
− ≤ =  + > 

Question 12:10 2 1, if 1 ( ) , if 1 x x f x x x
 − ≤ =  > 

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Question 13:Is the function defined by 5, if 1 ( ) 5, if 1 x x f x x x
+ ≤ =   − > a continuous function? Discuss the continuity of the function f, where f is defined by

Question 14:3, if 0 1 ( ) 4, if 1 3 5, if 3 10 x f x x x ≤ ≤
 = < <   ≤ ≤

Question 15:2 , if 0 ( ) 0, if 0 1 4 , if >1 x x f x x x x <
 = ≤ ≤ 

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Question 16: 2, if 1 ( ) 2 , if 1 1 2, if 1 x f x x x x − ≤ −
 = − < ≤   >

Question 17:Find the relationship between a and b so that the function f defined by 1, if 3 ( ) 3, if 3 ax x f x bx x
+ ≤ =   + > is continuous at x = 3.

Question 18:For what value of l is the function defined by 2 ( 2 ), if 0 ( ) 4 1, if 0 x x x f x x x
l − ≤ =  + >  continuous at x = 0? What about continuity at x = 1?

Question 19:Show that the function defined by g(x) = x – [x] is discontinuous at all integral points. Here [x] denotes the greatest integer less than or equal to x.

Exercise 5.1 class 12 Math ncert solutions Chapter 5

Question 20:Is the function defined by f (x) = x2 – sin x + 5 continuous at x = p?

Question 21:Discuss the continuity of the following functions: (a) f (x) = sin x + cos x (b) f (x) = sin x – cos x (c) f (x) = sin x . cos x

Question 22:Discuss the continuity of the cosine, cosecant, secant and cotangent functions.

1. Prove that the function f (x) = 5x 3 is contin

Question 23:Find all points of discontinuity of f, where sin , if 0 ( ) 1, if 0 x x f x x x x
 < =  + ³

Question 24:Determine if f defined by 2 1 sin , if 0 ( ) 0, if 0 x x f x x x
 ¹ =  = is a continuous function?

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Question 25:Examine the continuity of f, where f is defined by sin cos , if 0 ( ) 1, if 0 x x x f x x
− ¹ =  − = Find the values of k so that the function f is continuous at the indicated point in Exercises 26 to 29

Question 26: cos , if 2 2 ( ) 3, if 2 k x x x f x x
p ¹  π − =   π =  at x = 2 p

Question 27: 2 , if 2 ( ) 3, if 2 kx x f x x 
≤ =  >  at x = 2

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Question 28: 1, if ( ) cos , if kx x f x x x
+ ≤ p =   > p at x = p

Question 29: 1, if 5 ( ) 3 5, if 5 kx x f x x x
+ ≤ =   − > at x = 5

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Question 30:Find the values of a and b such that the function defined by 5, if 2 ( ) , if 2 10 21, if 10 x f x ax b x x ≤
 = + < <   ³ is a continuous function.

Question 31:Show that the function defined by f (x) = cos (x2) is a continuous function.

Question 32:Show that the function defined by f (x) = | cos x| is a continuous function.

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Question 33:Examine that sin | x| is a continuous function.

Question 34:Find all the points of discontinuity of f defined by f (x) = | x| – | x + 1|.