NCERT Solutions for Class 12 Maths Maths Part-1 Chapter 5

Continuity and Differentiability Class 12

Chapter 5 Continuity and Differentiability Exercise 5.1, 5.2, 5.3, 5.4, 5.5, 5.6, 5.7, 5.8, miscellaneous Solutions

<< Previous Chapter 4 : Determinants Next Chapter 6 : Application of Derivatives >>

Exercise 5.1 : Solutions of Questions on Page Number : 159

Q1 :  

Prove that the functionis continuous at


Answer :

Therefore, f is continuous at x= 0

Therefore, f is continuous at x= - 3

Therefore, f is continuous at x= 5

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Q2 :  

Examine the continuity of the function.


Answer :

Thus, f is continuous at x= 3

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Q3 :  

Examine the following functions for continuity.

(a) (b)

(c) (d)


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Q4 :  

Prove that the function is continuous at x= n, where n is a positive integer.


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Q5 :  

Is the function fdefined by

continuous at x= 0? At x= 1? At x= 2?


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Q6 :  

Find all points of discontinuity of f, where f is defined by


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Q7 :  

Find all points of discontinuity of f, where f is defined by


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Q8 :  

Find all points of discontinuity of f, where f is defined by


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Q9 :  

Find all points of discontinuity of f, where f is defined by


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Q10 :  

Find all points of discontinuity of f, where f is defined by


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Q11 :  

Find all points of discontinuity of f, where f is defined by


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Q12 :  

Find all points of discontinuity of f, where f is defined by


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Q13 :  

Is the function defined by

a continuous function?


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Q14 :  

Discuss the continuity of the function f, where f is defined by


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Q15 :  

Discuss the continuity of the function f, where f is defined by


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Q16 :  

Discuss the continuity of the function f, where f is defined by


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Q17 :  

Find the relationship between aand b so that the function fdefined by

is continuous at x = 3.


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Q18 :  

For what value of is the function defined by

continuous at x = 0? What about continuity at x= 1?


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Q19 :  

Show that the function defined by is discontinuous at all integral point. Here denotes the greatest integer less than or equal to x.


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Q20 :  

Is the function defined by continuous at x = π?


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Q21 :  

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 x cos x


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Q22 :  

Discuss the continuity of the cosine, cosecant, secant and cotangent functions,


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Q23 :  

Find the points of discontinuity of f, where


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Q24 :  

Determine if fdefined by

is a continuous function?


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Q25 :  

Examine the continuity of f, where f is defined by


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Q26 :  

Find the values of k so that the function fis continuous at the indicated point.


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Q27 :  

Find the values of k so that the function fis continuous at the indicated point.


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Q28 :  

Find the values of k so that the function fis continuous at the indicated point.


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Q29 :  

Find the values of k so that the function fis continuous at the indicated point.


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Q30 :  

Find the values of aand b such that the function defined by

is a continuous function.


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Q31 :  

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


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Q32 :  

Show that the function defined by is a continuous function.


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Q33 :  

Examine that is a continuous function.


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Q34 :  

Find all the points of discontinuity of f defined by.


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<< Previous Chapter 4 : Determinants Next Chapter 6 : Application of Derivatives >>

Exercise 5.2 : Solutions of Questions on Page Number : 166

Q1 :  

Differentiate the functions with respect to x.


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Q2 :  

Differentiate the functions with respect to x.


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Q3 :  

Differentiate the functions with respect to x.


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Q4 :  

Differentiate the functions with respect to x.


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Q5 :  

Differentiate the functions with respect to x.


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Q6 :  

Differentiate the functions with respect to x.


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Q7 :  

Differentiate the functions with respect to x.


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Q8 :  

Differentiate the functions with respect to x.


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Q9 :  

Prove that the function f given by

is notdifferentiable at x = 1.


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Q10 :  

Prove that the greatest integer function defined byis not

differentiable at x = 1 and x = 2.


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<< Previous Chapter 4 : Determinants Next Chapter 6 : Application of Derivatives >>

Exercise 5.3 : Solutions of Questions on Page Number : 169

Q1 :  

Find :


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Q2 :  

Find :


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Q3 :  

Find :


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Q4 :  

Find :


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Q5 :  

Find :


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Q6 :  

Find :


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Q7 :  

Find :


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Q8 :  

Find :


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Q9 :  

Find :


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Q10 :  

Find :


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Q11 :  

Find :


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Q12 :  

Find :


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Q13 :  

Find :


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Q14 :  

Find :


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Q15 :  

Find :


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<< Previous Chapter 4 : Determinants Next Chapter 6 : Application of Derivatives >>

Exercise 5.4 : Solutions of Questions on Page Number : 174

Q1 :  

Differentiate the following w.r.t. x:


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Q2 :  

Differentiate the following w.r.t. x:


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Q3 :  

Differentiate the following w.r.t. x:


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Q4 :  

Differentiate the following w.r.t. x:


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Q5 :  

Differentiate the following w.r.t. x:


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Q6 :  

Differentiate the following w.r.t. x:


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Q7 :  

Differentiate the following w.r.t. x:


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Q8 :  

Differentiate the following w.r.t. x:


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Q9 :  

Differentiate the following w.r.t. x:


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Q10 :  

Differentiate the following w.r.t. x:


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<< Previous Chapter 4 : Determinants Next Chapter 6 : Application of Derivatives >>

Exercise 5.5 : Solutions of Questions on Page Number : 178

Q1 :  

Differentiate the function with respect to x.


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Q2 :  

Differentiate the function with respect to x.


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Q3 :  

Differentiate the function with respect to x.


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Q4 :  

Differentiate the function with respect to x.


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Q5 :  

Differentiate the function with respect to x.


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Q6 :  

Differentiate the function with respect to x.


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Q7 :  

Differentiate the function with respect to x.


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Q8 :  

Differentiate the function with respect to x.


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Q9 :  

Differentiate the function with respect to x.


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Q10 :  

Differentiate the function with respect to x.


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Q11 :  

Differentiate the function with respect to x.


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Q12 :  

Find of function.


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Q13 :  

Find of function.


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Q14 :  

Find of function.


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Q15 :  

Find of function.


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Q16 :  

Find the derivative of the function given by and hence find.


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Q17 :  

Differentiate 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?


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Q18 :  

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.


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<< Previous Chapter 4 : Determinants Next Chapter 6 : Application of Derivatives >>

Exercise 5.6 : Solutions of Questions on Page Number : 181

Q1 :  

If x and y are connected parametrically by the equation, without eliminating the parameter, find.


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Q2 :  

If x and y are connected parametrically by the equation, without eliminating the parameter, find.

x = a cos θ, y = b cos θ


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Q3 :  

If x and y are connected parametrically by the equation, without eliminating the parameter, find.

x = sin t, y = cos 2t


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Q4 :  

If x and y are connected parametrically by the equation, without eliminating the parameter, find.


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Q5 :  

If x and y are connected parametrically by the equation, without eliminating the parameter, find.


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Q6 :  

If x and y are connected parametrically by the equation, without eliminating the parameter, find.


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Q7 :  

If x and y are connected parametrically by the equation, without eliminating the parameter, find.


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Q8 :  

If x and y are connected parametrically by the equation, without eliminating the parameter, find.


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Q9 :  

If x and y are connected parametrically by the equation, without eliminating the parameter, find.


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Q10 :  

If x and y are connected parametrically by the equation, without eliminating the parameter, find.


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Q11 :  

If


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<< Previous Chapter 4 : Determinants Next Chapter 6 : Application of Derivatives >>

Exercise 5.7 : Solutions of Questions on Page Number : 183

Q1 :  

Find the second order derivatives of the function.


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Q2 :  

Find the second order derivatives of the function.


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Q3 :  

Find the second order derivatives of the function.


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Q4 :  

Find the second order derivatives of the function.


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Q5 :  

Find the second order derivatives of the function.


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Q6 :  

Find the second order derivatives of the function.


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Q7 :  

Find the second order derivatives of the function.


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Q8 :  

Find the second order derivatives of the function.


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Q9 :  

Find the second order derivatives of the function.


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Q10 :  

Find the second order derivatives of the function.


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Q11 :  

If, prove that


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Q12 :  

If findin terms of y alone.


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Q13 :  

If, show that


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Q14 :  

Ifshow that


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Q15 :  

If, show that


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Q16 :  

If, show that


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Q17 :  

If, show that


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<< Previous Chapter 4 : Determinants Next Chapter 6 : Application of Derivatives >>

Exercise 5.8 : Solutions of Questions on Page Number : 186

Q1 :  

Verify Rolle's Theorem for the function


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Q2 :  

Examine if Rolle's Theorem is applicable to any of the following functions. Can you say some thing about the converse of Rolle's Theorem from these examples?

(i)

(ii)

(iii)


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Q3 :  

If is a differentiable function and if does not vanish anywhere, then prove that.


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Q4 :  

Verify Mean Value Theorem, if in the interval, where and.


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Q5 :  

Verify Mean Value Theorem, if in the interval [a, b], where a = 1 and b = 3. Find all for which


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Q6 :  

Examine the applicability of Mean Value Theorem for all three functions given in the above exercise 2.


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<< Previous Chapter 4 : Determinants Next Chapter 6 : Application of Derivatives >>

Exercise Miscellaneous : Solutions of Questions on Page Number : 191

Q1 :  


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Q2 :  


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Q4 :  


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Q5 :  


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Q6 :  


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Q7 :  


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Q8 :  

, for some constant aand b.


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Q9 :  


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Q10 :  

, for some fixed and


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Q11 :  

, for


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Q12 :  

Find, if


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Q13 :  

Find, if


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Q14 :  

If, for, - 1 < x<1, prove that


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Q15 :  

If, for some prove that

is a constant independent of aand b.


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Q16 :  

If with prove that


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Q17 :  

If and, find


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Q18 :  

If, show that exists for all real x, and find it.


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Q19 :  

Using mathematical induction prove that for all positive integers n.


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Q20 :  

Using the fact that sin (A + B) = sin Acos B + cos A sin B and the differentiation, obtain the sum formula for cosines.


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Q21 :   Does there exist a function which is continuos everywhere but not differentiable at exactly two points? Justify your answer ?


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Q22 :  

If, prove that


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Q23 :  

If, show that


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<< Previous Chapter 4 : Determinants Next Chapter 6 : Application of Derivatives >>

Maths Part-1 - Maths : CBSE NCERT Exercise Solutions for Class 12th for Continuity and Differentiability ( Exercise 5.1, 5.2, 5.3, 5.4, 5.5, 5.6, 5.7, 5.8, miscellaneous ) will be available online in PDF book form soon. The solutions are absolutely Free. Soon you will be able to download the solutions.

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Exercise 5.1
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Exercise 5.2
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Exercise 5.3
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Exercise 5.4
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Exercise 5.5
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Question 2
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Question 7
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Exercise 5.6
Question 1
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Question 3
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Exercise 5.7
Question 1
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Question 8
Question 9
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Question 11
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Exercise 5.8
Question 1
Question 2
Question 3
Question 4
Question 5
Question 6
Exercise Miscellaneous
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Question 2
Question 3
Question 4
Question 5
Question 6
Question 7
Question 8
Question 9
Question 10
Question 11
Question 12
Question 13
Question 14
Question 15
Question 16
Question 17
Question 18
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Question 21
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Question 23