Question BankApplied Mathematics-I

MEAN VALUE THEOREM

Q1. A function of is defined by , and , . Is Rolle’s theorem applicable? Q2. With the help of suitable theorem prove that

has a root between .

Q3. Verify the validity of the conditions and the conclusion of Rolle’s theorem for the Function f defined on the interval as given below:

a) on b) on c) on

Q4. Show that hence show that

Q5. Considering the functions and , prove that of Cauchy’s

Mean Value theorem is the geometric mean between and .

SUCCESSIVE DIFFERENTIATION

Q1. If then find .

Q2. Find the derivative of

a) b) c)

Q3. If , prove that

Q4. Prove that

Q5. prove that

Q6. Prove that the value of n differential coefficient of for is if n is

even and if is odd and greater than 1.

Q7. If , prove that

Q8. If , prove that

Q9. Prove that

INFINITE SERIES,EXPANSION OF FUNCTIONS & INDETERMINATE FORMS

Q1. Expand in power of x

a) b) c)

Q2. By using Taylor’s theorem arrange in power of x Q3. Calculate the value of to four places of decimals by using Taylor’s theorem.

Q4 Expand the following functions by Maclaurin’s theorem. a) b) in power of

Q5. Evaluate a) b) c)

Q6. Evaluate a) b) c)

Q7. Evaluate a) b)

c) d) e)

Q8 Test the convergence by either D’ Alembert’s ratio or Cauchy’s nth root test

a) b) >0) c) d)

PARTIAL DIFFERENTIATION

Q1. If , show that

Q2. If , prove that

Q3. If , prove that

Q4. If , prove that

, Hence deduce that

Q5. If , prove that hence show

that .

Q6. If , prove that

Q7. If satisfies the equation , prove that

.

Q8. If find

Q9. If , prove that

Q10. If prove that

where is a function of (2005)

Q11. If , show that

Q12. If , prove that

Q13. If show that

Q14. If where is a homogeneous function of degree n then show

that

Q15. If , prove that

Q16. If , prove that

Q17. Verify Euler’s theorem for

a) b) c)

Q18. If , find the value of

Q19. If , prove that

Q20. If is a homogeneous function of degree n then prove that

APPLICATION OF PARTIAL DIFFERENTIATION

Q1. Find the stationary value of Q2. Find the extreme values of the following functions: a) b) c) Q3. A rectangular box open at the top is to have volume of 108 cm. units. Find the dimensions of the box requiring least material.Q4. If measurements of radius of base and height of a right circular cone was incorrect by and then prove that there will be no volume.

Q5. The period of a simple pendulum is given by . If is computed using

. Find approximate error in if the value are .

Q6. Find the stationary points of the following functions: a) b)

Q7. If , prove that

Q8. Expand upto .

COMPLEX NUMBERS

Q1. Find the complex number if and 3

2)1arg(

z . (2002)

Q2. If tansin i , prove that

Q3. If and are two complex numbers such that , prove that

(2001)

Q4. If prove that

and

(2002)

Q5. If and are two complex numbers such that then and

Q6. If and then find .

Q7. If and are two non zero complex numbers of equal modulus and then

prove that is purely imaginary.

Q8. If and then show that

Q9. If , prove that .

Q10. Evaluate (2005)

Q11. If and Prove that

Q12. If and prove that .

Q13. If is a +ve integer and then prove that

Q14. Prove that

Q15. If prove that

Q16. If prove that

and (2005)

Q17. If are roots of equation prove that

Q18. Show that

Q19. Show that =Q20. If prove that

Q21.Using De-Moivre’s Theorem prove that

Q22.If is a cube root of unity prove that (2000)Q23. Find the roots common to and

Q24. Solve i) ii) Q25. Separate into real and imaginary parts

Q26. Prove that =

Q27. If find

Q28. Prove that

Q29. If Prove that (2005)

Q30. If prove that

Q31. Prove that Q32.If prove that (2000)

Q33. If prove that

Q34. If express and in terms of and .Hence show that and are roots of equation

Q35. If prove that

Q36. Prove that

Q37. Separate into real and imaginary parts i)

ii) Q38. Prove that the general value of is

Q39. Find the value of

Q40. Show that

Q41.If or prove that where is positive integer. (2000)

Q42. If prove that where (2003)

Q43. Prove that where

Q44. Separate into real and imaginary part of SOLUTIONS:

1.2.3.Let so VECTOR ALGEBRA & VECTOR CALCULUSQ1. Calculate the modulus and unit vector in the direction of the sum of the vectors and Q2. The coordinates of two points are and .Find the cosine of the angle between the vectors joining these points to origin.

Q3. If , , find .

Q4. Find the directional derivative of at (1,1,-2) in the direction of the tangent to the curve. at t=0Q5. Prove that

Q6.Find the angle between the surfaces at .

Q7.Find the value of where

Q8.If and compute the expression at

Q9.Find the directional derivative of at in the direction of the normal to the surface at .

Q10. Find the angle between the surfaces at .

Q11. Find the value of where

Q12.Find the directional derivative of at in the direction of the normal to the surface at .

Q13.Find the constant and such that the surface

at .

Q14. Prove that Q15. If where are constants, prove that

a) b)

Q16. Prove that the necessary and sufficient condition that a vector is of constant

direction is

Q17. If , show that

Q18.Show that the vector field given by is irrotational.Q19.Prove that

Q20.Prove that is irrotational.

Q21. Evaluate

Q22. Determine the constant , so that the vector field given by is solenoidal.

Q23. Show that the vector field given by is solenoidal.

Q24.Evaluate

Q25.Show that is irrotational as well as solenoidal.

Q26.

Q27.Find if is irrotational.Q28. Determine the constant , so that the vector field given by

is solenoidal.

Q29. Show that

Q30. Evaluate , where are constants.

Q31.Prove that is irrotational vector for any value of but is solenoidal only when .

Q32.Evaluate