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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