VU Midterm Past Papers
192 solved midterm past paper MCQs for MTH301 (Calculus II) at Virtual University, each with the correct answer marked. Use them to learn the VU question style and test your recall. An independent study tool — not affiliated with VU.
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Q1. Face of the parabola y = ax^2 + bx + c is opening upward when
Q2. If w = x + y while x = r + s and y = r - s then which of the following is partial derivative of w w.r.t s?
Q3. The surface z=f(x,y) falls (decreases) most rapidly at any point in the ---- of Gradient(z).
Q4. XY-Plane consist of all points where
Q5. What is the condition of the extreme value theorem?
Q6. If f(x,y,z) = x^2 + 2y - 3z^2 then f_x, f_y and f_z are
Q7. Can we evaluate the following integral in given order of integration?
Q8. The integral ∫∫ dz dy dx
Q9. If a vector of magnitude 4 is making angle 30-degree with Y-axis then its component vector along x-axis is
Q10. The function f(x,y,z) = (x^2 + yz)/(xy + z) is discontinuous when
Q11. Value of the function f(x,y,z) = √(x^2 + y^2 + z^2) at the point (0,3,0) is
Q12. ∫∫∫ dz dy dx =
Q13. Range of the function f(x,y,z) = √(x^2 + y^2 + z^2) is
Q14. Let f(x,y) be a function with continuous second order partial derivatives in some circle centered at a critical point (x0,y0) and let D = f_xx(x0,y0) f_yy(x0,y0) - [f_xy(x0,y0)]^2. If D>0 and f_xx(x0,y0)>0 then
Q15. x=3cosθ, y=3sinθ; 0<θ<π is a form of a curve in the plane.
Q16. If w = f(x,y) where x = g(r,s) then ∂w/∂r =
Q17. x-coordinate of vertex of parabola y = 3x^2 + 6x + 8 is
Q18. Length or magnitude of a unit vector is
Q19. Which of the following is the representation of spherical coordinates?
Q20. Position of any point in the plane can be obtained by the two perpendicular lines known as x and y axes and together we call it as coordinates system.
Q21. Cylindrical coordinate system is used to determine the position of any object in
Q22. Which coordinate system uses two distances and one angle?
Q23. If w = f(z) where z = g(r,s) then ∂w/∂r =
Q24. Partial derivative of f(x,y) = x^2 + y^2 + 2y with respect to y is
Q25. In rectangular coordinate system, the point (1,5) lies in
Q26. The differentiable functions are continuous ..... in its domain.
Q27. Double integral is used to find the ------ of solid.
Q28. f(x,y,z)=xy+z at the point (0,1,1) is
Q29. Value of the function f(x,y,z)=xy+z at the point (1,2,3) is
Q30. The function of the form f(x,y)=3x^2y is in the domain.
Q31. Every point in three dimensional space can be described by -------- coordinates.
Q32. What are the direction cosines for the line joining the points (1, 3, 2) and (7, -2, 3)?
Q33. The angles which a line makes with positive x ,y and z-axis are known as ------------
Q34. Which of the following is geometrical representation of the equation y = 4, in three dimensional space?
Q35. Domain of the function f(x, y, z) = sqrt(x^2 + y^2 + z^2) is
Q36. If f(x, y) = x^2 y - y^3 + ln x, then ∂²f/∂x² =
Q37. Suppose f(x, y) = xy - 2y^2 where x = 3t + 1 and y = 2t. Which one of the following is true?
Q38. For a function f(x, y, z), the equation ∂²f/∂x² + ∂²f/∂y² + ∂²f/∂z² = 0 is known as -------------
Q39. Magnitude of vector a is 2, magnitude of vector b is 3 and angle between them when placed tail to tail is 45 degrees. What is a . b?
Q40. Is the function f(x, y) = xy/(x^2 + y^2) continuous at origin? If not, why?
Q41. Let R be a closed region in two dimensional space. What does the double integral over R calculates?
Q42. Which of the following formula can be used to find the Volume of a parallelepiped with adjacent edges formed by the vectors a, b and c?
Q43. Two surfaces are said to be orthogonal at a point of their intersection if their normals at that point are ---------
Q44. Two surfaces are said to intersect orthogonally if their normals at every point common to them are ----------
Q45. Let the function f(x, y) has continuous second-order partial derivatives (fxx, fyy and fxy) in some circle centered at a critical point (x0, y0) and let D = fxx(x0,y0) fyy(x0,y0) - fxy^2(x0,y0). If D > 0 and fxx(x0,y0) < 0 then f has ---------------
Q46. Let the function f(x, y) has continuous second-order partial derivatives (fxx, fyy and fxy) in some circle centered at a critical point (x0, y0) and let D = fxx(x0,y0) fyy(x0,y0) - fxy^2(x0,y0). If D = 0 then ---------------
Q47. If R = {(x, y) / 0 ≤ x ≤ 2 and 1 ≤ y ≤ 4}, then ∬_R (6x^2 + 4xy^3) dA =
Q48. If R = {(x, y) / 0 ≤ x ≤ 2 and -1 ≤ y ≤ 1}, then ∬_R (x + 2y^2) dA =
Q49. If R = {(x, y) / 2 ≤ x ≤ 4 and 0 ≤ y ≤ 1}, then ∬_R (4xe^{2y}) dA =
Q50. If R = {(x, y) / 0 ≤ x ≤ 4 and 0 ≤ y ≤ 9}, then ∬_R (3x - 4x√(xy)) dA =
Q51. Every real number corresponds to on the co-ordinate line.
Q52. There is one-to-one correspondence between the set of points on co-ordinate line and .
Q53. Which of the following is associated to each point of three dimensional spaces?
Q54. All axes are positive in octant.
Q55. The spherical co-ordinates of a point are (3, π/3, π/2). What are its cylindrical co-ordinates?
Q56. Let w = f(x, y, z) and x = g(r, s), y = h(r, s), z = t(r, s) then by chain rule ∂w/∂r =
Q57. By Extreme Value Theorem, if a function f(x, y) is continuous on a closed and bounded set R, then f(x, y) has both on R.
Q58. If R = R1 ∪ R2, where R1 and R2 are non overlapping regions then ∬_R1 f(x,y) dA + ∬_R2 f(x,y) dA =
Q59. Which of the following number is associated to each point on a co-ordinate line?
Q60. If a > 0, then the parabola y = ax^2 + bx + c opens in which of the following direction?
Q61. Rectangular co-ordinate of a point is (1, 3, -2). What is its spherical co-ordinate?
Q62. If a function is not defined at some point, then its limit ------------ exist at that point.
Q63. Suppose f(x, y) = x^3 e^{xy}. Which one of the statements is correct?
Q64. Is the function f(x,y) continuous at origin? If not, why? f(x,y) = {0 if x≥0 and y≥0, 1 otherwise}
Q65. What is the relation between the direction of gradient at any point on the surface to the tangent plane at that point?
Q66. Plane is an example of ---------------------
Q67. If R = R1 ∪ R2, where R1 and R2 are no overlapping regions then ∫∫_R f(x,y) dA =
Q68. If R = {(x,y) / 0≤x≤2 and 0≤y≤3}, then ∫∫_R (1-ye^{xy}) dA =
Q69. Which of the following is geometrical representation of set of real numbers?
Q70. There is one-to-one correspondence between the set of points on a co-ordinate line and ------------
Q71. An ordered triple corresponds to ----------- in a three dimensional space.
Q72. If the positive directions of x and y axes are known then ------------ the positive direction of z-axis.
Q73. A composition of continuous functions...
Q74. According to the Euler's theorem, the order of partial differentiation can be changed, provided the function and all of its partial derivatives are. . . . . . . .
Q75. If x= f(r,s) and r = g(t), s = k(t), then derivative of x with respect to t is written as. . . . . . . . ..
Q76. A vector in a plane is always represented by a... ......
Q77. If two vectors a and b have the relation a = λb where λ is a non-zero scalar then a and b are ..........to each other.
Q78. The function decreases most rapidly in the direction of. . . . . . ..
Q79. The direction of gradient at any point on the surfaces... ............. to the tangent plane at that point.
Q80. 2x^2+y^2=4 is the... ........ form of equation of a curve.
Q81. If f(x,y) has a relative extremum at a point (x0,y0) and both the first partial derivatives of f exist at this point, then... .
Q82. For a function f(x), f'(x) is equal to some non-zero constant, then f(x) will have . . . ......
Q83. Let x, y, z be the length, width and height of an open rectangular box. The surface area of the box will be . . ...... ..
Q84. Double integral of a function f(x,y) represents . . . . . .... of the region between the surface defined by the function and the plane which contains its domain.
Q85. For the double integral ∫∫ f(x,y) dx dy, order of integration does not matter provided that f(x,y) is. ........ ..
Q86. IF R={(x,y): 0≤x≤4 and 0≤y≤9}, then ∫∫_R (3x-4x^2 y) dA =
Q87. An ordered pair corresponds to ----------- on the plane —
Q88. Plane is a special type of -------------.
Q89. What is the general equation of parabola whose axis of symmetry is parallel to y-axis? —
Q90. If the spherical coordinates of a point are (ρ,θ,φ), then z-coordinate in rectangular coordinate system is ...
Q91. The function z = 1/(x^2+y^2) is discontinuous at origin because at the point (0,0), it...
Q92. The function f(x,y)=√(y-x) is continuous in the region --------- and discontinuous elsewhere.
Q93. Let w=f(x,y,z) and x=g(r,s), y=h(r,s), z=k(r,s) then by chain rule ∂w/∂r = ...
Q94. Every differentiable function is always ............ —
Q95. The vector a×b is............ to both a and b.
Q96. Gradient of a scalar function always results in a ........... function —
Q97. The direction of gradient at any point on the surface is................ to the tangent plane at that point. —
Q98. f_x(x0)(x-x0)+f_y(x0)(y-y0)+f_z(z0)(z-z0)=0 is the equation of . . .. —
Q99. For a function z=f(x,y), the total differential is defined as ............. —
Q100. For a function f(x,y) to have both absolute maximum and minimum, it must be Continuous on... ...............set R —
Q101. The volume of parallelepiped with dimensions x,y,z is ........ —
Q102. Let x be the length, width and height of a cube. The area of bottom will be ........ —
Q103. A double integral and iterated integral become identical provided that the integrand is . . . .. . . over the given rectangular region. —
Q104. If R={(x,y):2≤x≤4 and 0≤y≤1}, then ∬_R (4xe^y) dA =
Q105. If R={(x,y):0≤x≤2 and 0≤y≤3} then ∬_R (1-ye^x) dA =
Q106. If the spherical coordinates of a point (279) , then z-coordinate in rectangular coordinate system is .. . .. .
Q107. The function z = =r is discontinuous at origin because at the pont (0,0), 1... ........
Q108. The function f(x, y) = 4 'y — x 1s continuous in the region --------- and discontinuous elsewhere.
Q109. Let w=1f{x, y, z) and x = g(r, 5), y = hit, s), z=t{r, s) then by chain rule LE
Q110. The vector aXbis............toboth aandd.
Q111. Tthe direction of gradient at any point on the surfaces... ............. tothe tangent plane at that point.
Q112. JP x =x) + £,(p)(y — yo) + fe(p)(z~2,) =0 isthe equationof . ............
Q113. For a function z = f(x, ), the total differential is definedas. ..............
Q114. The volume of parallelepiped with dimensions x,y,z 1s..........
Q115. IfR={(x)): 28 x<4and0< y <1}, then J] Cea Ce =
Q116. If R={(x,): 0x < 2and 0£ y <3}, then [Eros Ce =
Q117. Which of the following 1s geometrical representation of set of real numbers?
Q118. There 1s one-to-one correspondence between the set of points on a co-ordinate line and ------------
Q119. According to the Euler's theorem, the order of partial differentiation can be changed, provided the function and all of its partial denvatives are. . . .......
Q120. A vector in a plane 1s always represented by a... ......
Showing the first 120 of 192 MCQs.