1. The value of \(\oint_C \left[ (2xy - x^2)dx + (x^2 + y^2)dy \right]\), where \(C\) is the closed region bounded by \(y = x^2\) and \(y^2 = x\)
   1. \(4\pi\)
   2. \(2\pi\)
   3. 0
   4. \(\pi\)
   Answer: C
2. \( S \) is any closed surface enclosing a volume \( V \) and \(\overrightarrow{F} = x \overrightarrow{i} + 2y \overrightarrow{j} + 3z \overrightarrow{k}\), if \(\iint_S \vec{F} \cdot \vec{n} \, dS = kV\), then \( k = \)
   1. 3
   2. 4
   3. 5
   4. 6
   Answer: D
3. If \( S \) is the surface of unit sphere \( x^2 + y^2 + z^2 = 1 \), then \(\iint_S (yz \, dy dz + xz \, dz dx + xy \, dx dy) =\)
   1. 0
   2. \(\frac{4\pi}{3}\)
   3. \(4\pi\)
   4. \(\pi\)
   Answer: A
4. The area of a plane region bounded by a simple closed curve \( C : x = \phi(t), y = \psi(t) \) in the \( xy \)-plane is
   1. \(\oint_C (x \, dy - y \, dx)\)
   2. \(\frac{1}{2} \oint_C (x \, dy - y \, dx)\)
   3. \(\frac{1}{2} \oint_C (y \, dx - x \, dy)\)
   4. \(\oint_C (y \, dx - x \, dy)\)
   Answer: B
5. The vector integral theorem which transforms surface integral into volume integral is
   1. Green’s Theorem
   2. Stokes’ Theorem
   3. Gauss’s Divergence Theorem
   4. Line Integral Theorem
   Answer: C (Gauss’s Divergence Theorem)
6. If \( S : x^2 + y^2 + z^2 = 1 \) then \(\iint_S x\,dy\,dz + y\,dz\,dx + z\,dx\,dy =\)
   1. \(\pi\)
   2. \(2\pi\)
   3. \(4\pi\)
   4. \(\frac{4\pi}{3}\)
   Answer: C
7. If \( F = xy\,i - zy\,j + x^2\,k \) and \( C : x = t^2, y = 2t, z = t^3 \) from \( t = 0 \) to \( t = 1 \) then \(\int_C F \cdot d\tau =\)
   1. \(\frac{51}{70}\)
   2. \(\frac{50}{70}\)
   3. \(\frac{30}{70}\)
   4. \(\frac{60}{70}\)
   Answer: A
8. If \( F = 3xy\,i - y^2\,j \) and \( C \) is the curve \( y = 2x^2 \) in the \( xy \)-plane from (0,0) to (1,2) then \(\int_C F \cdot d\tau =\)
   1. \(\frac{1}{6}\)
   2. \(-\frac{1}{6}\)
   3. \(\frac{7}{6}\)
   4. \(-\frac{7}{6}\)
   Answer: D
9. The value of \(\int_C \frac{dx}{x + y}\) where \( C \) is the curve \( x = at^2, y = 2at, 0 \leq t \leq 2 \)
   1. \(2 \log 2\)
   2. \(3 \log 2\)
   3. \(4 \log 2\)
   4. \(\log 2\)
   Answer: A
10. If \( F = (4xy - 3x^2 z^2)\,i + 2x^2\,j - 2x^3 z\,k \) and \( C : x^2 + y^2 = 1, z = 0 \) then \(\int_C F \cdot d\tau =\)
    1. 0
    2. 1
    3. 2
    4. 3
    Answer: A
11. If \( \vec{r} = x\vec{i} + y\vec{j} + z\vec{k} \) and \( V \) is the volume bounded by the closed surface \( S \) then \(\iint_S \vec{r} \cdot d\vec{S} =\)
    1. \( V \)
    2. \( 2V \)
    3. \( 3V \)
    4. 0
    Answer: D
12. If \( \vec{A} = 2x\vec{i} + 3y\vec{j} + 4z\vec{k} \) and \(\oint_C \vec{A} \cdot d\vec{r} = K\) then \( K =\)
    1. 9
    2. 2
    3. 0
    4. 4
    Answer: C
13. If \(\iint_S (x+z)dy\,dz + (y+z)dz\,dx + (x+y)dx\,dy = K\) and \( S \) is the surface of the sphere \( x^2 + y^2 + z^2 = 4 \) then \( K =\)
    1. \(\frac{64\pi}{3}\)
    2. \(\frac{16\pi}{3}\)
    3. \(\frac{32\pi}{3}\)
    4. \(\frac{8\pi}{3}\)
    Answer: A
14. If \( C \) is the square formed by the lines \( x = \pm 1, y = \pm 1 \) then \(\oint_C (x^2 + xy)dx + (x^2 + y^2)dy =\)
    1. 0
    2. 1
    3. 2
    4. 3
    Answer: A
15. If \( S \) is the surface of \( x^2 + y^2 + z^2 = a^2 \) above the \( xy \)-plane and \( F = y\mathbf{i} + (x - 2xz)\mathbf{j} - xy\mathbf{k} \) then \(\iint_S (\nabla \times F) \cdot \mathbf{n}\,dS =\)
    1. 0
    2. \(\overline{0}\)
    3. 1
    4. 2
    Answer: A
16. If \( F = y\mathbf{i} + z\mathbf{j} + x\mathbf{k} \) and \( S \) is the surface of the sphere \( x^2 + y^2 + z^2 = 1 \) in the \( xy \)-plane then \(\iint_S (\nabla \times F) \cdot \mathbf{n}\,dS =\)
    1. 0
    2. \(\pi\)
    3. \(-\pi\)
    4. 1
    Answer: C
17. The area of a plane region bounded by the closed curve \( C : x = a \cos t, y = a \sin t \) is
    1. \( 2\pi a^2 \)
    2. \( \pi a^2 \)
    3. \( 4\pi a^2 \)
    4. \( \frac{4}{3}\pi a^2 \)
    Answer: B
18. The vector integral theorem which transforms surface integral into volume integral is
    1. Stokes's theorem
    2. Green's theorem
    3. Gauss's divergence theorem
    4. Cauchy's theorem
    Answer: C
19. The value of \(\oint_C (x^2 + y^2)ds\) where C is the circle \(x^2 + y^2 = 4\) is:
    1. \(8\pi\)
    2. \(16\pi\)
    3. \(32\pi\)
    4. \(64\pi\)
    Answer: B
20. If \(\vec{F} = (x^2 + y)\mathbf{i} + (y^2 + x)\mathbf{j}\), then \(\int_C \vec{F} \cdot d\vec{r}\) along the parabola \(y = x^2\) from (0,0) to (1,1) is:
    1. \(\frac{5}{6}\)
    2. \(\frac{7}{6}\)
    3. \(\frac{11}{6}\)
    4. \(\frac{13}{6}\)
    Answer: C
21. The flux of \(\vec{F} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k}\) through the surface \(x^2 + y^2 + z^2 = 9\) is:
    1. \(36\pi\)
    2. \(108\pi\)
    3. \(144\pi\)
    4. \(288\pi\)
    Answer: B
22. The work done by \(\vec{F} = y\mathbf{i} + z\mathbf{j} + x\mathbf{k}\) around the triangle with vertices (1,0,0), (0,1,0), (0,0,1) is:
    1. \(\frac{1}{2}\)
    2. \(-\frac{1}{2}\)
    3. \(\frac{3}{2}\)
    4. \(-\frac{3}{2}\)
    Answer: D
23. The value of \(\iint_S (x dy dz + y dz dx + z dx dy)\) where S is the surface of the cube \(0 \leq x,y,z \leq 1\) is:
    1. 0
    2. 1
    3. 2
    4. 3
    Answer: D
24. If \(\vec{F} = (2xy + z^3)\mathbf{i} + x^2\mathbf{j} + 3xz^2\mathbf{k}\), then \(\int_C \vec{F} \cdot d\vec{r}\) from (1,1,1) to (2,1,1) along \(y = 1\), \(z = 1\) is:
    1. 3
    2. 4
    3. 5
    4. 6
    Answer: C
25. The circulation of \(\vec{F} = -y\mathbf{i} + x\mathbf{j}\) around the ellipse \(\frac{x^2}{4} + \frac{y^2}{9} = 1\) is:
    1. \(6\pi\)
    2. \(12\pi\)
    3. \(24\pi\)
    4. \(36\pi\)
    Answer: B
26. The flux of \(\vec{F} = x^2\mathbf{i} + y^2\mathbf{j} + z^2\mathbf{k}\) through the cylinder \(x^2 + y^2 = 4\), \(0 \leq z \leq 5\) is:
    1. \(40\pi\)
    2. \(80\pi\)
    3. \(120\pi\)
    4. \(160\pi\)
    Answer: D
27. The value of \(\oint_C (x^2 - y^2)dx + 2xy dy\) where C is the square with vertices (0,0), (1,0), (1,1), (0,1) is:
    1. 0
    2. 1
    3. 2
    4. 4
    Answer: A
28. The surface integral \(\iint_S (x + y + z)dS\) over the plane \(x + y + z = 1\) in the first octant is:
    1. \(\frac{\sqrt{3}}{6}\)
    2. \(\frac{\sqrt{3}}{3}\)
    3. \(\frac{\sqrt{3}}{2}\)
    4. \(\sqrt{3}\)
    Answer: C