VU Midterm Past Papers
54 solved midterm past paper MCQs for MTH632 (Complex Analysis & Differential Geometry) 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. Using the Cauchy-Goursat theorem when contour C is unit circle ∫_C dz/(z^2+2z+2) =
Q2. By using the Extension of Cauchy Integral formula, ∫_C f(z)/(z-z0) dz =
Q3. The restriction of a curve to a finite interval is known as a
Q4. If p and q are points in R^n then ___ is the symmetry property of dot product
Q5. The principal value of (-5)^i is
Q6. Let f(z) be analytic function and C is contour enclosing point z0, then Cauchy Integral formula states f(z0) =
Q7. The differential of f is df = ∂f/∂x dx + ∂f/∂y dy + ∂f/∂z dz, where df calculates the small changes in the value of f when there is small change in
Q8. Let f(z)=z be a continuous complex valued function, then ∫_C f(z) dz =
Q9. The circle z = e^{iθ}, 0≤θ<2π is oriented in which direction?
Q10. For hyperbolic functions of complex number, cosh(ix) =
Q11. Let Z be a complex valued function then tanh Z =
Q12. The points at which f(z) = 1/(z^2+1) is not analytic are
Q13. Harmonic conjugate of u(x,y) = e^x cos y is
Q14. If u and v are harmonic functions then f(z) = u + iv is
Q15. A point at which a function ceases to be analytic is called a
Q16. Any polynomial p(z) = a_0 + a_1 z + ... + a_n z^n (a_n ≠ 0) of degree n ≥ 1 has at least one zero. That is, there exists at least one point z0 such that p(z0)=0.
Q17. If a function f(z) = u(x,y) + iv(x,y) is analytic in a domain D, then its component functions u and v are
Q18. The logarithmic function of e^w = z, where z = re^{iθ} and w = u + iv then
Q19. In the complex valued function f(z) = sin z, the value of u =
Q20. Let f(z) = 2z^2 + 2z - 7, then f'(z) =
Q21. The absolute value of the number z = 1 - i is
Q22. The magnitude of exp(-2+3i) is
Q23. The conjugate of complex number z = 3 - 3i is
Q24. The modulus of z = 3 + 2i is
Q25. The sum of two complex numbers, z = 3 + i, w = 1 + 2i, z + w =
Q26. If f approaches two complex numbers L1 ≠ L2 for two different curves or paths through z0 then the limit of f(z) ...
Q27. The Cauchy-Riemann equations on a pair of real-valued functions u(x,y) and v(x,y) are u_x = v_y and ...
Q28. Let z = 1 + i then arg(z) = ...
Q29. The product of complex numbers (3+2i) and (1+7i) is :
Q30. By definition, lim_{z→z0} f(z) = L means that for every ε > 0 there exists a δ > 0 such that ...
Q31. Suppose that lim_{z→z0} f(z) = A and lim_{z→z0} g(z) = B, then choose the correct option
Q32. If two complex-valued functions f and g satisfy conditions g(f(z)) = z and f(g(w)) = w, then these functions are ______ of each other
Q33. A complex-valued linear transformation is always ________ on the entire complex plane.
Q34. The complex number (4i)^(1/2) have ______ number of roots.
Q35. Let z = 3, then arg(z) =
Q36. The Jordan Curve Theorem guarantees that a simple closed curve must enclose a region.
Q37. A complex-valued linear transformation could be a composition of
Q38. If -8i = 8 exp[i(-π/2+2kπ)], then the root of (-8i)^(1/2) for k=1 is
Q39. If f(z) = 1/z, g(z) = z^2, then g(f(5)) =
Q40. If -8i = 8 exp[i(-π/2+2kπ)], then the root of (-8i)^(1/2) for k=0 is
Q41. Harmonic conjugate of u(x,y) = e^y cos x is
Q42. A ______ is a way of visualizing a given function.
Q43. Any polynomial p(z) = a0 + a1z + a2z^2 + ... + an z^n (an ≠ 0), of degree n (n ≥ 1) has at least one zero, i.e. there exists at least ______ z0 such that p(z0) = 0.
Q44. A point at which a function ceases to be analytic is called a ______ point.
Q45. Let z = 1 + i then r = ____
Q46. If f(z) = z^2, g(z) = 1/z, then g(f(5)) =
Q47. If f(z)=2z, then f(x+iy)=
Q48. If a function f(z)=u(x,y)+iv(x,y) is analytic in a domain D, then its component functions u and v are ________ in D.
Q49. Let z=7i, then r=____
Q50. If Cauchy-Riemann equations are not satisfied at a point z, then function is not ________ at z.
Q51. Consider lim f(z)=A and lim g(z)=B. Then lim [f(z)+g(z)] =
Q52. The Cauchy-Riemann equations on a pair of real-valued functions of two real variables u(x,y) and v(x,y) are u_x = v_y and
Q53. Product of complex numbers (3+5i) and (4-3i) is _______.
Q54. If f(z)=7, g(z)=1, then g(f(z)) =