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1. Expand e^(x + 1/2) in powers of x – 1. (2)
2. Differentiate ∫₁ˣ³ cos t dt with respect to 'x'. (2)
3. If the normals at two points of the parabola y² = 4x intersect on the curve, find the product of the ordinates of the two points. (2)
4. A straight line and a conic are described in polar forms as l/r = 3 cos θ + sin θ and l/r = 1 + e cos θ respectively. If the line touches the conic, find the eccentricity e and identify the conic. (2)
5. Find the values of log (-1) and log (-1). (2)
6. A particle moving along the curve C has an instantaneous velocity ż = cosec² t. Obtain the path C described by the particle, given that it passes through the point (π/4, 0). (4)
7. Compute the area between the curve y = sin 2x and the x-axis from x = 0 to x = 2π. (4)
8. Evaluate ∫₀^√π ∫ₓ²^π (sin y/√y) dy dx. (4)
9. Find the eccentricity of the ellipse whose one pair of conjugate diameters are y = x + 3 and 3y+ 2x + 5 = 0. (4)
10. Identify the points in the region R: 0 ≤ x ≤ π, 0 ≤ y ≤ 1, where the complex function f(z) = sin z has a maximum value. (4)
11. Determine the range and kernel of the linear transformation T: ℝ³ → ℝ³ defined by T(x, y, z) = (x + z, x + y + 2z, 2x + y + 3z). (5)
12. Determine the volume of the cone cut from the unit solid sphere by the cone φ = π/3, where (ρ, φ, θ) is any point in space in spherical coordinates. (5)
13. Find the partial differential equation satisfied by the set of all spheres of radius 'r' with their centers on the xy-plane. (5)
14. Construct all the distinct possible composition tables for the group (G, *), where G = {e, a, b, c}, 'e' being the identity element for the binary composition '*'. (5)
15. Prove that all the values of i^(i^(-4i)) are real. (5)
16. Let G be a positively oriented simple closed contour in the complex plane and 'z' is a point inside C. Find the value of g(z) = ∮_C (s³ + 2s)/(s – z)³ ds. (5)
17. Prove that the function f(x) = sin x is uniformly continuous on [0, ∞). (5)
18. Sum the series: 1 + (1/2) cos 2θ – (1/(2·4)) cos 4θ + ((1·3)/(2·4·6)) cos 6θ – …, where -π/2 < θ < π/2. (7)
19. Let S₂ denote the set of all symmetric matrices in M(2, ℝ), the set of all 2 × 2 matrices with entries as real numbers. Show that S₂ is a vector subspace of M(2, ℝ) over ℝ. Determine the dimension of S₂ as a vector space over ℝ. (7)
20. Show that the real part of f(z) = i/z² is harmonic in the xy-plane that doesn't contain the origin. (7)
21. Expand f(z) = -1/((z – 1)(z – 2)) using Laurent series in the domains D₁: 1 < |z| < 2 and D₂: |z| > 2. (7)
22. Let z be a complex variable and define f(z) = (z̄)²/z if z ≠ 0, and f(z) = 0 if z = 0. Show that the Cauchy–Riemann equations are satisfied at (0, 0), but f is not differentiable there. (7)
23. Evaluate ∮_C F · dr if F = (x + y)i + (2x − z)j + (y + z)k, where C is the boundary of the triangle with vertices (2, 0, 0), (0, 3, 0) and (0, 0, 6). (10)
24. Determine whether the vector field F = [ln x + sec²(x + y)]i + [sec²(x + y) + y/(y² + z²)]j + [y/(y² + z²)]k is conservative and find a potential function for it. (10)
25. Let 'p' be the permutation (a b c d e / b d e a e). Find the cyclic group generated by 'p' with permutation multiplication as composition. Also determine the inverses of the elements of this cyclic group. (10)
26. Describe the group ℤ₁₈. Determine all the subgroups of ℤ₁₈ and draw the subgroup diagram. (10)
27. Evaluate lim_(x→0) (tan x/x)^(1/x²). (10)
28. Evaluate ∫₁³ (2x + 1) dx as the limit of sums using a partition of [1, 3]. (10)
29. Evaluate ∮_C (cosh πz)/(z(z² + 1)) dz, where C is the circle in the z-plane |z| = 2 described in the anticlockwise direction. (10)
30. Solve the differential equation x³y‴ + x²y″ − 2xy′ + 2y = 0. (10)
31. Find two linearly independent series solutions in powers of x of y″ − 2xy′ + 2py = 0, where p is a constant. (10)
32. Find the Cauchy's Principal Value of the integral ∫₋∞^∞ (x sin x)/(x² + 2x + 2) dx. (10)
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