CSS 2018 Pure Mathematics Past Paper

Paper details

Examination

CSS

Year

2018

Subject

Pure Mathematics

Paper

Single Paper

Subject type

Optional

Mode

Subjective

Questions recorded from this paper

These 8 question wordings are transcribed from CSS Vista’s structured Pure Mathematics past-paper record for CSS 2018.

1. Q1 · Normal subgroups(a) Let H and K be normal subgroups of a group G. Show that HK is a normal (10) subgroup of G. Let H and K be normal subgroups of a group G such that H ⊆ K. Then show that (b) (10) (20) (G / H ) ≅ G/K (K / H )
2. Q3 · Vector spaces(a) Find condition on a,b,c so that vector (a,b,c) in R3 belongs to (10) W= span {u 1 ,u 2 ,u 3 } where u 1 = (1,2,0), u 2 = (-1,1,2), u 3 = (3,0,-4). (b) Let W 1 and W2 be finite dimensional subspaces of a vector space V. Show that dimW 1 + dimW2 = dim ( W1  W 2 ) + dim ( W 1 + W 2 ) x 2 if x ≤ 1 (10)
3. Q2 · Solution of a system of homogeneous and non-homogeneous linear equations(a) Show that every finite integral domain is a field. (10) (b) Consider the following linear system, (10) (20) x + 2y + z = 3 ay + 5 z = 10 2 x + 7 y + az = b (i) Find the values of a for which the system has unique solution. (ii) Find the values of the pair (a, b) for which the system has more than one solution.
4. Q4 · Mean value theorems(a) Let f ( x ) = { x if x > 1 1  Does the Mean Value Theorem hold for f on  ,2 . 2  lnsin3x (b) Calculate the. lim lnsinx (10) (20) x →0
5. Q6 · The plane(a) Find the area of the region bounded by the cycloid (10) x = a (θ − sin θ ), y = a (1 − cosθ ) and its base. (b) Find the equation of a plane through (5,-1,4) and perpendicular to each of the (10) (20) planes x + y − 2 z − 3 = 0 and 2 x − 3 y + z = 0
6. Q7 · Cauchy residue theorem and contour integration(a) Express cos5 θ sin3 θ in a series of sines of multiples of θ . (10) 5z − 2 (b) Use Cauchy’s Residue Theorem to evaluate the integral ∫ dz where C (10) (20) c Z ( Z − 1) is the circle z = 2 , described counter clock wise. z +1
7. Q8 · Fourier series and Fourier transforms(a) Find the Laurent series that represent the function f ( z ) = in the domain (10) z −1 1< z < ∞ . (b) Expand f(x) = sinx in a Fourier cosine series in the interval 0 ≤ x ≤ π . (10) (20)
8. Q5 · Integrated / Cross-Syllabus Questions(a) Evaluate ∫ x − 2 dx. (10) −1 (b) Prove that f xy (0,0) ≠ f yx (0,0) if (10) (20)  2 1 f ( x, y ) =  x y sin when x, y are not both 0 x  0 when x, y are both 0

Topic-wise Pure Mathematics past-paper analysis.