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FPSC · CSS 2018

Pure Mathematics

100 marks · 3 hours · 7 questions
This paper Pure Mathematics · all yearsQ. 1 · Let H and K be…Q. 2 · Show that every finite integral…Q. 3 · Find condition on a,b,c so…Q. 5 · Evaluate ∫ − 1 3…Q. 6 · Find the area of the…Q. 7 · Express cos ⁡ 5 θ…Q. 8 · Find the Laurent series that…With this paper← 20172019 →
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Section A

Q. 1
  1. (a)Let H and K be normal subgroups of a group G. Show that HK is a normal subgroup of G. [10]
  2. (b)Let H and K be normal subgroups of a group G such that H < K. Then show that (G/H)/(K/H)=G/K(G/H)/(K/H) = G/K [10]
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Q. 2
  1. (a)Show that every finite integral domain is a field. [10]
  2. (b)Consider the following linear system, x + 2 y + z = 3 x + 2y + z = 3 x + 2 y + z = 3 ay+5z=1ay + 5z = 1 2x+7y+az=b2x + 7y + az = b [10]
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Q. 3
  1. (a)Find condition on a,b,c so that vector (a,b,c) in R 3 belongs to } W= span \{u_1,u_2,u_3\} where u1=(1,2,0),u2=(−1,1,2),u3=(3,0,−4)u_1 = (1,2,0), u_2 = (-1,1,2), u_3 = (3,0,-4) . [10]
  2. (b)Let W 1 and W 2 be finite dimensional subspaces of a vector space V. Show that dimW1+dimW2=dim(W1∩W2)+dim(W1+W2)dimW_1 + dimW_2 = dim ( W_1 \cap W_2) + dim ( W_1 + W_2) [10]
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Section B

Q. 5
  1. (a)Evaluate ∫ − 1 3 ∣ x − 2 ∣ d x \int_{-1}^{3}|x-2|dx ∫ − 1 3 ​ ∣ x − 2∣ d x . [10]
  2. (b)Prove that f x y ( 0 , 0 ) ≠ f y x ( 0 , 0 ) f_{xy}(0,0) \neq f_{yx}(0,0) f x y ​ ( 0 , 0 )  = f y x ​ ( 0 , 0 ) if f ( x , y ) = { x 2 y sin ⁡ 1 x x 2 + y 2 when x , y are not both 0 0 when x , y are both 0 f(x, y) = \begin{cases} \frac{x^2 y \sin\frac{1}{x}}{x^2+y^2} & \text{when } x, y \text{ are not both } 0 \\ 0 & \text{when } x, y \text{ are both } 0 \end{cases} f ( x , y ) = { x 2 + y 2 x 2 y s i n x 1 ​ ​ 0 ​ when x , y are not both 0 when x , y are both 0 ​ [10]
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Q. 6
  1. (a)Find the area of the region bounded by the cycloid x = a ( θ − sin ⁡ θ ) , y = a ( 1 − cos ⁡ θ ) x = a(\theta-\sin\theta), y = a(1-\cos\theta) x = a ( θ − sin θ ) , y = a ( 1 − cos θ ) and its base. [10]
  2. (b)Find the equation of a plane through (5,-1,4) and perpendicular to each of the planes x+y−2z−3=0x+y-2z-3=0 and 2x−3y+z=02x-3y+z=0 . [10]
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Section C

Q. 7
  1. (a)Express cos ⁡ 5 θ sin ⁡ 3 θ \cos^5 \theta \sin^3 \theta cos 5 θ sin 3 θ in a series of sines of multiples of θ\theta . [10]
  2. (b)Use Cauchy's Residue Theorem to evaluate the integral ∮C5z−2z(z−1)dz\oint_C \frac{5z-2}{z(z-1)} dz where C is the circle ∣ z ∣ = 2 |z| = 2 ∣ z ∣ = 2 , described counter clock wise. [10]
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Q. 8
  1. (a)Find the Laurent series that represent the function f ( z ) = f(z)=z+1z−1f(z) = \frac{z+1}{z-1} in the domain 1 < ∣ z ∣ < ∞ 1<|z|<\infty 1 < ∣ z ∣ < ∞ . [10]
  2. (b)Expand f ( x ) = sin ⁡ x f(x) = \sin x f ( x ) = sin x in a Fourier cosine series in the interval 0 0≤x≤π0 \le x \le \pi . [10]
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About this paper

The 2018 CSS Pure Mathematics paper set by the FPSC. Question wording only; questions marked “Not yet checked” have not been compared with the official paper yet.

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