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FPSC · CSS 2015

Pure Mathematics, Paper I

100 marks · 3 hours · 8 questions
This paper Pure Mathematics · all yearsQ. 1 · Let be a subgroup of…Q. 2 · Write three non-isomorphic groups of…Q. 3 · Construct Cayley's table for Multiplication…Q. 4 · Show that W = {…Q. 5 · Define eigen value of a…Q. 6 · Find equations of tangent plane…Q. 7 · In spherical coordinates.Q. 8 · Find curvature of the space…With this paper← 20142015 →
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Section I

Q. 1
  1. (a)Let HH be a subgroup of a group GG . Prove that the normalizer of HH in GG (i.e. N(H)N(H) ) is a subgroup of GG . [10]
  2. (b)Prove that a group of prime order is cyclic. [10]
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Q. 2
  1. (a)Write three non-isomorphic groups of order 12. [10]
  2. (b)) \text{lcm}(\text{ord}(a), \text{ord}(b)) lcm ( ord ( a ) , ord ( b )) , which for this group is lcm(6,2)=6\text{lcm}(6,2) = 6 . This property clearly distinguishes it from Z12\mathbb{Z}_{12} . 3. The Dihedral Group D6D_6 The dihedral group D6D_6 represents the symmetries of a regular hexagon, and it is a non-abelian group of order 12. It consists of 6 rotations and 6 reflections. It can be formally presented as ⟨ r , s ∣ r 6 = e , s 2 = e , s r s − 1 = r − 1 ⟩ \langle r, s \mid r^6 = e, s^2 = e, srs^{-1} = r^{-1} \rangle ⟨ r , s ∣ r 6 = e , s 2 = e , sr s − 1 = r − 1 ⟩ , where rr is a rotation by 2π6\frac{2\pi}{6} and ss is a reflection. Its non-abelian nature (e.g., r s ≠ s r rs \neq sr r s  = sr ) immediately distinguishes it from both Z12\mathbb{Z}_{12} and Z6×Z2\mathbb{Z}_6 \times \mathbb{Z}_2 , which are abelian groups. (b) Prove that a group GG is isomorphic to a subgroup of group of automorphisms of GG . [10]
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Q. 3

Construct Cayley's table for Multiplication Modulo 7 of } Z_7 = \{0, 1, 2, 3, 4, 5, 6\} .

  1. (a)Show that Z7Z_7 is an integral domain. (You may use Cayley's table.) [8]
  2. (b)Give an example of zero divisor in } Z_6 = \{0,1,2,3,4,5\} . [5]
  3. (c)Is Z7Z_7 a field? Justify your answer. [6]
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Q. 4
  1. (a)Show that W = { a b c 0 } : a , b , c ∈ R W = \begin{Bmatrix} a & b \\ c & 0 \end{Bmatrix} : a, b, c ∈ \mathbb{R} W = { a c ​ b 0 ​ } : a , b , c ∈ R is a subspace of the vector space M2(R)M_2(\mathbb{R}) consisting of all 2 2×22 \times 2 matrices over R\mathbb{R} . [10]
  2. (b)Prove that if a subset { } \{v_1, v_2, \dots, v_k\} of a vector space VV is linearly dependent then one vector among v1,v2,…,vkv_1, v_2, \dots, v_k is linear combination of the remaining vectors. [4]
  3. (c)(1) What is dimension of R3\mathbb{R}^3 ? (2) Write a basis of R3\mathbb{R}^3 . (3) Is { {(1,1,0),(1,1,2),(1,0,1),(0,1,2)}⊆R3\{(1,1,0), (1,1,2), (1,0,1), (0,1,2)\} \subseteq \mathbb{R}^3 linearly dependent or independent? Justify your answer. (4) Is { {(0,0,0),(1,1,2),(1,0,1)}⊆R3\{(0,0,0), (1,1,2), (1,0,1)\} \subseteq \mathbb{R}^3 linearly dependent or independent? Justify your answer. [6]
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Section II

Q. 5
  1. (a)Define eigen value of a square matrix. [10]
  2. (b)Find eigen values and eigen vectors of A = [ 1 0 1 1 ] A = \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix} A = [ 1 1 ​ 0 1 ​ ] . [10]
  3. (c)Find reduced echelon form of the matrix A = [ 4 3 7 1 1 5 4 5 7 ] A=\begin{bmatrix} 4 & 3 & 7 \\ 1 & 1 & 5 \\ 4 & 5 & 7 \end{bmatrix} A = ​ 4 1 4 ​ 3 1 5 ​ 7 5 7 ​ ​ . [10]
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Q. 6
  1. (a)Find equations of tangent plane and normal line at a point (x1,y1,z1)(x_1, y_1, z_1) of ellipsoid x24+y29+z24=1\frac{x^2}{4} + \frac{y^2}{9} + \frac{z^2}{4} = 1 . [10]
  2. (b)Find equation of the ellipse centered at the origin, a focus at (3,0)(3,0) and vertex at (3,0)(3,0) . [5]
  3. (c)Find the polar equation of a parabola x = x=8y2x = 8y^2 . [5]
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Q. 7

In spherical coordinates.

  1. (a)Find the equation of elliptic paraboloid x = x=y2+z2x = y^2 + z^2 . [10]
  2. (b)Convert the following equation of quadratic surface to standard form. What is this surface? 4x2+y2+4z2−16x−2y+17=44x^2+y^2+4z^2-16x-2y+17=4 [10]
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Q. 8

Find curvature of the space curve

  1. (a)r ⃗ ( t ) = 2 t i ^ + t 2 j ^ + t 3 k ^ \vec{r}(t) = 2t\hat{i} + t^2\hat{j} + t^3\hat{k} r ( t ) = 2 t i ^ + t 2 j ^ ​ + t 3 k ^ [10]
  2. (b)(1) Find first fundamental form of the surface r ⃗ ( u , v ) = ( c o s u , s i n u , v ) \vec{r}(u, v) = (cosu, sinu, v) r ( u , v ) = ( cos u , s in u , v ) (2) Write formulae for normal and Guassian curvature of a surface r ⃗ = r ⃗ ( u , v ) \vec{r} = \vec{r}(u,v) r = r ( u , v ) [10]
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The 2015 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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