Not yet checked 5 of 8 questions have not yet been compared with the official paper.
Q. 1
(a)If order of an element b is n. Then show that the elements b 0 , b 1 , b 2 , . , b n − 1 b^0, b^1, b^2, ..., b^{n-1} b 0 , b 1 , b 2 , ... , b n − 1 are all distinct and bk=e iff k is divisible by n. [10]
(b)Show that the only idempotent element in a group G is its identity. [10]
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Q. 2
(a)Prove that every sub group of a cyclic group is itself cyclic. [10]
(b)Show that any two cyclic groups of the same order are isomorphic to each other. [10]
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Q. 3
(a)Define an integral domain. If p is a prime number, then show that ring of integers mod p is an integral domain. [10]
(b)Let F be a field of real numbers. Then set of all real valued functions whose nth derivative exist for n = 1,2,... is a subspace of all real valued continuous function on [0,1]. [10]
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Q. 4
(a)Let U and W be 2-dimensional subspaces of R3 . Show that U ∩ W ≠ { 0 } U \cap W \neq \{0\} U ∩ W = { 0 } . [10]
(b)Give at the least three examples of infinite dimensional vector spaces. [10]
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Q. 5
(a)Show that the intersection of any number of topologies is also a topology and is coarser than each of the given topologies. [10]
(b)Show that a subspace of a topological space is itself a topological space. [10]
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Q. 6
(a)Solve { 1 2 x 1 + 1 2 x 2 + 1 2 x 3 = 40 4 x 1 + x 2 + x 3 = 50 4 x 1 + x 2 + 3 x 3 = 50 4 x 1 + x 2 + 1 2 x 3 = 60 \begin{cases} \frac{1}{2}x_1 + \frac{1}{2}x_2 + \frac{1}{2}x_3 = 40 \\ 4x_1 + x_2 + x_3 = 50 \\ 4x_1 + x_2 + 3x_3 = 50 \\ 4x_1 + x_2 + \frac{1}{2}x_3 = 60 \end{cases} ⎩ ⎨ ⎧ 2 1 x 1 + 2 1 x 2 + 2 1 x 3 = 40 4 x 1 + x 2 + x 3 = 50 4 x 1 + x 2 + 3 x 3 = 50 4 x 1 + x 2 + 2 1 x 3 = 60 [12]
(b)Show that the system Ax=b has a unique solution if A is non-singular matrix. [8]
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Q. 7
(a)Prove that ∣ 1 + a 1 1 1 1 1 + b 1 1 1 1 1 + c 1 1 1 1 1 + d ∣ = a b c d ( 1 + 1 a + 1 b + 1 c + 1 d ) \begin{vmatrix} 1+a & 1 & 1 & 1 \\ 1 & 1+b & 1 & 1 \\ 1 & 1 & 1+c & 1 \\ 1 & 1 & 1 & 1+d \end{vmatrix} = abcd \left( 1 + \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} \right) 1 + a 1 1 1 1 1 + b 1 1 1 1 1 + c 1 1 1 1 1 + d = ab c d ( 1 + a 1 + b 1 + c 1 + d 1 ) [10]
(a)Find the solution of the given system of equations by reducing it to reduced echelon form { 6 X 1 − 6 X 2 + 6 X 3 = 6 2 X 1 − 4 X 2 − 6 X 3 = 12 10 X 1 − 5 X 2 + 5 X 3 = 30 \begin{cases} 6X_1 - 6X_2 + 6X_3 = 6 \\ 2X_1 - 4X_2 - 6X_3 = 12 \\ 10X_1 - 5X_2 + 5X_3 = 30 \end{cases} ⎩ ⎨ ⎧ 6 X 1 − 6 X 2 + 6 X 3 = 6 2 X 1 − 4 X 2 − 6 X 3 = 12 10 X 1 − 5 X 2 + 5 X 3 = 30 [10]
(b)For what Value of λ do the following homogeneous equations have non trivial solutions? Find these solutions { ( 3 − λ ) X 1 − X 2 + X 3 = 0 X 1 − ( 1 − λ ) X 2 + X 3 = 0 X 1 − X 2 + ( 1 − λ ) X 3 = 0 \begin{cases} (3-\lambda)X_1 - X_2 + X_3 = 0 \\ X_1 - (1-\lambda)X_2 + X_3 = 0 \\ X_1 - X_2 + (1-\lambda)X_3 = 0 \end{cases} ⎩ ⎨ ⎧ ( 3 − λ ) X 1 − X 2 + X 3 = 0 X 1 − ( 1 − λ ) X 2 + X 3 = 0 X 1 − X 2 + ( 1 − λ ) X 3 = 0 [10]
The 2022 PMS Punjab Mathematics paper set by the PPSC. Question wording only; questions marked “Not yet checked” have not been compared with the official paper yet.
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