CSS Pure Mathematics Past Paper 2023 PDF

Federal Public Service Commission (FPSC)
Competitive Examination for Recruitment to BPS-17 Posts under the Federal Government

 Time Allowed: 3 Hours 

 PURE MATHEMATICS 

 MAXIMUM MARKS: 100 Marks 

Attempt FIVE questions in all by selecting at least TWO Questions From SECTION-A & B and ONE Questions from SECTION-C, All questions carry EQUAL marks. 

 Use of Scientific Calculator is Allowed. 

 SECTION-A 

Q. 1. (a) Find the centre of S₃.
(b) Using the row operations, show that the matrix
[ 3 2 -1 ]
[ 1 2 -1 ]
[ 3 2 -5 ]
has no inverse.

Q. 2. (a) For any group G, show that G / {e} ≅ G and G / G ≅ {e}.

(b) Suppose U and W are distinct four-dimensional subspaces of a vector space V of dimension six. Find the possible dimensions of U ∩ W.

Q. 3. (a) For what value of α is the matrix
[ α+1 α+2 α-3 ]
[ 1 2 3 ]
[ 1 1 2-α ]
singular?

(b) Define T : ℝ³ → ℝ³ by T(x₁, x₂, x₃) = (x₃ – x₁, x₁ + x₃, x₁). Find N(T). Is T one-to-one?

 SECTION-B 

Q. 4. (a) Find the value of θ and the limit in order that lim_{x→0} ( sin 2x + θ sin x ) / x³ be finite.

(b) Show that x – x³/6 < sin x < x, for 0 < x < 1.

Q. 5. (a) Given that U = 1 / √(x² + y² + z²). Verify that U_xx + U_yy + U_zz = 0.
Answer:
(b) Evaluate ∬_D (x² + y²) dx dy, over the domain bounded by y² = x and x² = y.

Q. 6. (a) Evaluate ∬_R (x² + y²) dx dy, over the region bounded by xy = 1, y = 0, y = x and x = 2.

(b) Find an equation of a normal to the hyperbola x²/a² – y²/b² = 1 in the form ax cos θ + by cot θ = a² + b². Prove that the normal is the external bisector of the angle between the focal distances of its foot.

 SECTION-C 

Q. 7. (a) Determine k such that U = e^(ky) cos 2x is harmonic and find a conjugate harmonic.

(b) Evaluate ∫_C (z⁵ + 3) / (z – 1) dz from 1 to -1 along the upper arc of the unit circle.

Q. 8. (a) Find the Laurent Series of 1 / (z² – 1) in the region 0 < |z – 1| < 2.

(b) Find the residues at the singular points of (z³ – 2z² + 8z – 5) / (z⁴ – z²) which lie inside the circle |z| = 2.