Federal Public Service Commission (FPSC)
Competitive Examination for Recruitment to BPS-17 Posts under the Federal Government
Time Allowed: 3 Hours
PURE MATHEMATICS, PAPER-I
MAXIMUM MARKS: 100 Marks
Attempt FIVE questions in all by selecting at least THREE Questions From SECTION-A and TWO Questions from SECTION-B, All questions carry EQUAL marks.
Use of Scientific Calculator is Allowed.
SECTION-A
Q. 1. (a) Let H be a normal subgroup and K a subgroup of a group G. Prove that HK is a
subgroup of G and H ∩ K is normal in K and K / (H ∩ K) ≅ HK / H. (12)
(b) Show that the number of elements in a Conjugacy class Cₐ of an element ‘a’ in a group
G is equal to the index of its normaliser. (8)
Q. 2. (a) Prove that if G is an Abelian group, then for all a, b ∈ G and integers n, (ab)ⁿ = aⁿ bⁿ. (6)
(b) Show that a subgroup of Index 2 in a group G is normal. (7)
(c) If H is a subgroup of a group G, let N(H) = {a ∈ G | a H a⁻¹ = H}. Prove that N(H) is a
subgroup of G and contains H. (7)
Q. 3. (a) Show that the set ℂ of complex numbers is a field. (6)
(b) Prove that a finite integral domain is a field. (6)
(c) Show that ℤ₆ = {0, 1, 2, 3, 4, 5} is a ring under addition mod 6 and multiplication mod 6
but not a field. Find the divisors of Zero in ℤ₆. (8)
Q. 4. (a) Let F be a field of real numbers, show that the set V of real valued continuous
functions on the closed interval [0,1] is a vector space over F and the subset Y of V
containing all functions whose nth derivatives exist, forms a subspace of V. (10)
(b) Prove that any finite dimensional vector space is isomorphic to Fⁿ. (10)
Q. 5. (a) State and prove Cayley-Hamilton theorem. (10)
(b) Use Cramer’s rule to solve the following system of linear equations: (10)
x + y + z + w = 1
x + 2y + 3z + 4w = 0
x + y + 4z + 5w = 1
x + y + 5z + 6w = 0
SECTION-B
Q. 6. (a) Prove that an equation of normal to the astroid x^(2/3) + y^(2/3) = a^(2/3) can be written in the form:
y cos θ – x sin θ = a cos 2θ
Hence show that the evolute of the curve is
(x + y)^(2/3) + (x – y)^(2/3) = 2a^(2/3) (10)
(b) If ρ₁ and ρ₂ are radii of curvature at the extremities of any chord of the Cardioid
r = a(1 + cos θ) which passes through the pole, then prove that ρ₁² + ρ₂² = (16a²) / 9. (10)
Q. 7. (a) Find an equation of the normal at any point of the curve with parametric equations: (10)
x = a(cos t + t sin t), y = a(sin t – t cos t).
Hence deduce that an equation of its evolute is x² + y² = a².
(b) Find equations of the planes bisecting the angle between the planes (10)
3x + 2y – 6z + 1 = 0 and 2x + y + 2z – 5 = 0.
Q. 8. (a) Define a surface of revolution. Write equation of a right elliptic-cone with vertex at origin. (6)
(b) Identify and sketch the surface defined by (6)
x² + y² = 2z – z².
(c) If y=f(x) has continuous derivative on [a,b] and S denotes the length of the arc of (8)
y=f(x) between the lines x=a and x=b, prove that
S = ∫_a^b √[1 + (dy/dx)²] dx.
Find the length of the parabolas y²=4ax
(i) From vertex to an extremity of the latus rectum.
(ii) Cut off by the latus rectum.
PURE MATHEMATICS, PAPER-II
Attempt FIVE questions in all by selecting at least THREE Questions From SECTION-A and TWO Questions from SECTION-B, All questions carry EQUAL marks.
Use of Scientific Calculator is Allowed.
SECTION-A
Q. 1. (a) State and prove Taylor’s theorem with Cauchy’s form of remainder. (8)
(b) Evaluate (i) lim_{x→0+} (1 / x)^(tan x)
(ii) ∫ e^(ax) sin(bx + c) dx (6)
(c) Show that ∫₀^{π/2} sinᵖ x cosᵠ x dx = ( Γ((p+1)/2) Γ((q+1)/2) ) / ( 2 Γ((p+q)/2 + 1) ). (6)
Q. 2. (a) Sketch the graph of the curve r² = a² sin 2θ, a > 0. Also write pedal equation for this curve. (8)
(b) Show that the hyperbola x²/a² – y²/b² = 1 has asymptotes y = (b/a)x and y = -(b/a)x. (6)
(c) Define extrema (local and global) of a function of two variables. Find three positive
numbers whose sum is 48 and whose product is as large as possible. (6)
Q. 3. (a) Find the volume of the tetrahedron bounded by the coordinate planes and the plane
x/a + y/b + z/c = 1, a, b, c > 0. (8)
(b) Evaluate ∫₀^{π/2} ln(sin x) dx. (6)
(c) Determine the values of x for which the power series ∑_{n=2}^∞ xⁿ / ln n converges absolutely,
converges conditionally and diverges. (6)
Q. 4. (a) Define a metric on a non-empty set X. If d is a metric on X, show that if (5+3+2
d'(x,y) = d(x,y) / (1 + d(x,y)) then d’ is also a metric on X. Also write open and closed =10)
balls (spheres) in the discrete metric space (X, d₀) with radius 1 and 1.1 centered at
some x ∈ X.
(b) Define limit point of a subset A of a metric space X. Show that an open sphere (10)
containing a limit point x of A contains infinitely many points of A other than x.
Q. 5. (a) Show that ℝⁿ is a complete metric space under the metric defined by
d(x,y) = √[ ∑_{i=1}ⁿ (ξ_i – η_i)² ], x, y ∈ ℝⁿ
Where x = (ξ₁, ξ₂, ……, ξₙ) and y = (η₁, η₂, ……, ηₙ). (7)
(b) Show that a function f : (X, d) → (Y, d′) is continuous if and only if for an open subset V
of Y, f⁻¹(V) is an open subset of X. (7)
(c) Find the radius of convergence and interval of convergence of the power series: (6)
∑_{n=0}^∞ ( (-1)ⁿ⁻¹ (x + 1)²ⁿ ) / ( (n + 1)² 5ⁿ )
SECTION-B
Q. 6. (a) If C is a continuous curve and f(z) is defined on each point of C, then prove that (10)
| ∮_C f(z) dz | ≤ ML
Where M = max |f(z)| and L is length of curve C.
(b) Suppose f(z) = U(x,y) + iV (x,y) is differentiable at a point z = x + iy, then at z the (10)
first order partial derivatives of U and V exist and satisfy Cauchy-Riemann equations:
∂U/∂x = ∂V/∂y , ∂U/∂y = -∂V/∂x.
Verify Cauchy-Riemann equations for the function f(z) = e⁻ˣ cos y – i e⁻ˣ sin y.
Q. 7. (a) Define singularity of a function f(z). Investigate for the pole, singularities and zeros, (6)
the function f(z) = z⁷.
(b) Let D be simply connected domain and f(z) be analytic in D. Let f′(z) exist and is (6)
continuous at each point of D then prove that ∮_C f(z) dz = 0, where C is any closed
contour in D.
(c) State De Moivre’s theorem and hence prove that (8)
(i) cos 5θ = 16 cos⁵ θ – 20 cos³ θ + 5 cos θ
(ii) sinⁿ θ = (-1)^{(n-1)/2} 1 / 2ⁿ⁻¹ [ sin nθ – sin(n – 2)θ + (n(n – 1) / 2) sin(n – 4)θ – …… ]
Q. 8. (a) Solve the equation z¹² – 1 = 0 and find which of its roots satisfy the equation z⁴ + z² + 1 = 0. (6)
(b) Show that multiplication of a vector z by e^{iα} where α is a real number, rotates the (6)
vector counter clockwise through an angle of measure α.
(c) Sum the series (8)
n sin θ + (n(n + 1) / 2!) sin 2θ + (n(n + 1)(n + 2) / 3!) sin 3θ + ……
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