HPAS 2015 Maths Optional Paper-2 Question 1(a)
Show that the set:
G=\{0,1,2,3,4\}
is a finite group of order 5 with respect to addition modulo 5.
Solution:
HPAS 2015 Maths Optional Paper-2 Question 1(b)
Prove that the sequence \{x_{n}\}, where:
x_{n}=\frac{2n-7}{3n+2},\forall n\in N
is bounded monotonically increasing and convergent.
Solution:
HPAS 2015 Maths Optional Paper-2 Question 1(c)
Prove that in a metric space every open sphere is an open set.
Solution:
HPAS 2015 Maths Optional Paper-2 Question 1(d)
Form a partial differential equation by eliminating the functions from the equation:
Z=f(x+iy)+\phi(x-iy), \text{ where } i=\sqrt{-1}
Solution:
HPAS 2015 Maths Optional Paper-2 Question 1(e)
Using method of false position, find the real root of the equation:
x^{3}-2x-5=0
Solution:
HPAS 2015 Maths Optional Paper-2 Question 1(f)
Draw a flow chart to find the largest number from two numbers.
Solution:
HPAS 2015 Maths Optional Paper-2 Question 2(a)
Prove that the order of an element of a group is always equal to the order of its inverse.
Solution:
HPAS 2015 Maths Optional Paper-2 Question 2(b)
Prove that every homomorphic image of group G is isomorphic to some quotient group of G.
Solution:
HPAS 2015 Maths Optional Paper-2 Question 3(a)
State and prove Darboux theorem.
Solution:
HPAS 2015 Maths Optional Paper-2 Question 3(b)
If function:
f(x)=\sin x, x\in[0,\frac{\pi}{2}] and P=\{0,\frac{\pi}{2n},\frac{2\pi}{2n},\dots,\frac{n\pi}{2n}\}
is the partition of [0,\frac{\pi}{2}] then prove that f\in R[0,\frac{\pi}{2}] .
Also find L(f, P), U(f, P), \sup \{L(f, P)\} and \inf \{U(f, P)\}.
Solution:
HPAS 2015 Maths Optional Paper-2 Question 4(a)
Find whether the following series is convergent or divergent:
x+\frac{1}{2}\cdot\frac{x^{3}}{3}+\frac{1 \cdot 3}{2 \cdot 4}\cdot\frac{x^{5}}{5}+\frac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6}\cdot\frac{x^{7}}{7}+\dots
Solution:
HPAS 2015 Maths Optional Paper-2 Question 4(b)
Let f and g be complex continuous functions on a metric space (A, d) then f+g, fg and af are continuous on (A, d). Prove it. In the last case a is real or complex.
Solution:
HPAS 2015 Maths Optional Paper-2 Question 5(a)
Verify Cauchy’s theorem for the function 5 \sin 2z if C is the square with vertices 1\pm i and -1\pm i, where i=\sqrt{-1} and C: closed contour.
Solution:
HPAS 2015 Maths Optional Paper-2 Question 5(b)
Show that the transformation:
w=\frac{2z+3}{z-4}
maps the circle:
x^{2}+y^{2}-4x=0
into the straight line 4u+3=0.
Solution:
HPAS 2015 Maths Optional Paper-2 Question 6(a)
Solve:
(y+z)p+(z+x)q=(x+y)
Solution:
HPAS 2015 Maths Optional Paper-2 Question 6(b)
Solve:
r+s-6t=y \cos x
Solution:
HPAS 2015 Maths Optional Paper-2 Question 7(a)
Find the Laplace transform of \sin\sqrt{t}. Also show that:
L\left\{\frac{\cos\sqrt{t}}{\sqrt{t}}\right\}=\sqrt{\frac{\pi}{p}}\cdot e^{-\frac{1}{4p}}
Solution:
HPAS 2015 Maths Optional Paper-2 Question 7(b)
Find the extremum curve for the functional:
I[y(x)]=\int_{0}^{x_{2}}\sqrt{\frac{1+(y^{\prime})^{2}}{y}}dx
given that:
y(0)=0 and y_{2}=x_{2}+5.
Solution:
HPAS 2015 Maths Optional Paper-2 Question 8(a)
Find the polynomial of the lowest possible degree which assumes the values 3, 12, 15, -21; when x has values 3, 2, 1, -1 respectively.
Solution:
HPAS 2015 Maths Optional Paper-2 Question 8(b)
Using Euler’s method with step-size 0.1, find the value of y(0.5) from the following differential equation:
\frac{dy}{dx}=x^{2}+y^{2}, y(0)=0