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.For solution: Click here
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.For solution: Click here
HPAS 2015 Maths Optional Paper-2 Question 1(c)
Prove that in a metric space every open sphere is an open set.For solution: Click here
HPAS 2015 Maths Optional Paper-2 Question 1(d)
Form a partial differential equation by eliminating the functions from the equation : \( Z=f(x+iy)+\phi(x-iy), \text{ where } i=\sqrt{(-1)} \)For solution: Click here
HPAS 2015 Maths Optional Paper-2 Question 1(e)
Using method of false position, find the real root of the equation : \( x^{3}-2x-5=0 \)For solution: Click here
HPAS 2015 Maths Optional Paper-2 Question 1(f)
Draw a flow chart to find the largest number from two numbers.For solution: Click here
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.For solution: Click here
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.For solution: Click here
HPAS 2015 Maths Optional Paper-2 Question 3(a)
State and prove Darboux theorem.For solution: Click here
HPAS 2015 Maths Optional Paper-2 Question 3(b)
If function : \( f(x)=\sin x, x\in[0,\frac{\pi}{2}] \) and \( P=\{0,\frac{\pi}{2n},\frac{2\pi}{2n},….,\frac{n\pi}{2n}\} \) 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)}.For solution: Click here
HPAS 2015 Maths Optional Paper-2 Question 4(a)
Find whether the following series is convergent or divergent: \( x+\frac{1}{2}\cdot\frac{x^{3}}{3}+\frac{1.3}{2.4}\cdot\frac{x^{5}}{5}+\frac{1.3.5}{2.4.6}\cdot\frac{x^{7}}{7}+…. \)For solution: Click here
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.For solution: Click here
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.For solution: Click here
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 \(4u+3=0\)For solution: Click here
HPAS 2015 Maths Optional Paper-2 Question 6(a)
Solve : \( (y+z)p+(z+x)q=(x+y) \)For solution: Click here
HPAS 2015 Maths Optional Paper-2 Question 6(b)
Solve : \( r+s-6t=y~\cos~x \)For solution: Click here
HPAS 2015 Maths Optional Paper-2 Question 7(a)
Find the Laplace transform of \(\sin\sqrt{t}\). Also show that: \( L\left\{\frac{\cos\sqrt{t}}{\sqrt{t}}\right\}=\sqrt{\frac{\pi}{p}}\cdot e^{-\frac{1}{4p}} \)For solution: Click here
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 \(y_{2}=x_{2}+5\) .For solution: Click here
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.For solution: Click here
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 : \( \frac{dy}{dx}=x^{2}+y^{2}, y(0)=0 \)For solution: Click here