Question 1(a)

Let G be the group of all 2 \times 2 non-singular matrices over the reals. Find the centre of G.

Question 2(a)

Show that a group of order p^2 is either cyclic, or the product of two cyclic groups of order p, where p is a prime number.

Question 2(c)

For every n \ge 5, show that the alternating group A_{n} is a simple group.

Question 2(a)

Let G be a group. Let Aut(G) denote the set of all automorphisms of G and let A(G) be the group of all permutations of G. Show that Aut(G) is a subgroup of A(G).

Question 1(a)

Suppose G is a finite group of order pq, where p and q are prime numbers such that p > q. Show that G has at most one subgroup of order p.

Question 2(a)

Show that every finitely generated subgroup of \langle \mathbb{Q}, +\rangle is cyclic, where \mathbb{Q} is the set of rational numbers.

Question 2(b)

Show that a subgroup of an infinite cyclic group is infinite.

Question 2(c)

Give an example of an infinite group in which every element is of finite order. Justify your answer.

Question 1(c)

Give an example of a finite abelian group which is not cyclic.

Question 2(a)

Show that a finite group having more than two elements has a non-trivial automorphism.

Question 2(b)

Prove that every quotient group of a cyclic group is cyclic. Does the converse of this statement hold? Justify your answer with an example.

Question 1(b)

Determine all group homomorphisms from S_3 to \mathbb{Z}_3.

Question 2(a)

Prove that any subgroup of a cyclic group is cyclic.

Question 2(b)

Define the order of an element in a group. Let G be a finite group of even order. Show that G has an element of order 2 and that the number of elements of order 2 in G is odd.

Question 3(a)

Show that a cyclic group is necessarily abelian. Show by an example that the converse may not be true.

Question 3(b)

Examine if there exists a one-to-one correspondence between the right and left cosets of H in G, if H is any subgroup of G.

Question 3(c)

If f:G \to G' is a homomorphism, then prove that Im(f) is a subgroup of G'.

Question 1(a)

Let X be any non-empty set and S(X) be the set of all bijections of X onto itself. Then prove that (S(X), \circ) is an abelian group if and only if X is a set with one or two elements, where \circ is the operation of composition of functions.

Question 2(a)

Let G be a group that has two subgroups of orders 45 and 75. If |G| < 400, find |G|.

Question 2(b)

Let f:G \to H be a group homomorphism with kernel K. If the orders of G, H, and K are 75, 45, and 15 respectively, find the order of the image f(G).

Question 1(a)

If a \ne e is the only element of order 2 in a group G, then prove that ax = xa for all x \in G.

Question 2(a)

Prove that a finite group of prime order does not have any proper subgroup.

Question 2(b)

Prove that the kernel of a homomorphism f from a group G to a group G’ is a normal subgroup of G.

Question 1(d)

Let G be a group of all 2\times2 non-singular matrices with real entries. Determine the centre of G.

Question 3(a)

Show that the converse of Lagrange’s theorem holds in a finite cyclic group.

Question 3(b)

Give an example of two subgroups H and K which are not normal, but HK is a subgroup.

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.

Question 2(a)

Prove that the order of an element of a group is always equal to the order of its inverse.

Question 2(b)

Prove that every homomorphic image of group G is isomorphic to some quotient group of G.

Question 1(a)

If

is a permutation on five symbols, then find A^{3} and order of A.

Question 2(a)

State and prove Cayley’s Theorem.

Question 2(b)

Let I be the additive group of Integers. Let H be the subgroup of I such that :
H=\{mx:x\in I\} ,
where m is a fixed integer. Write the element of the quotient group \frac{I}{H}.
Also prepare a composition table for \frac{I}{H} when m=5.

Question 2(b)

Let \mathcal{R}[a, b] be the set of all Riemann integrable functions on the interval [a, b]. If f:[a,b] \to \mathbb{R} is a monotone function on [a, b], then show that f \in \mathcal{R}[a, b].

Question 8(b)

Let f and g be integrable functions on [a, b]. Then, show that:
(i) f \cdot g is integrable on [a, b].
(ii) \max(f,g) and \min(f,g) are integrable on [a, b].

Question 1(b)

If f is a Riemann integrable function on the interval [a, b], then show that f^2 is also a Riemann integrable function.

Question 3(a)

Show that the function

is not Riemann integrable in the interval [a, b], but |f| is Riemann integrable.

Question 1(d)

Show by an example that if functions f and g are not Riemann integrable, then their product f \cdot g can be Riemann integrable.

Question 2(c)

If f: [a,b] \to \mathbb{R} is a step function, then show that f is Riemann integrable.

Question 3(b)

State the second mean value theorem of integral calculus. Let f:[0,1] \to \mathbb{R} be a continuous function such that \int_{0}^{1} |f(x)| dx = 0. Show that f is identically zero on [0,1].

Question 8(c)

Let f be a continuous function. Then show that f is Riemann integrable.

Question 5(a)

Let

If \int_{0}^{1} f(x) dx = \frac{7}{16}, find the value of c.

Question 5(b)

Let f:[0,1] \to \mathbb{R} be a continuous function. For any partition P of [0, 1], let L(P,f) and U(P,f) denote the lower and upper Darboux sums respectively. Let P = \{0, 0.01, 0.02, \dots, 1\} and Q = \{0, 0.001, 0.002, \dots, 1\} be two partitions of [0, 1]. Then prove that L(P,f) \le L(Q,f) \le U(Q,f) \le U(P,f).

Question 1(b)

Let p_1 and p_2 be two partitions of [a, b]. Show that L(f, p_1) \le U(f, p_2) and L(f, p_2) \le U(f, p_1).

Question 3(a)

Let a real-valued function f be defined on [0, 1] by f(x)=x for all x \in [0, 1]. Show that f is Riemann-integrable over [0, 1] and that \int_{0}^{1} f(x) dx = \frac{1}{2}.

Question 5(b)

If f:[a,b]\to R is continuous on [a, b] then show that the function is Riemann-integrable on [a, b].

Question 6(a)

Show that the identity:

is not always valid, with the help of an example.

Question 3(a)

State and prove Darboux theorem.

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)\}.

Question 3(a)

Let f be bounded on [a, b]. Then prove that f is R-integrable over [a, b] iff given \epsilon > 0 there exists a partition P of [a, b] such that :
0\le U(f,P)-L(f,P) < \epsilon

Question 3(b)

For the functions:
f(x)=x , g(x)=e^{x} ,
then verify the second mean value theorem in the interval [-1, 1].

Question 3(b)

For what values of m and n is the integral

convergent?

Question 2(c)

State Abel’s test for convergence of improper integrals. Test the convergence of

Question 4(b)

Show that the integral

is divergent.

Question 3(a)

Determine the values of s for which the improper integral \int_{0}^{\infty} e^{-sx} dx converges.

Question 3(b)

Test the convergence of the following integral:

Question 1(b)

Test the convergence of the integral :

Question 1(c)

Let Y=(y_{n}) defined inductively by y_{1}=1, y_{n+1}=(2y_{n}+3)/4 for n \ge 1.
Then show that \lim_{n\rightarrow\infty}Y=3/2.

Question 3(b)

If a series \sum x_{n} is convergent, then show that any series obtained from it by grouping the terms is also convergent and to the same value.

Question 1(a)

If the sequences \langle a_n \rangle and \langle b_n \rangle converge to finite limits a and b, respectively, then show that

Question 3(a)

If \langle S_n \rangle is a sequence of positive real numbers such that S_n = \frac{1}{2}(S_{n-1} + S_{n-2}) for all n > 2, then show that \langle S_n \rangle converges and find \lim_{n\to\infty} S_n.

Question 4(a)

Let \langle a_n \rangle be a sequence such that \lim_{n\to\infty} a_n = l. Then show that

Question 4(a)

Test the series for convergence:

Question 4(b)

Show that the sequence (s_n) defined by:

is divergent.

Question 4(c)

If the partial sums of the series \sum a_n are bounded, show that the series

converges for t > 0.

Question 1(a)

Show that

is a conditionally convergent series.

Question 3(a)

State the logarithmic test for convergence of a series. If \{a_n\} is a sequence of real numbers such that \{n^2a_n\} is a convergent sequence, show that \sum_{n=1}^{\infty} a_n is an absolutely convergent series.

Question 1(a)

Discuss the convergence of the series:

Question 1(d)

Find the \limsup and \liminf of the sequence (-1)^n + \frac{1}{n}.

Question 7(a)

Test the convergence of the series

Question 7(b)

Test the convergence of the series

Question 4(b)

Test the convergence of the series:

Question 1(a)

Show that

Question 4(a)

Find out whether the series:

is convergent or divergent for x\in R^{+}?

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.

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

Question 1(c)

Prove that every Cauchy sequence is bounded.

Question 4(a)

Find whether the following series is convergent or divergent :
x^{2}+\frac{2^{2}\cdot x^{4}}{3 \cdot 4}+\frac{2^{2}\cdot 4^{2}}{3 \cdot 4 \cdot 5 \cdot 6}\cdot x^{6}+\frac{2^{2}\cdot 4^{2}\cdot 6^{2}}{3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8}\cdot x^{8}+\dots

Question 1(d)

Show that the function f(z)=1/z^{2} is not uniformly continuous in the region |z| < 1
but is uniformly continuous in the region (1/2) \le |z| \le 1.

Question 2(b)

Show that a subset K of \mathbb{R}^n (n \ge 1) is compact if and only if K is closed and bounded.

Question 4(a)

Let C[0,1] = \{f : f:[0,1]\rightarrow\mathbb{R} \text{ is continuous on } [0,1]\}, then show that C[0,1] is a complete metric space.

Question 3(b)

Let R_\infty be the extended set of real numbers. The function d is defined by d(x,y) = |f(x) – f(y)| for all x, y \in R_\infty, where f(x) = \frac{x}{1+|x|} when -\infty < x < \infty, f(x)=1 when x=\infty, and f(x)=-1 when x=-\infty. Show that (R_\infty, d) is a bounded metric space.

Question 4(a)

Let (X, d) be a complete metric space and Y be a subspace of X. Show that Y is complete if and only if it is closed in (X, d).

Question 1(e)

Show that every closed sphere is a closed set.

Question 4(b)

Show that every compact subset F of a metric space (X, d) is closed.

Question 4(c)

Let (X, d) be a metric space. Then show that any disjoint pair of closed sets in X can be separated by disjoint open sets in X.

Question 1(a)

Show that every closed subspace of a complete metric space (X, d) is complete.

Question 3(a)

Let X be the set of all continuous real-valued functions on [0, 1], and let:

Show that the metric space (X, d) is not complete.

Question 4(c)

Define a connected metric space. Prove that every closed interval of the real line \mathbb{R} is connected.

Question 5(a)

Define a continuous function between metric spaces. Prove that any real-valued continuous function on a compact metric space is bounded.

Question 4(a)

Define a compact metric space. Prove that a closed subset of a compact space is compact.

Question 1(b)

Let (X, d) be a metric space and let A \subseteq X. Then prove that \overline{A} = \{x \in X : d(x,A) = 0\}.

Question 6(a)

Define an open sphere in a metric space. Describe with a figure the open sphere of unit radius centered at (0,0) for the metric d(z_1, z_2) = \sqrt{(x_1-x_2)^2 + (y_1-y_2)^2} defined on \mathbb{R}^2, where z_1 = (x_1, y_1) and z_2 = (x_2, y_2) are any two points of \mathbb{R}^2.

Question 6(b)

Define a closed sphere in a metric space. Describe with a figure the closed sphere of unit radius centered at (0,0) for the metric d(z_1, z_2) = |x_1-x_2| + |y_1-y_2| defined on \mathbb{R}^2, where z_1 = (x_1, y_1) and z_2 = (x_2, y_2) are any two points of \mathbb{R}^2.

Question 4(a)

If (A, d) is a metric space, then show that:

Question 6(b)

Let (X, d) be a metric space. Show that the function d^{*} defined by:

for all x, y\in X is a metric on X.

Question 1(c)

Prove that in a metric space every open sphere is an open set.

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.

Question 1(d)

Prove that the set of real numbers R and the function defined as follows :
d:d(x,y)=|x-y| \quad \forall x,y\in R
form a metric space.

Question 4(b)

Prove that the metric space (R, d) is complete, where d is the usual metric for the set of real numbers R.

Question 3(a)

Compute an upper bound for the value of the integral I=\int_{C}e^{z}dz where C is the line segment joining the points (0,0) and (1,2\sqrt{2}).

Question 3(c)

Give an example of a function which has non-removable singularity at z=0 and whose residue at that point is zero.

Question 4(b)

Determine the points at which the mapping w=z+\frac{1}{z} is not conformal and demonstrate the same by considering the image in the w-plane of the real axis in the z-plane.

Question 1(b)

Evaluate the contour integral

Where C is the square with vertices at \pm 1, \pm i.

Question 4(b)

Let G be a region and suppose that f:G \to \mathbb{C} is analytic such that f(G) is a subset of a circle. Then show that f is constant.

Question 8(a)

Let C be the unit circle z = e^{i\theta}, for -\pi \le \theta \le \pi. First, show that for any real constant a,

Then, write this integral in terms of \theta to derive the integration formula:

Question 1(c)

Find the equation whose roots are 2\cos\frac{\pi}{7}, 2\cos\frac{3\pi}{7}, and 2\cos\frac{5\pi}{7}.

Question 5(a)

Determine the analytic function f(z) = u+iv if

and f\left(\frac{\pi}{2}\right) = 0.

Question 5(b)

Show that the transformation w = z + \frac{1}{z} converts the straight line \text{arg}(z) = \alpha (where |\alpha| < \frac{\pi}{2}) into a branch of a hyperbola with eccentricity \sec\alpha.

Question 1(b)

Find a transformation w = f(z) which maps the real axis of the z-plane onto the real axis in the w-plane.

Question 3(b)

Using the concept of residue, determine the value of the integral:

when m > 0.

Question 3(c)

Let f(z) = \frac{z-1}{z+1} and let L be the line in the z-plane through z=0 and z=1+i. Find the image of L under f.

Question 4(a)

Find the bilinear transformation which maps the points 0, i, 1+i onto 2i, -1, and 0, respectively.

Question 4(b)

Prove that every bounded entire function is constant. Is f(z) = \sin z bounded on \mathbb{C}? Support your answer.

Question 1(b)

Examine if the function u = e^{2xy}\sin(x^2 – y^2) is harmonic. Find the complex function f(z) in terms of z, where f(z) = u+iv.

Question 2(a)

Find the transformation which maps the semi-infinite strip of width \pi, bounded by the lines v=0, v=\pi, and u=0, into the upper half of the z-plane.

Question 2(b)

Evaluate

where C is the circle |z-1|=1.

Question 1(c)

If X = \sqrt{-1}, then find the value of X^X.

Question 5(a)

Prove that the function u(x,y) = x^3 – 3xy^2 is harmonic and obtain its conjugate.

Question 5(b)

Evaluate:

where C is the circle |z-1|=4.

Question 1(b)

Show that :
f(z)=|z|^{2}=x^{2}+y^{2}
has a derivative at the origin.

Question 2(b)

Evaluate the integral :

Question 4(b)

By using Cauchy Residue theorem, evaluate the integral:

Question 5(a)

Determine the bilinear transformation that maps the points z=0, -i, 2i into the points w=5i, \infty, -i/3 respectively. What is the image of |z| < 1 under this transformation?

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}

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.

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.

Question 5(a)

If f(z)=u+iv is an analytic function of z=x+iy and
u-v=e^{x}(\cos y-\sin y)
find f(z) in terms of z.

Question 5(b)

If w=f(z) represents a conformal transformation of a domain D in the z-plane into a domain D’ of the w-plane, then prove that f(z) is an analytic function of z in D.

Question 1(b)

Find the complete integral of the partial differential equation

where p=\frac{\partial z}{\partial x}, q=\frac{\partial z}{\partial y}.

Question 5(a)

Find the equation of the surface which cuts orthogonally the system of surfaces 2xz+3yz=a(z+2), where a is an arbitrary constant, and passes through the circle z=0, x^{2}+y^{2}=9.

Question 8(a)

Solve the partial differential equation:

where D=\partial/\partial x and D^{\prime}=\partial/\partial y.

Question 8(b)

Solve the following non-linear partial differential equation of the second order by Monge’s method:

where p=\frac{\partial z}{\partial x}, q=\frac{\partial z}{\partial y}, s=\frac{\partial^2 z}{\partial x \partial y}, and t=\frac{\partial^2 z}{\partial y^2}.

Question 8(c)

A rod of length L has its ends A and B kept at 0^\circ C and 100^\circ C respectively until steady state conditions prevail. If the temperature of end B is suddenly reduced to 0^\circ C and kept so while end A is maintained at 0^\circ C, find the temperature distribution u(x,t).

Question 5(a)

Solve the partial differential equation:

where p = \frac{\partial z}{\partial x} and q = \frac{\partial z}{\partial y}.

Question 5(b)

Using Monge’s method, solve the wave equation r = a^2t (for a > 0), where r = \frac{\partial^2 z}{\partial x^2} and t = \frac{\partial^2 z}{\partial y^2}.

Question 6(a)

Solve the partial differential equation

where p = \frac{\partial z}{\partial x} and q = \frac{\partial z}{\partial y}.

Question 6(b)

Using Charpit’s method, find the solution of the partial differential equation p^2x + q^2y = z, where p = \frac{\partial z}{\partial x} and q = \frac{\partial z}{\partial y}.

Question 6(c)

Solve the partial differential equation (D^2 – DD' + D' – 1)z = \cos(x+2y) + e^y, where D = \frac{\partial}{\partial x} and D' = \frac{\partial}{\partial y}.

Question 5(a)

Solve the following partial differential equation using the Lagrange method:

where p = \frac{\partial z}{\partial x} and q = \frac{\partial z}{\partial y}.

Question 5(c)

Find the complete integral of the partial differential equation:

where p = \frac{\partial z}{\partial x} and q = \frac{\partial z}{\partial y}.

Question 6(a)

Solve the partial differential equation:

where D = \frac{\partial}{\partial x} and D' = \frac{\partial}{\partial y}.

Question 6(b)

Solve the partial differential equation using Monge’s method:

where p = \frac{\partial z}{\partial x}, r = \frac{\partial^2 z}{\partial x^2}, and t = \frac{\partial^2 z}{\partial y^2}.

Question 1(c)

Solve:
x\frac{\partial z}{\partial x} + y\frac{\partial z}{\partial y} = -xy

Question 7(a)

Find the general solution of the following partial differential equation:

where D \equiv \frac{\partial}{\partial x} and D' \equiv \frac{\partial}{\partial y}.

Question 7(b)

Apply Charpit’s method to find the complete integral of the partial differential equation xp^2 + yq^2 = z, where p = \frac{\partial z}{\partial x} and q = \frac{\partial z}{\partial y}.

Question 7(a)

Find the series solution of the following differential equation:

Question 7(b)

Solve the partial differential equation

where p = \frac{\partial z}{\partial x} and q = \frac{\partial z}{\partial y}.

Question 4(a)

Obtain the partial differential equation for the set of all right circular cones whose axes coincide with the z-axis.

Question 4(b)

Prove that any sufficiently differentiable function of the form F(x+kt) satisfies the wave equation F_{xx} = (1/k^2)F_{tt}.

Question 6(a)

Solve the following equation by Charpit’s method:
(p^2+q^2)y = qz

Question 6(b)

Solve:
x^2r + 2xys + y^2t = 0

Question 1(e)

Form a partial differential equation by eliminating the arbitrary function from the equation :
lx+my+nz=\phi(x^{2}+y^{2}+z^{2}).

Question 8(b)

Find a complete integral of the partial differential equation (p + q) (px+ qy) = 1.

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}

Question 6(a)

Solve:
(y+z)p+(z+x)q=(x+y)

Question 6(b)

Solve:
r+s-6t=y \cos x

Question 1(e)

Solve the following partial differential equation:
p^{2}+q^{2}-2px-2qy+1=0

Question 6(a)

Solve :
pxy+pq+qy=yz

Question 6(b)

Solve by Monge’s method :
pq=x(ps-qr)

Question 5(b)

Using Laplace transform, find the solution of the initial value problem:

where y^{\prime}=\frac{dy}{dt}.

Question 7(b)

Using the convolution theorem of Laplace transform, show that

where J_{0} denotes Bessel’s function of order zero.

Question 1(c)

Find the Laplace transform of:

Question 7(a)

If C(t) = \int_{t}^{\infty} \frac{\cos x}{x} dx , show that the Laplace transform of C(t) is \log\frac{\sqrt{s^2+1}}{s} .

Question 1(d)

Determine the inverse Laplace Transform of the function:

Question 1(e)

Determine the inverse Laplace transform of:

where c and d are constants.

Question 7(a)

Determine the inverse Laplace transform of:

Question 7(b)

Using the concept of Laplace Transform, find the solution of the initial value problem:

with y(0) = 1, and y'(0) is arbitrary.

Question 6(b)

Solve the following initial value problem using the Laplace transform:

with y(0)=2 and y'(0)=1.

Question 6(c)

Find the Laplace transform of the following periodic function f(x) with period 2\pi:

Question 1(c)

Obtain the Laplace transform of the function f(t) = e^{-3t}u(t-2).

Question 6(a)

Obtain the inverse Laplace Transform of f(s) = \log\frac{s+1}{s-1}.

Question 6(b)

Find the Laplace transform of the following periodic function f(t) with period 2c:

Question 6(c)

Solve the following Initial Value Problem using the Laplace transform:

with initial conditions x(0)=0 and x'(0)=1.

Question 3(b)

Find the value of L\{x^{7/2}\}.

Question 1(c)

Evaluate the Laplace transform of the function f(x) = (x+2)^2 e^x.

Question 7(a)

Show that:

Question 7(b)

Apply the convolution theorem to find:

Question 1(c)

Determine the inverse Laplace transform of \frac{e^{-1/s}}{s}

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}}

Question 7(a)

If
L_{n}(x)=\frac{e^{x}}{n!}\cdot\frac{d^{n}}{dx^{n}}(e^{-x}\cdot x^{n})
then find Laplace transform L\{L_{n}(x); p\}, p > 1.

Question 6(a)

Find the extremals of I[y(x)]=\int_{0}^{1}yy^{\prime 3}dx subject to the boundary conditions y(0)=0, y(1)=1 (where y^{\prime}=\frac{dy}{dx}).

Question 6(a)

Is the Jacobi condition fulfilled for the extremal of the functional

with fixed boundaries A(0,0) and B(a,0)? (where y' = dy/dx)

Question 7(a)

On which curve can the functional

with boundary conditions y(0)=0 and y(\frac{\pi}{2})=0, be extremized? (where y' = \frac{dy}{dx})

Question 7(c)

Test for an extremum of the functional:

with boundary conditions y(0) = 1, y(1) = 2, where y' = \frac{dy}{dx}.

Question 6(a)

Find the shortest distance between the parabola y=x^2 and the straight line x-y=5.

Question 8(b)

Find the extremal of the following functional

that satisfies the conditions y(0)=0, y'(0)=1, y(1)=1, and y'(1)=1.

Question 8(b)

Using the Euler equation, find the extremal of the following functional:

Question 1(d)

On which curves can the functional

with boundary conditions y(0)=0 and y(1)=1, be extremized?

Question 8(a)

Find the path on which a particle, in the absence of friction, will slide from one fixed point to another point not in the same vertical line in the shortest time under the action of gravity.

Question 2(a)

On which curve can the functional:

with y(0)=0 and y(\pi/2)=0 be extremized?

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.

Question 7(b)

Find the extremal curve of the functional :
I[y(x),z(x)]=\int_{0}^{\pi/2}\{(y^{\prime})^{2}+(z^{\prime})^{2}+2yz\}dx
given that :
y(0)=0 , y(\pi/2)=-1; z(0)=0, z(\pi/2)=1.

Question 1(e)

Solve the system of equations by Gauss elimination method:

Question 6(b)

Write the algorithm of the Regula falsi method for finding a real root of the equation f(x)=0 in the interval [a,b]. Further, develop a simple program in C language for finding a real root of the equation 4e^{-x}\sin x-1=0 by the Regula falsi method.

Question 7(a)

Using Newton’s forward difference formula, find the value of S_{n} where

Question 7(c)

Evaluate the integral \int_{0}^{1}\frac{dx}{1+x} using Simpson’s 1/3 rule with a step size of h=0.25. Compare the result with the exact analytical value and find the absolute error.

Question 1(d)

Show that the Newton-Raphson process has a quadratic convergence.

Question 6(b)

Using Euler’s method, solve the initial value problem \frac{dy}{dt} = 1 – t + 4y with y(0)=1, in the interval 0 \le t \le 0.5 with h=0.1. If the exact solution is y = -\frac{9}{16} + \frac{1}{4}t + \frac{19}{16}e^{4t}, then compute the error and the percentage error.

Question 7(b)

Write the algorithm for the Bisection method for finding a real root of the equation f(x)=0 which lies in the interval [a, b]. Further, develop a simple program in C language for finding a real root of the equation x^3 – 2x – 1 = 0 using the Bisection method.

Question 7(b)

By applying Gauss’s quadrature formula, compute the integral

Also find the error.

Question 7(c)

Show that the rate of convergence of the Newton-Raphson method is quadratic, and determine a root of the equation x^{10}-1=0 with the initial point x_0 = 0.5.

Question 8(a)

Let the polynomial \phi(x) be of the form \phi(x) = \sum_{i=0}^{n} L_i(x)y_i, where each Lagrangian function L_i(x) is a polynomial in x of degree less than or equal to n. Then show that \sum_{i=0}^{n} L_i(x) = 1.

Question 8(b)

Using the 4th order Runge-Kutta method, solve the differential equation \frac{dy}{dx} = -xy^2 with y(0)=1. Taking a step size of h=0.2, determine y(0.4).

Question 5(b)

Find the root of the equation x \sin x + \cos x = 0 by using the Newton-Raphson method.

Question 8(a)

Determine the maximum error in evaluating the integral:

by both the Trapezoidal and Simpson’s rules using four subintervals.

Question 8(b)

Apply Euler’s modified method to find the value of y at x = 0.1 correct to five decimal places, given:

with the initial condition y(0) = 0.94.

Question 1(d)

Find the value of y(0.5) for the initial value problem \frac{dy}{dx}=y, y(0)=1, using Euler’s method with a step size of h=0.1.

Question 5(b)

Obtain the interpolating polynomial by Newton’s divided difference formula for the following data. Also, find the value of f(3.5).

x:     -3,   -1,   0,   3,       5

f(x): -30, -22, -12, 330, 3458

Question 7(c)

Find a real root of the equation xe^x – \cos x = 0 using the Regula-Falsi method.

Question 8(a)

Evaluate the integral

correct to three decimal places by using the trapezoidal rule and Simpson’s one-third rule, taking h=0.25.

Question 8(c)

Write a flow chart in C language for the Runge-Kutta method of the fourth order.

Question 1(d)

Obtain the smallest positive root of the equation x \log_{10}x – 1.2 = 0, correct to 2 decimal places, using the Newton-Raphson method.

Question 5(a)

Solve the following system of equations by the Gauss-Seidel method, correct to 2 decimal places:

Question 5(b)

From the following table, obtain f(2.07) using the best formula:

x:     2.00,     2.05,     2.10,     2.15,     2.20,     2.25

f(x): 0.69315, 0.71784, 0.74194, 0.76547, 0.78846, 0.81093

Question 8(a)

What do the following explain in the C Language:
(i) ++nc
(ii) else if (condition)
(iii) tolower(c)
(iv) Pointer

Question 8(a)

Write an algorithm and draw a flow chart for integrating \int_{a}^{b} f(x) dx by the Trapezoidal rule, taking a step size h.

Question 8(b)

Write an algorithm and draw a flow chart for finding the value of y at x=x_n for the differential equation \frac{dy}{dx}=f(x,y), taking a step size h, when the initial values of x and y are given, by Euler’s method.

Question 8(b)

Find a root of the equation x \log_{10}x – 1.2 = 0 correct to four places of decimals.

Question 7(a)

By using the Newton-Raphson method, find a root of the equation:
x \sin x+\cos x=0.

Question 7(b)

Determine the cubic polynomial which takes the following values :
y(0)=1, y(1)=0, y(2)=1, y(3)=10

Question 8(a)

Draw a flowchart to find the roots of a quadratic equation ax^{2}+bx+c=0.

Question 1(e)

Using method of false position, find the real root of the equation:
x^{3}-2x-5=0

Question 1(f)

Draw a flow chart to find the largest number from two numbers.

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.

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

Question 1(f)

Draw a flow chart to print all even numbers between 1 and 50.

Question 8(a)

Find the roots of the quadratic equation :
x^{2}-5x+2=0
correct to four decimal places by the Newton-Raphson method.

Question 8(b)

Evaluate :
\int_{0}^{1}\frac{dx}{1+x^{2}}
using Simpson’s \frac{1}{3} and \frac{3}{8} rule. Hence obtain the approximate value of \pi in each case.