hpas MATHS p2

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 \text{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\}. 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)

If f:[a,b] \to R is a monotone function on [a,b], then show that f is Riemann integrable (f \in 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 f(x)=x (when x is rational) and f(x)=-x (when x is not rational) 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 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 R be a continuous function such that \int_{0}^{1} |f(x)| dx = 0. Show that f is identically zero on \mathbb{R}.

Question 8(c)

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

Question 5(b)

For any partition P of [a,b], let L(P,f) and U(P,f) denote the lower and upper Darboux sums. Let P and Q be two partitions of [a,b]. 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 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: \int_{a}^{b} f'(x) dx = f(b) - f(a) 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 is a given 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] if and only if 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 \int_{0}^{1} x^{m-1} (1-x)^{n-1} \log x dx convergent?

Question 2(c)

State Abel’s test for convergence of improper integrals. Test the convergence of \int_{0}^{\infty} \frac{\sin x}{\log(x+2)} dx.

Question 4(b)

Show that the integral \int_{a}^{\infty} (1+x^4) \frac{\sin^2 x}{x} dx 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: \int_{0}^{\infty} e^{-ax} \frac{\sin x}{x} dx \quad (a \ge 0).

Question 1(b)

Test the convergence of the integral: \int_{0}^{\infty} e^{-x^2} dx.

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 \lim_{n\to\infty} \frac{a_1b_n + a_2b_{n-1} + \dots + a_nb_1}{n} = ab. (This involves limits of sequences and is related to the Cauchy Product limit).

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 \lim_{n\to\infty} \frac{a_1+a_2+a_3+\dots+a_n}{n} = l. (This is Cesàro Mean Theorem for sequence limits).

Question 4(a)

Test the series for convergence: \sum_{n=1}^{\infty} (-1)^{n+1} \frac{\log n}{n}. (This addresses alternating series, requiring the use of Leibnitz’s test).

Question 4(b)

Show that the sequence (s_n) defined by: s_n = 1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n} is divergent. (This concerns the harmonic series/sequence divergence).

Question 4(c)

If the partial sums of the series \sum a_n are bounded, show that the series \sum_{n=1}^{\infty} a_n e^{-nt} converges for t > 0.

Question 1(a)

Show that \sum_{n=1}^{\infty} \frac{(-1)^n}{n-\log n} is a conditionally convergent series. (This relates to absolute and conditional convergence).

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. (This combines sequence algebra with comparison tests for absolute convergence).

Question 1(a)

Discuss the convergence of the series: 1 + \frac{x}{2} + \frac{2!}{3}x^2 + \frac{3!}{4}x^3 + \dots. (Requires testing for convergence, likely using the Ratio Test).

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: \sum_{n=1}^{\infty} \frac{1}{\sqrt{n!}}.

Question 7(b)

Test the convergence of the series: \sum_{n=1}^{\infty} \left(1+\frac{1}{n}\right)^{-n^2}. (This is typically solved using the Root Test).

Question 4(b)

Test the convergence of the series: x^2(\log 2)^q + x^3(\log 3)^q + x^4(\log 4)^q + \dots.

Question 1(a)

Show that \lim_{n\to\infty}\left\{\frac{1}{\sqrt{n^{2}+1}}+\frac{1}{\sqrt{n^{2}+2}}+\frac{1}{\sqrt{n^{2}+3}}+\dots+\frac{1}{\sqrt{n^{2}+n}}\right\}=1. (This concerns the limit of a sequence defined by a sum).

Question 4(a)

Find out whether the series: 1+\frac{x}{1!}+\frac{2^{2}x^{2}}{2!}+\frac{3^{3}x^{3}}{3!}+\dots 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. (This covers convergence and properties of sequences).

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. (This directly addresses Cauchy’s convergence criterion).

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 3(b)

Show that (R_{\infty}, d) defined using d(x,y)=|f(x)-f(y)| (where f(x)=x/(1+|x|)) 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)

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)

Show that the metric space (X,d) defined by d(x,y)=\int_{0}^{1} |x(t)-y(t)| dt (on continuous functions) 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 A \subseteq X. Then prove that \bar{A}=\{x \in X: d(x,A)=0\}.

Question 6(a)

Define an open sphere in a metric space. Describe the open sphere of unit radius centered at (0,0) for the Euclidean metric.

Question 6(b)

Define a closed sphere in a metric space. Describe 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|.

Question 4(a)

If (A, d) is a metric space, then show that: |d(x_1,y_1)-d(x_2,y_2)| \le d(x_1,x_2)+d(y_1,y_2).

Question 6(b)

Show that the function d^* defined by: d^*(x,y)=\frac{d(x,y)}{1+d(x,y)} 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).

Question 1(d)

Prove that the set of real numbers \mathbb{R} and the function d(x,y)=|x-y| form a metric space.

Question 4(b)

Prove that the metric space (\mathbb{R}, d) is complete, where d is the usual metric.

Question 1(b)

Evaluate the contour integral \oint_{C} \frac{e^{3z}}{(z-\log 2)^4} dz Where C is the square with vertices at \pm 1, \pm i. (This typically requires Cauchy’s Integral Formula for derivatives).

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. (This relates to the open mapping theorem or general properties of analytic functions).

Question 8(a)

Let C be the unit circle z = e^{i\theta}. First, show that for any real constant a, \int_{C} \frac{e^{az}}{z} dz = 2\pi i. Then, write this integral in terms of \theta to derive the integration formula: \int_{0}^{\pi} e^{a \cos\theta} \cos(a \sin\theta) d\theta = \pi. (This combines Cauchy’s Integral Formula with definite integral derivation).

Question 5(a)

Determine the analytic function f(z) = u+iv if u-v = \frac{\cos x + \sin x - e^{-y}}{2(\cos x - \cosh y)} and f\left(\frac{\pi}{2}\right) = 0. (This uses the Milne-Thomson method or related techniques to construct an analytic function from a linear combination of its components).

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. (This is a question on conformal mapping).

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. (This involves properties of transformations, likely bilinear/Möbius).

Question 3(b)

Using the concept of residue, determine the value of the integral: \int_{0}^{\infty} \frac{\sin(mx)}{x} dx when m > 0. (This requires the application of the Cauchy Residue Theorem to evaluate a real improper integral).

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. (This involves complex mapping and transformations).

Question 4(a)

Find the bilinear transformation which maps the points 0, i, 1+i onto 2i, -1, \text{ 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. (This directly tests Liouville’s Theorem and knowledge of standard complex functions).

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. (This requires checking the Laplace equation and constructing the analytic function).

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. (This is a conformal mapping problem).

Question 2(b)

Evaluate \int_C \frac{3z^2+z}{z^2-1} dz where C is the circle |z-1|=1. (This requires the use of Cauchy’s Integral Formula or the Residue Theorem).

Question 1(c)

If X = \sqrt{-1}, then find the value of X^X. (This is a complex number arithmetic question).

Question 5(a)

Prove that the function u(x,y) = x^3 - 3xy^2 is harmonic and obtain its conjugate. (This involves checking the Laplace equation and finding the harmonic conjugate v).

Question 5(b)

Evaluate: \int_C \frac{e^{3z}}{z-\pi i} dz where C is the circle |z-1|=4. (This requires checking if \pi i lies inside C and applying Cauchy’s Integral Formula).

Question 1(b)

Show that f(z)=|z|^{2}=x^{2}+y^{2} has a derivative at the origin. (This is a classic example testing the limits of the Cauchy-Riemann equations).

Question 2(b)

Evaluate the integral: \int_{C}\sqrt{z}dz, where C: z=z(t)=e^{it}, 0\le t\le2\pi. (This is a contour integration problem).

Question 4(b)

By using Cauchy Residue theorem, evaluate the integral: \int_{0}^{\infty}\frac{x^{2}}{x^{6}+1}dx. (Application of the Residue Theorem to real integrals).

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), where i=\sqrt{(-1)}. (This uses complex variables in the context of PDE formation).

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. (This links conformal mapping directly to the property of analyticity).

Question 5(a)

Solve the partial differential equation (Lagrange’s method type):

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 (Lagrange’s method type):

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 (Linear PDE with constant coefficients): (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: p(z+e^x) + q(z+e^y) = z^2 - e^{x+y}.

Question 6(a)

Solve the partial differential equation (Linear Homogeneous PDE): (D^2 + DD' - 6D'^2)z = y \cos x, where D = \frac{\partial}{\partial x} and D' = \frac{\partial}{\partial y}.

Question 6(b)

Solve the partial differential equation using Monge’s method: r - t \sin^2 x - p \cot x = 0.

Question 6(c)

Find the complete integral of the partial differential equation: 2\sqrt{p} + 3\sqrt{q} = 6x + 2y.

Question 1(c)

Solve (Lagrange’s method type): x\frac{\partial z}{\partial x} + y\frac{\partial z}{\partial y} = -xy.

Question 7(a)

Find the general solution of the partial differential equation (Linear PDE with constant coefficients): D(D^2 - D' + 1)(D + 2D' + 1)^3 z = 0.

Question 7(b)

Apply Charpit’s method to find the complete integral of the partial differential equation xp^2 + yq^2 = z.

Question 7(b)

Solve the partial differential equation (Lagrange’s method type): (x^2 - y^2 - z^2)p + 2xyq = 2xz.

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 (Homogeneous Linear PDE): 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).

Question 6(a)

Solve (Lagrange’s method type): (y+z)p+(z+x)q=(x+y).

Question 6(b)

Solve (Linear Non-Homogeneous PDE): 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 (Non-linear PDE): pxy+pq+qy=yz.

Question 6(b)

Solve by Monge’s method: pq=x(ps-qr).

Question 1(c)

Find the Laplace transform of: \frac{1}{t} \int_{0}^{t} e^{u} \sin u \,du.

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: \tan^{-1}\left(\frac{2}{s^2}\right).

Question 1(e)

Determine the inverse Laplace transform of: \log\frac{s+c}{s+d}, where c and d are constants.

Question 7(a)

Determine the inverse Laplace transform of: \frac{1}{s^2 - e^{-as}}.

Question 7(b)

Using the concept of Laplace Transform, find the solution of the initial value problem: t\frac{d^2y}{dt^2} + 2t\frac{dy}{dt} + 2y = 2 with y(0) = 1, and y'(0) is arbitrary.

Question 6(b)

Solve the following initial value problem using the Laplace transform: \frac{d^2y}{dx^2} - 3\frac{dy}{dx} - 4y = x^2 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: x'' + 2x' + 5x = e^{-t}\sin t 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: L\left\{ \int_{0}^{x} \frac{1-e^{-u}}{u} du \right\}(p) = \frac{1}{p}\log\left(1+\frac{1}{p}\right).

Question 7(b)

Apply the convolution theorem to find: L^{-1}\left\{ \frac{p^2}{(p^2+a^2)^2} \right\}(x).

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\}, with p>1.

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). (Note: a \ne \frac{n\pi}{2} is given in the context of the question).

Question 7(a)

On which curve can the functional:

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

Question 6(c)

Test for an extremum of the functional:

with boundary conditions y(0) = 1 and 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 (The Brachistochrone problem).

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:

given that: y(0)=0 and y_{2}=x_{2}+5.

Question 7(b)

Find the extremal curve of the functional:

given that: y(0)=0, y(\pi/2)=-1; z(0)=0, z(\pi/2)=1.

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 step size h=0.1. If the exact solution is given, compute the error and the percentage error.

Question 7(b)

Write the algorithm for the Bisection method for finding a real root of f(x)=0 which lies in [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 \int_{5}^{12} \frac{1}{x} dx and 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 (Lagrangian interpolation). 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 \int_{0}^{\pi/2} \cos x \,dx 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 \frac{dy}{dx} = x^2 + y 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 given data. Also, find the value of f(3.5).

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 I = \int_{0}^{1} \frac{dx}{1+x} 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 (Interpolation).

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 (numerical root finding).

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 values y(0)=1, y(1)=0, y(2)=1, y(3)=10 (Interpolation).

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 (Regula-Falsi), 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 (Interpolation).

Question 8(b)

Using Euler’s method with step-size 0.1, find the value of y(0.5) from the 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.