6)(-8 19,%11%(=%67%5-& 7)7)1%8,+,7/%2(-;% APPLIED MATH, PAPER-II

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1 6)(-8 9,%%(=%67%5-& 7)7)%8,+,7/%(-;% APPLIED MATH, PAPER-II *&]EVWEUMF FEDERAL PUBLIC SERVICE COMMISSION COMPETITIVE EXAMINATION FOR RECRUITMENT TO POSTS IN BPS-7 UNDER THE FEDERAL GOVERNMENT, 9 APPLIED MATH, PAPER-II S.No. R.No. TIME ALLOWED: HOURS MAXIMUM MARKS: NOTE: (i) Attempt FIVE question in all by selecting at least TWO questions from SECTION A, ONE question from SECTION B and TWO questions from SECTION C. All questions carry EQUAL marks. (ii) Use of Scientific Calculator is allowed. SECTION A Q.. (a) Using method of variation of parameters, find the general solution of the differential equation. e y y + y =. Find the recurrence formula for the power series solution around = for the differential equation + y + y = e. Q.. (a) Find the solution of the problem u + 6u + 9u = u() =, u () = Find the integral curve of the equation z z z + yz = ( + y ). y Q.. (a) Using method of separation of variables, solve u u < < = 9 t t >, subject to the conditions u(, t) = u(, t) = u u(,) = t Find the solution of u u u y + = 4e + cos. y y = sin 4π. t= Q.4. (a) SECTION B Define alternating symbol ijk and Kronecker delta δ. ij Also prove that = δ δ δ δ. ijk lmk il jm Using ϖ the ϖ tensor ϖ notation, ϖ ϖ prove ϖ that ϖ ϖ ϖ ϖ A B = A B B A + B A A ( ) ( ) ( ) ( ) ( )B im jl Page of

2 APPLIED MATH, PAPER-II Q.5. (a) Show that the transformation matri T = is orthogonal and right-handed. Prove that lik l jk = δ ij where l ik is the cosine of the angle between ith-ais of the system K and jth-ais of the system K. SECTION C Q.6. (a) Use Newton s method to find the solution accurate to within -4 for the equation 5 =, [, 4]. Solve the following system of equations, using Gauss-Siedal iteration method 4 + = 8, =, =. Q.7. (a) Approimate the following integral, using Simpson s rules e Approimate the following integral, using Trapezoidal rule π / 4 e d. sin d. Q.8. (a) The polynomial f() = has one real zero in [-, ]. Attempt approimate this zero to within -6, using the Regula Falsi method. Using Lagrange interpolation, approimate. f(.5), if f() =.6847, f(.) = , f(.) =.99796, f(.) =.4989, f(.4) =.674 ******************** Page of

3 FEDERAL PUBLIC SERVICE COMMISSION COMPETITIVE EXAMINATION FOR RECRUITMENT TO POSTS IN BPS-7 UNDER THE FEDERAL GOVERNMENT, Roll Number APPLIED MATH, PAPER-II TIME ALLOWED: HOURS MAXIMUM MARKS: NOTE: (i) Attempt FIVE question in all by selecting at least TWO questions from SECTION A, ONE question from SECTION B and TWO questions from SECTION C. All questions carry EQUAL marks. (ii) Use of Scientific Calculator is allowed. Q.. SECTION A Solve the following equations: (a) d y/d + 5 dy/d + 6y = d y/d + 5 y = e Q.. (a) Derive Cauchy Rieman partial differential equations. Derive Lapace Equation. Q.. Solve: (a) y ( / / y / y ) u 4 e u + 6u + 9=; Given that u()= and u ()=. Q.4. (a) SECTION B Discuss the following supported by eamples: Tensor, ijk lmk Scaler Fields for a continuously differentiable function f=f(,y,z) (5) Can we call a vector as Tensor, discuss. What is difference between a vector and a tensor? What happens if we permute the subscripts of a tensor? (5) Q.5. (a) Discuss the simplest and efficient method of finding the inverse of a square matri a ij of order. Apply any efficient method to compute the inverse of the following matri A: 5 A = 4 SECTION C Q.6. (a) Develop Gauss Siedal iterative Method for solving a linear system of equations A = b, where A is the coefficient matri. Apply Gauss Siedal iterative Method to solve the following equations: 5X + X + X = 69 X + X + X = 6 X + X + X = 4 Q.7. (a) Derive Simpson s Rule for finding out the integral of a function f() from limits =a to =b for n=6 subintervals (i.e. steps). Apply Simpson s Rule for n=6 to evaluate: f()d where f() /( ). Q.8. (a) Derive Lagrange Interpolation Formula for 4 points: A curve passes through the following points: (,),(,),(,5),(,). Apply this Lagrange Formula to interpolate the polynomial. (5) (5) ********************

4 FEDERAL PUBLIC SERVICE COMMISSION COMPETITIVE EXAMINATION FOR RECRUITMENT TO POSTS IN BS-7 UNDER THE FEDERAL GOVERNMENT, Roll Number APPLIED MATHEMATICS, PAPER-II TIME ALLOWED: THREE HOURS MAXIMUM MARKS: NOTE: (i) (ii) (iii) Attempt FIVE questions in all by selecting THREE questions from SECTION A and TWO questions from SECTION B. All questions carry equal marks. Use of Scientific Calculator is allowed. Etra attempt of any question or any part of the attempted question will not be considered. SECTION - A Q.. (a) Q.. (a) Q.. (a) Solve by method of variation of parameter Solve first order non-linear differential equation Solve d y dy y e n d d dy y y n d c u u(, t) u( l, t) u tt u(,) Sin l u (,) t z z Solve y ( y) z y Work out the two dimensional metric tensor for the coordinates p and q given by. / p ( y), q y Prove that d dc g g g g ab ac, b bc, a ab, c Page of

5 APPLIED MATHEMATICS, PAPER-II Q.4. (a) Work out the Christoffel symbols for the following metric tensor g ab Work out the covariant derivative of the tensor with components r cos sin sin cos r asin ar sin asin cos Q.5. (a) Find recurrence relations and power series solution of ( ) y y 4 Solve the Cauchy Euler s equation y y y y SECTION B ar a Q.6. (a) Find the positive solution of the following equation by Newton Raphson method sin = Solve the following system by Jacobi method: - 8 = = = 8 Q.7. (a) Evaluate the following by using the trapezoidal rule. ) ( d Q.8. (a) Evaluate the following integral by using Simpson s rule 4 e d Solve the following equation by regular falsi method: Calculate the Lagrange interpolating polynomial using the following table: also calculate f (.5). f() - ********* Page of

6 FEDERAL PUBLIC SERVICE COMMISSION COMPETITIVE EXAMINATION FOR RECRUITMENT TO POSTS IN BS-7 UNDER THE FEDERAL GOVERNMENT, Roll Number APPLIED MATHS, PAPER-II TIME ALLOWED: THREE HOURS MAXIMUM MARKS: NOTE:(i) (ii) (iii) (iv) Candidate must write Q. No. in the Answer Book in accordance with Q. No. in the Q. Paper. Attempt FIVE questions in all by selecting TWO questions from SECTION-A and ONE question from SECTION-B and TWO questions from SECTION-C. ALL questions carry EQUAL marks. Etra attempt of any question or any part of the attempted question will not be considered. Use of Scientific Calculator is allowed. SECTION-A Q.. Solve the following differential equations: e (a) y y y e y y y ye e / y / y / y e Q.. (a) Find the series solution of the following differential equation: y y Use the method of Fourier integrals to find the solution of initial value problem with the partial differential equation. u u c ; t u, f And with initial condition Q.. (a) Solve y y 5y sinln Find the solution of wave equation u u c with boundary and initial conditions t u u, t ul, t, u, f, t, t g SECTION-B Q. 4. Discuss the following terms: (i) Tensors (ii) Kronecker delta (iii) Contraction (iv) Metric Tensor (v) Contravariant tensor of order two (54=) Q. 5. (a) Prove that i i ij (log g ) Prove that n n n mnp and Hence prove that m p ijk m p mnp m p im jn in jm ijk mnp mi ni pi mj nj pj mk nk pk Page of

7 APPLIED MATHS, PAPER-II SECTION-C Q. 6. (a) (i) What is the difference between secant and false position method? Show also graphically. (ii) Prove that f ( n) n n f n Solve the following system by Jacobi method. (Up to four decimal places). 8 + y z = 8 + y + 9z = 8y + z = 5 (5+5=) Q. 7. (a) Evaluate by 8 Simpson s rule d ; with n = 6 Also calculate the absolute error. The amount A of a substance remaining in a reacting system after an interval of time t in a certain chemical eperiment is given by following data: A: t: Find t when A=8. Q. 8. (a) If f() =, show that f(a,b,c) = a + b + c Solve by trapezoidal rule sin d ; with n = 8 ***********

8 FEDERAL PUBLIC SERVICE COMMISSION COMPETITIVE EXAMINATION FOR RECRUITMENT TO POSTS IN BS-7 UNDER THE FEDERAL GOVERNMENT, Roll Number APPLIED MATHEMATICS, PAPER-II TIME ALLOWED: THREE HOURS MAXIMUM MARKS: NOTE: (i) (ii) Q.No.. (iii) (iv) Candidate must write Q. No. in the Answer Book in accordance with Q. No. in the Q. Paper. Attempt FIVE questions in all by selecting TWO questions from SECTION-A and ONE question from SECTION-B and TWO questions from SECTION-C ALL questions carry EQUAL marks. Etra attempt of any question or any part of the attempted question will not be considered. Use of Calculator is allowed. Solve the following equations: (a) d y dy Sec d d dy Cos y d y SECTION-A Q.No.. (a) Find the power series solution of the differential equation // ( ) y / y y, about the point =. Z Z Solve Z(+y) Z( y) ( y ). y Q.No.. (a) Classify the following equations: (i) Z Z Z y (ii) Z Z y Z y y y 4 u u Solve:,, t t u u u(-, t) = u(, t); (, t) (, t) for t > u(, o) = +, -< <. (5) (5) SECTION-B Q.No.4. (a) Highlight the difference between a vector and a tensor. What happens if we permute the subscripts of a tensor? Transform g ab into Cartesian coordinates. r (5) (5) Page of

9 APPLIED MATHEMATICS, PAPER-II Q.No.5. (a) a Workout the Christoffel symbols for the metric tensor g ab a sin Workout the two dimensional metric tensor for the coordinates p and q given by p ( y), q y SECTION-C Q.No.6. (a) Solve the following system of equations by Jacobi iteration method: + y z = y +z = y +5z = 54.8 Solve Sin = + Using Newton-Raphson method. Q.No.7. (a) Find the root of e = by regular falsi method correct to three decimal places. d Evaluate using (5+5) (i) Trapezoidal rule and (ii) Simpson s rule. Q.No.8. (a) Find the real root of the equation Cos = correct to seven decimal places by the iterative method. Use Lagrange s interpolation formula to find the value of y when =, if the values of and y are given below: X Y 4 6 ************* Page of

10 FEDERAL PUBLIC SERVICE COMMISSION COMPETITIVE EXAMINATION FOR RECRUITMENT TO POSTS IN BS-7 UNDER THE FEDERAL GOVERNMENT, 4 APPLIED MATHEMATICS, PAPER-II Roll Number TIME ALLOWED: THREE HOURS MAXIMUM MARKS: NOTE:(i) Candidate must write Q.No. in the Answer Book in accordance with Q.No. in the Q.Paper. (ii) Attempt FIVE questions in all by selecting TWO questions from SECTION-A, ONE question from SECTION-B and TWO questions from SECTION-C. ALL questions carry EQUAL marks. (iii) No Page/Space be left blank between the answers. All the blank pages of Answer Book must be crossed. (iv) Etra attempt of any question or any part of the attempted question will not be considered. (v) Use of Calculator is allowed. Q. No.. (a) Solve the initial-value problem dy ;( ) y. d y SECTION-A Initially there were milligrams of a radioactive substance present. After 6 hours the mass decreased by %. If the rate of decay is proportional to the amount of the substance present at any time, find the amount remaining after 4 hours. Q. No.. (a) Solve. ( ) y y y Obtain the partial differential equation by elimination of arbitrary functions, asin bcos y z (take z as dependent variable). Q. No.. (a) Solve the partial differential equation u u yy u t, u(, y,)( t, u,) a y t subject to the conditions u(,,)( t, u,) a t and the initial condition, u(, y,)(,) y. Solve r () a b s abt y by Monge s method. SECTION-B Q. No. 4. (a) Prove that if Ai, B j, and Ck are three first order tensors, then their product A B C ( i, j, k,,) is a tensor of order, while i j k A B C ( i, j,,) form a first order tensor. i j k A... If i i i in is a tensor of order n, then its partial derivative with respect to p that is A i... i i in is also a tensor of order n+. p Page of

11 APPLIED MATHEMATICS, PAPER-II Q. No. 5. (a) 6 Show that the transformation 6 7 is orthogonal and 6 right-handed. A second order tensor Aij is defined in the system O by A i, j,,. Evaluate its components at the point P where ij i j,. Also evaluate the component A of the tensor at P. The Christofell symbols of the second kind denoted by m are defined ij m mk g ij,( k,, i j, k,...). n ij m m m Prove that (i), (ii) [ ij, k] gmk, ij ji ij ij g im i jm i (iii) g g k. km km SECTION-C Q. No. 6. (a) Apply Newton-Raphson s method to determine a root of the equation * 8 f () cos e such that f (),where * is the approimation to the root. Consider the system of the equations 7, Solve the system by using Gauss-Seidel iterative method and perform three iterations. Q. No. 7. (a) Use the trapezoidal and Simpson s rules to estimate the integral f ()( d 7 5) d. Find the approimate root of the equation f (). Q. No. 8. (a) Find a 5 th degree polynomial which passes through the 6 points given below f() Determine the optimal solution graphically to the linear programming problem, Minimize z 6 subject to 4, ************* Page of

12 FEDERAL PUBLIC SERVICE COMMISSION COMPETITIVE EXAMINATION FOR RECRUITMENT TO POSTS IN BS-7 UNDER THE FEDERAL GOVERNMENT, 5 APPLIED MATHEMATICS, PAPER-II Roll Number TIME ALLOWED: MAXIMUM MARKS: THREE HOURS NOTE:(i) Attempt FIVE questions in all by selecting TWO questions from SECTION-A, ONE question from SECTION-B and TWO questions from SECTION-C. ALL questions carry EQUAL marks. (ii) Candidate must write Q.No. in the Answer Book in accordance with Q.No. in the Q.Paper. (iii) All the parts (if any) of each Question must be attempted at one place instead of at different places. (iv) Candidate must write Q. No. in the Answer Book in accordance with Q. No. in the Q.Paper. (v) No Page/Space be left blank between the answers. All the blank pages of Answer Book must be crossed. (vi) Etra attempt of any question or any part of the attempted question will not be considered. (vii) Use of Calculator is allowed. Q. No.. (a) Solve the initial value problem. dy y, y() d y SECTION-A Solve y 4y' 4y e Q. No.. Solve the following equations: d y dy (a) ( ) y d d d y d dy d cosec Q. No.. (a) Classify the following: (5 each) (i) U ( a y ) U,, a y a yy (ii) U 6U 9U y Solve u u 5 t u(, t) u(5, t) u(,) ( 5) (,) u t y yy SECTION-B Q. No. 4. (a) Prove that if Ai and B j are two first order tensors, then their product A B ( i, j,,) is a second order tensor. i j If,, ) is a scalar point function then (c) ( order tensor. Find the invariant of the following second order tensor are the components of a first i (7) (7) (6) Page of

13 APPLIED MATHEMATICS, PAPER-II Q. No. 5. (a) Verify that the transformation (5 4 ) 5 ( ) ( ) 5 Is orthogonal and right handed. A vector field A is defined in the system O by A Evaluate the components A, j, A A of the vector field in the new system O. Prove that any second order tensor Aij can be written as the sum of a deviator and an isotropic tensor. (c) If aij a ji are constants. Calculate. X X k m ( a X ij i X j ) (7) (7) (6) SECTION-C Q. No. 6. (a) Find the real root of the equation by using Newton Raphson s method. cos Solve the following system of equations by Gauss-Seidel method. Take initial approimation as,,. Perform Iterations Q. No. 7. (a) Find the real root of the equation 49 by Regular falsi method. Take the interval of the root as (,) and perform 4 iterations.. Find a polynomial which possess through the following points: : f ( ) : 5 Q. No. 8. (a) Use the langrage s Interpolation formula to find the value f() if the values of and f() are given below : 5 7 f() Evaluate d using Simpson s rule and trapezoidal rule for n = 6 ************* Page of

14 FEDERAL PUBLIC SERVICE COMMISSION COMPETITIVE EXAMINATION FOR RECRUITMENT TO POSTS IN BS-7 UNDER THE FEDERAL GOVERNMENT, Roll Number APPLIED MATHEMATICS, PAPER-I TIME ALLOWED: THREE HOURS MAXIMUM MARKS: NOTE: (i) (ii) (iii) Attempt FIVE questions in all by selecting THREE questions from SECTION A and TWO questions from SECTION B. All questions carry equal marks. Use of Scientific Calculator is allowed. Etra attempt of any question or any part of the attempted question will not be considered. SECTION - A Q.. (a) Find the divergence and curl f If f yziˆ z y) ˆ ( j ( y z ) kˆ Also find a function such that f Q.. (a) Find the volume R y dawhere R is the region bounded by the line y = and the parabola y 6. Q.. (a) Q.4. (a) Evaluate the following line intergral: c y d dy where c c is the line segment joining the points (-5, -) to (, ), and c = c is the arc of the parabola = 4 - y. Three forces P, Q and R act at a point parallel to the sides of a triangle ABC taken in the same order. Show that the magnitude of the resultant is p Q R QRcos A RP cos B PQ cosc A hemispherical shell rests on a rough inclined plane whose angle of friction is. Show that the inclination of the plane base to the horizontal cannot be greater than arcsin( sin ) A uniform square lamina of side a rests in a vertical plane with two of its sides in contact with two smooth pegs distant b apart and in the same horizontal line. Show that if b a, a non symmetric position of equilibrium is possible in which b(sin cos ) a Find the centre of mass of a semi circular lamina of radius a whose density varies as the square of the distance from the centre. Page of

15 APPLIED MATHEMATICS, PAPER-I Q.5. (a) Q.6. (a) Q.7. (a) Q.8. (a) Evaluate the integral ( y ) dyd also show that the order of integration is immaterial. Find the directional derivative of the function at the point P along z ais f (, y) 4z y z, P (,,) SECTION B A particle is moving along the parabola = 4ay with constant speed v. Determine the tangential and the normal components of its acceleration when it reaches the point whose abscissa is 5a Find the distance travelled and the velocity attained by a particle moving in a straight line at t t sin t e any time t, if it starts from rest at t = and is subject to an acceleration A particle moves in the y plane under the influence of a force field which is parallel to the ais of y and varies as the distance from ais. Show that, if the force is repulsive, the path of the particle supposed not straight and then where a and b are constants. y = a cosh n + a sinh n Discuss the motion of a particle moving in a straight line with an acceleration, where is the distance of the particle from a fied point O on the line, if it starts at t = from a point = c with the velocity c /. A battleship is steaming ahead with speed V and a gun is mounted on the battleship so as to point straight backwards and is set at angle of elevation. If v is the speed of projection v (relative to the gun) show that the range is sin( v cos V ) g n n Show that the law of force towards the pole of a particle describing the survey r a cos n n ( n ) h a is given by f where h is a constant. n r ********* Page of

16 FEDERAL PUBLIC SERVICE COMMISSION COMPETITIVE EXAMINATION FOR RECRUITMENT TO POSTS IN BS-7 UNDER THE FEDERAL GOVERNMENT, Roll Number TIME ALLOWED: THREE HOURS MAXIMUM MARKS: NOTE: (i) (ii) APPLIED MATHEMATICS, PAPER-I Candidate must write Q.No. in the Answer Book in accordance with Q.No. in the Q.Paper. Attempt FIVE questions in all by selecting THREE questions from SECTION-A and TWO questions from SECTION-B. All questions carry EQUAL marks. (iii) Use of Calculator is allowed. (iv) Etra attempt of any question or any part of the attempted question will not be considered. SECTION-A Q.. (a) Find a function φ such that φ= f Prove that f ˆ yj ˆ kˆ n n n Q.. (a) Show that for any vectors a and b Prove that a b a b a b b c a b c a a b c Q.. (a) Q.4. (a) The greatest result that two forces can have is of magnitude P and the least is of magnitude Q. Show That when they act an angle α their resultant is of magnitude P cos / Q sin / A uniform rod of length a rests in equilibrium against a smooth vertical wall and upon a smooth peg at a distance b from the wall. Show that in the position of equilibrium the rod is inclined to the wall at an angle sin b a Three forces P, Q and R act along the BC, CA and AB respectively of triangle ABC. Prove that if P cos A+Q cos B + R cos C =, then the line of action of the resultant passes through the circum center of the triangle. A sphere of weight W and radius a is suspended by a string of length l from a point P and a weight w is also suspended from P by a string sufficiently long for the weight to hang below the sphere. Show that the inclination of the first string to the vertical is wa sin ( W w)( al) Page of

17 APPLIED MATHS, PAPER-I Q.5. (a) Find the volume R 4 y da where R is the region bounded by the parabola y and the line y =. Evaluate the following line integral c bonded by the triangle having the vertices (-,) to (,), and (,) dy Q.6. Q.7. (a) Q.8. (a) SECTION-B (a) The position of a particle moving along an ellipse is given by r a cos t ˆ b sin tˆj. If a > b, find the position of the particle where its velocity has maimum or minimum magnitude. Prove that the speed at any point of a central orbit is given by: vp = h, When h is the areal speed and p is the perpendicular distance from the centre of force, of the tangent at the point, Find the epression for v when a particle subject to the inverse square law of force describes an ellipse, a parabolic and hyperbolic orbit. A particle is moving with the uniform speed v along the curve y a a Show that its acceleration has the maimum value at 5 v 9a An aeroplane is flying with uniform speed v in an arc of a vertical circle of radius a, whose centre is a height h vertically above a point O of the ground. If a bomb is dropped from the aeroplane when at a height Y and strikes the ground at O, show that Y satisfies the equations KY Y a hk K h a, ga where K h v Find the tangential and normal components of the acceleration of a particle describing the ellipse y b a With uniform speed v when the particle is a a > b Find the velocity acquired by a block of wood of mass M lb., which is free to recoil when it is struck by a bullet of mass m lb. moving with velocity v in a direction passing through the centre of gravity. If the bullet is embedded a ft., show that the resistance of Mm the wood to the bullet, supposed uniform, is lb.wt. and that the time of ( M m) ga a ma penetration is sec., during which time the block will move ft. v m M ********* Page of

18 FEDERAL PUBLIC SERVICE COMMISSION COMPETITIVE EXAMINATION FOR RECRUITMENT TO POSTS IN BS-7 UNDER THE FEDERAL GOVERNMENT, 4 APPLIED MATHEMATICS, PAPER-I Roll Number TIME ALLOWED: THREE HOURS MAXIMUM MARKS: NOTE:(i) Candidate must write Q.No. in the Answer Book in accordance with Q.No. in the Q.Paper. (ii) Attempt FIVE questions in all by selecting THREE questions from SECTION-A and TWO questions from SECTION-B. ALL questions carry EQUAL marks. (iii) No Page/Space be left blank between the answers. All the blank pages of Answer Book must be crossed. (iv) Etra attempt of any question or any part of the attempted question will not be considered. (v) Use of Calculator is allowed. SECTION-A Q. No.. (a) prove that curl( F ) ( grad) F. If F is irrotational and (, y, z) is a scalar function. Determine whether the line integral: (yz d ( z zcosyz) dy ( yz ycosyz) dz is independent of the c path of integration? If so, then compute it from (,,) to (,,). Q. No.. (a) State and prove Stoke s Theorem. Verify Stoke s Theorem for the function F i y j integrated round the square in the plane z= and bounded by the lines = y =, = y = a. Q. No.. (a) Three forces act perpendicularly to the sides of a triangle at their middle points and are proportional to the sides. Prove that they are in equilibrium. Three forces P, Q, R act along the sides BC, CA, AB respectively of a triangle ABC. Prove that, if P Sec A + Q Sec B + R Sec C =, then the line of action of the resultant passes through the orthocentre of the triangle. Q. No. 4. (a) Find the centroid of the surface formed by the revolution of the cardioide r a ( Cos ) about the initial line. A uniform ladder rests with its upper end against a smooth vertical wall and its foot on rough horizontal ground. Show that the force of friction at the ground is tan vertical. Q. No. 5. (a) Define briefly laws of friction give atleast one eample of each law. A uniform semi-circular wire hangs on a rough peg, the line joining its etremities making an angle of 45 o with the horizontal. If it is just on the point of slipping, find the coefficient of friction between the wire and the peg. Page of

19 APPLIED MATHEMATICS, PAPER-I SECTION-B Q. No. 6. (a) If a point P moves with a velocity v given by v =n (a +b+c), show that P eecutes a simple harmonic motion. Find the center, the amplitude and the time-period of the motion. A particle P moves in a plane in such a way that at any time t its distance from a fied point O is r= at+bt and the line connecting O and P makes an angle ct with a fied line OA. Find the radial and transverse components of the velocity and acceleration of the particle at t =. Q. No. 7. (a) A particle of mass m moves under the influence of the force F a ( i Sin t j Cos t). If the particles is initially at rest on the origin, a prove that the work done upto time t is given by ( Cos t), and that m a the instantaneous power applied is Sin t. m A battleship is streaming ahead with speed V, and a gun is mounted on the battleship so as to point straight backwards, and is set an angle of elevation a, if v is the speed of projection relative to the gun, show that the range is v g o o Sin ( v Cos V ). Also prove that the angle of elevation for maimum o range is V V 8v arccos. 4v Q. No. 8. (a) Show that the law of force towards the pole, of a particle describing the curve n n ( n ) h a r a Cos n is given by f. n r A bar ft. long of mass Ib., lies on a smooth horizontal table. It is struck horizontally at a distance of 6 inches from one end, the blow being perpendicular to the bar. The magnitude of the blow is such that it would impart a velocity of ft./sec. to a mass of Ib. Find the velocities of the ends of the bar just after it is struck. ************* Page of

20 FEDERAL PUBLIC SERVICE COMMISSION COMPETITIVE EXAMINATION FOR RECRUITMENT TO POSTS IN BS-7 UNDER THE FEDERAL GOVERNMENT, 5 APPLIED MATHEMATICS, PAPER-I Roll Number TIME ALLOWED: THREE HOURS MAXIMUM MARKS = NOTE: (i) Attempt ONLY FIVE questions in all, by selecting THREE questions from SECTION-I and TWO questions from SECTION-II. ALL questions carry EQUAL marks. (ii) All the parts (if any) of each Question must be attempted at one place instead of at different places. (iii) Candidate must write Q. No. in the Answer Book in accordance with Q. No. in the Q.Paper. (iv) No Page/Space be left blank between the answers. All the blank pages of Answer Book must be crossed. (v) Etra attempt of any question or any part of the attempted question will not be considered. (vi) Use of Calculator is allowed. SECTION-I Q. No. (a) Prove that ()()() A B [()] B C C A A B C. If A ( )( y iˆ ) y ˆj, evaluate A dr y. where c is an ellipse 9 4 c in the y- plane traversed in the positive direction. Q. No. (a) Determine the epression for divergence in orthogonal curvilinear coordinates. Determine the unit vectors in spherical coordinate system. Q. No. (a) A particle moves from rest at a distance a from a fied point O where the acceleration at distance is by an equation of the form 5 4 a t A. Show that the time taken to arrive at O is given, where A is a number. Three forces P, Q, R acting at a point, are in equilibrium, and the angle between P and Q is double of the angle between P and R. Prove that R Q() Q P. Q. No. 4 (a) AB and AC are similar uniform rods, of length a, smoothly joined at A.BD is a weightless bar, of length b, smoothly joined at B, and fastened at D to a smooth ring sliding on AC. The system is hung on a small smooth pin at A. Show that the b rod AC makes with the vertical an angle tan a a b. Find the centroid of the arc of the curve y a lying in the first quadrant. Q. No. 5 (a) A hemispherical shell rests on a rough inclined plane whose angle of friction is. Show that the inclination of the plane base to the horizontal cannot be greater than sin(sin). A regular octahedron formed of twelve equal rods, each of weight w, freely jointed together is suspended from one corner. Show that the thrust in each horizontal rod is w. Page of

21 APPLIED MATHEMATICS, PAPER-I SECTION-II a Q. No. 6 (a) A particle is moving with uniform speed v along the curve y a(). 5 v Show that its acceleration has the maimum value 9a. Discuss the motion of a particle moving in a straight line if it starts from rest at a distance a from a point O and moves with an acceleration equal to times its distance from O. Q. No. 7 (a) Prove that the force field F ( y )( yz )(6 i ) y y j z yz k is conservative, and determine its potential. The components of velocity along and perpendicular to the radius vector form a fied origin are respectively r and. Find the polar equation of the path of the particle in terms of r and. Q. No. 8 (a) A particle is projected horizontally from the lowest point of a rough sphere of radius a. After describing an arc less than a quadrant, it returns and comes to rest at the lowest point. Show that the initial speed must be (sin) () ag ( ) Where is the coefficient of friction and a is the arc through which the particle moves. The law of force is Mu and a particle is projected from or apse at distance a. Find M the orbit when the velocity of the projection is. a, Page of

22 FEDERAL PUBLIC SERVICE COMMISSION COMPETITIVE EXAMINATION-6 FOR RECRUITMENT TO POSTS IN BS-7 UNDER THE FEDERAL GOVERNMENT Roll Number APPLIED MATHEMATICS TIME ALLOWED: THREE HOURS MAXIMUM MARKS = NOTE:(i) Attempt ONLY FIVE questions. ALL questions carry EQUAL marks (ii) All the parts (if any) of each Question must be attempted at one place instead of at different places. (iii) Candidate must write Q. No. in the Answer Book in accordance with Q. No. in the Q.Paper. (iv) No Page/Space be left blank between the answers. All the blank pages of Answer Book must be crossed. (vi) Etra attempt of any question or any part of the attempted question will not be considered. (v) Use of Calculator is allowed. Q. No.. Q. No.. f (r) r = r f ( r ) + f ( r ) (a) Prove that. Verify Stokes theorem for = ( y) i y z j y z k, where S is the upper half surface of the sphere + y + z = and C is its boundary. (a) Forces P, Q, R act at a point parallel to the sides of a triangle ABC taken in the r same order. Show that the magnitude of the resultant force is P + Q + R Q R cos A R P cos B P Q cos C Find the distance from the cusp of the centroid of the region bounded by the cardioid r = a ( + cos ). Q. No.. Q. No. 4. Q. No. 5. (a) A particle describes simple harmonic motion in such a way that its velocity and acceleration at a point P are u and f respectively and the corresponding quantities at another point Q are v and g. Find the distance PQ. Derive the radial and transverse components of velocity and acceleration of a particle. Solve the following differential equations: (a) dy y 4 + = y d (D 5D + 6) y = e (a) Solve the differential equation using the method of variation of parameters d y + y = tan, << d Solve the Euler Cauchy differential equation y y + 4y = ln. Q. No. 6. (a) Find the Fourier series of the following function: if < < f() = if < < Solve the initial - boundary value problem:

23 APPLIED MATHEMATICS Q. No. 7. (a) Apply Newton Raphson method to find the smaller positive root of the equation 4 + = Solve the following system of equations by Gauss Seidel iterative method by taking the initial approimation as =, =, = : 5 + = = = Q. No. 8. (a) Approimate d + using (i) Trapezoidal rule with n = 4 (ii) Simpson s rule with n = 4 Also compare the results with the eact value of the integral. Apply the improved Euler method to solve the initial value problem: y = + y, y () = by choosing h =. and computing y,, y 5. ************* Page of