04 Math Optional IMS with Solution[@Upsc Materials] PDF

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A CONSOLIDATED QUESTION PAPER-CUM-ANSWER BOOKLET

MAINS TEST SERIES-2019 (JUNE-2019 to SEPT.-2019)

Under the guidance of K. Venkanna

MATHEMATICS PAPER - II : PDE, NUMERICAL & COMP. PROG. AND MECHANICS & FD TEST CODE: TEST- 4: IAS(M)/30-JUNE.-2019

Time: 3 Hours

Maximum Ma

48 This question paper-cum-answer booklet has _______ pages and has

R E A D I N S TR U C LEFT SIDE CAREFULLY

30 PART/SUBPART questions. Please ensure that the copy of the question _________________

Name

INSTRUCTIONS 1.

TH E PAGE

paper-cum-answer booklet you have received contains all the questions. 2.

Write your Name, Roll Number, Name of the Test Centre and Medium in the appropriate space provided on the right side.

3.

A consolidated Question Paper-cum-Answer Booklet, having space below each part/sub part of a question shall be provided to them fo writing the answers. Candidates shall be required to attempt answe the part/sub-part of a question strictly within the pre-defined space attempt outside the pre-defined space shall not be evaluated.

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Answer must be written in the medium specified in th Certificate issued to you, which must be stated clearly o No marks will be given for the answers written in a that specified in the Admission Certificate.

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Candidates should attempt Question Nos. 1 an and any THREE of the remaining quest question from each Section.

6.

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10. All rough work should be d finally.

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ntre

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ated at the end of essary and indicate

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IMPORTANT NOTE: Whenever a question is being attempted, all its parts/ sub-parts must be attempted contiguously. This means that before moving on to the next question to be attempted, candidates must finish attempting all parts/ sub-parts of the previous question attempted. This is to be strictly followed. Pages left blank in the answer-book are to be clearly struck out in ink. Any answers that follow pages left blank may not be given credit.

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INDEX TABLE QUESTION 1

No.

PAGE NO.

MAX. MARKS

MARKS OBTAINED

(a) (b) (c) (d) (e)

2

(a) (b) (c) (d)

3

(a) (b) (c) (d)

4

(a) (b) (c) (d)

5

(a) (b) (c) (d) (e)

6

(a) (b) (c) (d)

7

(a) (b) (c)

8

( (b) (c) (d)

Total Marks

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SECTION – A 1.

(a) Solve x2p2 + y2q2 = z2.

[10]

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1.

(b) Solve (D2 – 4D′2)z = (4x/y2) – (y/x2).

[10]

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1.

(c) Obtain the Newton-Raphson extended formula 2

x1 = x 0 −

f ( x 0 ) 1 {f ( x0 )} f ′′( x0 ) − f ′( x0 ) 2 {f ′( x 0 )}3

for the root of the equation f(x) = 0.

[10]

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1.

(d) (i) Simplify the expression AB + AC + ABC (AB + C). (ii) Simplify the given Boolean expression Y = ABC + ABC + ABC + ABC.

[10]

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1.

(e) Find the M.I. of a rectangular parallelopiped about an edge. [10]

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2.

(a) Reduce the equation uxx + xuyy = 0,

x ≠0

for all x, y to canonical form.

[16]

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2.

(b) A plank of mass M is initially at rest along a lin plane inclined at an angle α to the horizon, an the upper end, walks down the plank so tha to the other end in time

 2M′a   ( M + M′) g

slope of a smooth mass M′ starting from move, show that he gets a is the length of the plane. [17]

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2.

(c) Four equidistant values u–1, u0, u1 and u2 being given, a value is interpolated by Lagrange’s formula. Show that it may be written in the form u x = yu 0 + xu 1

(y +y

where x + y = 1.

2

− 1)

3!

x ( x − 1) 2

2

∆ u −1 +

3!

∆2u 0 , [16]

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3.

(a) Find the characteristics of the PDE p2 + q2 = 2 and determine the integral surface which passe

0, z = y.

[15]

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3. (b) Ifthevelocityofanincompressiblefluidatthepoint(x,y,z)isgivenby  3xz 3yz 3z 2 − r 2  2 2 2 2  5 , 5 , ,r = x + y + z , r5  r  r

then prove that the liquid motion is possible and that the velocity potential is Further, determine the streamlines.

z . r3

[18]

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3.

(c) Solve the following system 17 12  56  3

65 −13 50  x 1   84 16 37 18  x 2   25 = −19  x 3   36 23 11     −5 47 10  x 4  18 

by Gauss-Seidel method and do c upto 10 iterations.

to two decimal places and obtain [17]

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4.

(a) A thermally conducting solid bounded by two concentric spheres of radii a and b, a < b, is such that the internal boundary is kept at T1 and the outer boundary at T2 (1 – cos θ). Find the steady state temperature in the solid. [20]

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4.

(b) When a pair of equal and opposite rectilinear vortice cylinder at equal distance from its axis, show th the equation. (r2 sin2 θ – b2) (r2 – a2)2 = 4a2b2r2 sin2 θ, θ being measured from the line throu the vortices.

in a long circular ch vortex is given by

perpendicular to the join of [15]

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4.

(c) Determine u(t) at t = 0.2, 0.4, using the classical fourth order Runge-Kutta method, given that u′ = (t/u), u(0) = 1.





Comparetheresultswiththeexactsolution.findtheerroratt=0.4.

[15]

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5.

SECTION – B (a) Find the integral surface of the equation (x – y)y2p + (y – x)x2q = (x2 + y2)z, through t

a3, y = 0.

[10]

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5.

(b) Solve 10 x − 7 y + 3 z + 5u = 6, −6 x + 8 y − z − 4u = 5,  3x + y + 4z + 11u = 2, 5x − 9y − 2z + 4u = 7 by Gauss’s elimination method.

[10]

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5.

(c) Realise (a) Y = A + BCD using NAND gates and (b) Y = ( A + C ) ( A + D) ( A + B + C ) using NOR ga

[10]

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5. (d) Use Lagrange’s equations to find the equation pendulum which oscillates in a vertical plane

of the compound horizontal axis. [10]

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5.

(e) Find the stream lines and paths of the particles fo field: x , v = y, w = 0. u= 1+ t

ensional velocity

[10]

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6.

(a) Form a partial differential equation by eliminating 2

2

z = f(x – y) + g(x + y).

nction f and g from [06]

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6.

(b) Solve (D2 – D′2 – 3D + 3D′)z = xy + ex + 2y.

[08]

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6.

(c) The temperature of a bar 50 cm long with insul end and 100° at the other end until steady co then suddenly insulated so that the temp thereafter. Find the temperature distrib

kept at 0° at one ail. The two ends are ent is zero at each end [16]

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6. (d) ApianostringoflengthLisfixedatbothends.T ρand isunder tensionτ. At time t = 0, the s equilibrium position at its mid-point so that then released (s ≤ L). Find the subsequen

as...