Page 1 No part of this publication may be reproduced or distributed in any form or any means, electronic, mechanical, photocopying, or otherwise without the prior permission of the author. GATE SOLVED PAPER Mechanical Engineering 2010 Copyright © By NODIA & COMPANY Information contained in this book has been obtained by authors, from sources believes to be reliable. However, neither Nodia nor its authors guarantee the accuracy or completeness of any information herein, and Nodia nor its authors shall be responsible for any error, omissions, or damages arising out of use of this information. This book is published with the understanding that Nodia and its authors are supplying information but are not attempting to render engineering or other professional services. NODIA AND COMPANY B-8, Dhanshree Tower Ist, Central Spine, Vidyadhar Nagar, Jaipur 302039 Ph : +91 - 141 - 2101150 www.nodia.co.in email : enquiry@nodia.co.in Page 2 No part of this publication may be reproduced or distributed in any form or any means, electronic, mechanical, photocopying, or otherwise without the prior permission of the author. GATE SOLVED PAPER Mechanical Engineering 2010 Copyright © By NODIA & COMPANY Information contained in this book has been obtained by authors, from sources believes to be reliable. However, neither Nodia nor its authors guarantee the accuracy or completeness of any information herein, and Nodia nor its authors shall be responsible for any error, omissions, or damages arising out of use of this information. This book is published with the understanding that Nodia and its authors are supplying information but are not attempting to render engineering or other professional services. NODIA AND COMPANY B-8, Dhanshree Tower Ist, Central Spine, Vidyadhar Nagar, Jaipur 302039 Ph : +91 - 141 - 2101150 www.nodia.co.in email : enquiry@nodia.co.in GATE SOLVED PAPER - ME 2010 © www.nodia.co.in Q. 1 The parabolic arc yx = , x 12 ## is revolved around the x -axis. The volume of the solid of revolution is (A) /4 p (B) /2 p (C) / 34 p (D) / 32 p Sol. 1 Option (D) is correct. We know that the volume of a solid generated by revolution about x -axis bounded by the function () fx & limits between a to b is given by the equation. V ydx a b 2 p =# Given y x = & a 1 = , b 2 = Therefore, V () xdx 2 1 2 p =# xdx 1 2 p = # On integrating above equation, we get x 2 2 1 2 p = :D Substitute the limits, we get V 2 4 2 1 2 3 p p =- = :D Q. 2 The Blasius equation, d df f d df 2 0 3 3 2 2 h h += , is a (A) second order nonlinear ordinary differential equation (B) third order nonlinear ordinary differential equation (C) third order linear ordinary differential equation (D) mixed order nonlinear ordinary differential equation Sol. 2 Option (B) is correct. Given: d df f d df 2 3 3 2 2 h h + 0 = Order " It is determined by the order of the highest derivation present in it. So, It is third order equation but it is a nonlinear equation because in linear equation, the product of f with / dfd 22 h is not allow. Therefore, it is a third order non-linear ordinary differential equation. Q. 3 The value of the integral x dx 1 2 + 3 3 - # is (A) p - (B) /2 p - (C) /2 p (D) p Sol. 3 Option (D) is correct. Let I x dx 1 2 = + 3 3 - # I tan x 1 = 3 3 - - 6@ GATE ME 2010 ONE MARK GATE ME 2010 ONE MARK GATE ME 2010 ONE MARK Page 3 No part of this publication may be reproduced or distributed in any form or any means, electronic, mechanical, photocopying, or otherwise without the prior permission of the author. GATE SOLVED PAPER Mechanical Engineering 2010 Copyright © By NODIA & COMPANY Information contained in this book has been obtained by authors, from sources believes to be reliable. However, neither Nodia nor its authors guarantee the accuracy or completeness of any information herein, and Nodia nor its authors shall be responsible for any error, omissions, or damages arising out of use of this information. This book is published with the understanding that Nodia and its authors are supplying information but are not attempting to render engineering or other professional services. NODIA AND COMPANY B-8, Dhanshree Tower Ist, Central Spine, Vidyadhar Nagar, Jaipur 302039 Ph : +91 - 141 - 2101150 www.nodia.co.in email : enquiry@nodia.co.in GATE SOLVED PAPER - ME 2010 © www.nodia.co.in Q. 1 The parabolic arc yx = , x 12 ## is revolved around the x -axis. The volume of the solid of revolution is (A) /4 p (B) /2 p (C) / 34 p (D) / 32 p Sol. 1 Option (D) is correct. We know that the volume of a solid generated by revolution about x -axis bounded by the function () fx & limits between a to b is given by the equation. V ydx a b 2 p =# Given y x = & a 1 = , b 2 = Therefore, V () xdx 2 1 2 p =# xdx 1 2 p = # On integrating above equation, we get x 2 2 1 2 p = :D Substitute the limits, we get V 2 4 2 1 2 3 p p =- = :D Q. 2 The Blasius equation, d df f d df 2 0 3 3 2 2 h h += , is a (A) second order nonlinear ordinary differential equation (B) third order nonlinear ordinary differential equation (C) third order linear ordinary differential equation (D) mixed order nonlinear ordinary differential equation Sol. 2 Option (B) is correct. Given: d df f d df 2 3 3 2 2 h h + 0 = Order " It is determined by the order of the highest derivation present in it. So, It is third order equation but it is a nonlinear equation because in linear equation, the product of f with / dfd 22 h is not allow. Therefore, it is a third order non-linear ordinary differential equation. Q. 3 The value of the integral x dx 1 2 + 3 3 - # is (A) p - (B) /2 p - (C) /2 p (D) p Sol. 3 Option (D) is correct. Let I x dx 1 2 = + 3 3 - # I tan x 1 = 3 3 - - 6@ GATE ME 2010 ONE MARK GATE ME 2010 ONE MARK GATE ME 2010 ONE MARK © www.nodia.co.in GATE SOLVED PAPER - ME 2010 I [( ) ( )] tan tan 11 33 =+ - - -- I 22 pp p =-- = ak () () tan tan 11 qq -=- -- Q. 4 The modulus of the complex number i i 12 34 - + bl is (A) 5 (B) 5 (C) / 15 (D) 1/5 Sol. 4 Option (B) is correct. Let, z i i 12 34 = - + Divide & multiply z by the conjugate of () i 12 - to convert it in the form of abi + . So, z i i i i 12 34 12 12 # = - + + + () ( ) ()() i ii 12 34 12 22 = - ++ i ii 14 310 8 2 2 = - ++ () i 14 310 8 = -- +- i i 5 510 12 = -+ =- + z () () 12 5 22 =- + = aib a b 22 += + Q. 5 The function yx 23 =- (A) is continuous xR 6 ! and differentiable xR 6 ! (B) is continuous xR 6 ! and differentiable xR 6 ! except at / x 32 = (C) is continuous xR 6 ! and differentiable xR 6 ! except at / x 23 = (D) is continuous xR 6 ! except x 3 = and differentiable xR 6 ! Sol. 5 Option (C) is correct. () yfx = 23 0 (2 3 ) if if if xx x xx 3 2 3 2 3 2 < > = - = -- Z [ \ ] ] ] ] ] ] Checking the continuity of the function. at x 3 2 = , () Lf x limfh 3 2 h 0 =- " bl 2 lim h 23 3 h 0 =- - " bl lim h 223 h 0 =-+ " 0 = and () Rf x limfh 3 2 h 0 =+ " bl lim h 3 3 2 2 h 0 =+- " bl lim h 23 2 0 h 0 =+ -= " Since () lim Lfx h 0 " () lim Rfx h 0 = " So, function is continuous xR 6 ! GATE ME 2010 ONE MARK GATE ME 2010 ONE MARK Page 4 No part of this publication may be reproduced or distributed in any form or any means, electronic, mechanical, photocopying, or otherwise without the prior permission of the author. GATE SOLVED PAPER Mechanical Engineering 2010 Copyright © By NODIA & COMPANY Information contained in this book has been obtained by authors, from sources believes to be reliable. However, neither Nodia nor its authors guarantee the accuracy or completeness of any information herein, and Nodia nor its authors shall be responsible for any error, omissions, or damages arising out of use of this information. This book is published with the understanding that Nodia and its authors are supplying information but are not attempting to render engineering or other professional services. NODIA AND COMPANY B-8, Dhanshree Tower Ist, Central Spine, Vidyadhar Nagar, Jaipur 302039 Ph : +91 - 141 - 2101150 www.nodia.co.in email : enquiry@nodia.co.in GATE SOLVED PAPER - ME 2010 © www.nodia.co.in Q. 1 The parabolic arc yx = , x 12 ## is revolved around the x -axis. The volume of the solid of revolution is (A) /4 p (B) /2 p (C) / 34 p (D) / 32 p Sol. 1 Option (D) is correct. We know that the volume of a solid generated by revolution about x -axis bounded by the function () fx & limits between a to b is given by the equation. V ydx a b 2 p =# Given y x = & a 1 = , b 2 = Therefore, V () xdx 2 1 2 p =# xdx 1 2 p = # On integrating above equation, we get x 2 2 1 2 p = :D Substitute the limits, we get V 2 4 2 1 2 3 p p =- = :D Q. 2 The Blasius equation, d df f d df 2 0 3 3 2 2 h h += , is a (A) second order nonlinear ordinary differential equation (B) third order nonlinear ordinary differential equation (C) third order linear ordinary differential equation (D) mixed order nonlinear ordinary differential equation Sol. 2 Option (B) is correct. Given: d df f d df 2 3 3 2 2 h h + 0 = Order " It is determined by the order of the highest derivation present in it. So, It is third order equation but it is a nonlinear equation because in linear equation, the product of f with / dfd 22 h is not allow. Therefore, it is a third order non-linear ordinary differential equation. Q. 3 The value of the integral x dx 1 2 + 3 3 - # is (A) p - (B) /2 p - (C) /2 p (D) p Sol. 3 Option (D) is correct. Let I x dx 1 2 = + 3 3 - # I tan x 1 = 3 3 - - 6@ GATE ME 2010 ONE MARK GATE ME 2010 ONE MARK GATE ME 2010 ONE MARK © www.nodia.co.in GATE SOLVED PAPER - ME 2010 I [( ) ( )] tan tan 11 33 =+ - - -- I 22 pp p =-- = ak () () tan tan 11 qq -=- -- Q. 4 The modulus of the complex number i i 12 34 - + bl is (A) 5 (B) 5 (C) / 15 (D) 1/5 Sol. 4 Option (B) is correct. Let, z i i 12 34 = - + Divide & multiply z by the conjugate of () i 12 - to convert it in the form of abi + . So, z i i i i 12 34 12 12 # = - + + + () ( ) ()() i ii 12 34 12 22 = - ++ i ii 14 310 8 2 2 = - ++ () i 14 310 8 = -- +- i i 5 510 12 = -+ =- + z () () 12 5 22 =- + = aib a b 22 += + Q. 5 The function yx 23 =- (A) is continuous xR 6 ! and differentiable xR 6 ! (B) is continuous xR 6 ! and differentiable xR 6 ! except at / x 32 = (C) is continuous xR 6 ! and differentiable xR 6 ! except at / x 23 = (D) is continuous xR 6 ! except x 3 = and differentiable xR 6 ! Sol. 5 Option (C) is correct. () yfx = 23 0 (2 3 ) if if if xx x xx 3 2 3 2 3 2 < > = - = -- Z [ \ ] ] ] ] ] ] Checking the continuity of the function. at x 3 2 = , () Lf x limfh 3 2 h 0 =- " bl 2 lim h 23 3 h 0 =- - " bl lim h 223 h 0 =-+ " 0 = and () Rf x limfh 3 2 h 0 =+ " bl lim h 3 3 2 2 h 0 =+- " bl lim h 23 2 0 h 0 =+ -= " Since () lim Lfx h 0 " () lim Rfx h 0 = " So, function is continuous xR 6 ! GATE ME 2010 ONE MARK GATE ME 2010 ONE MARK © www.nodia.co.in GATE SOLVED PAPER - ME 2010 Now checking the differentiability : () Lf x l lim h fh f 3 2 3 2 h 0 = - -- " bb ll lim h h 23 3 2 0 h 0 = - -- - " bl lim lim h h h h 223 3 3 hh 00 = - -+ = - =- "" And () Rf x l lim h fh f 3 2 3 2 h 0 = +- " bb ll lim lim h h h h 3 3 2 20 23 2 hh 00 = +- - = +- "" bl 3 = Since Lf 3 2 l bl Rf 3 2 ! l bl , () fx is not differentiable at x 3 2 = . Q. 6 Mobility of a statically indeterminate structure is (A) 1 # - (B) 0 (C) 1 (D) 2 $ Sol. 6 Option (A) is correct. Given figure shows the six bar mechanism. We know movability or degree of freedom is 3( 1) 2 nl jh =- - - The mechanism shown in figure has six links and eight binary joints (because there are four ternary joints ,, & ABC D, i.e. , l 6 = j 8 = h 0 = So, n () 36 1 2 8 # =- - 1 =- Therefore, when n 1 =- or less, then there are redundant constraints in the chain, and it forms a statically indeterminate structure. So, From the Given options (A) satisfy the statically indeterminate structure n 1 # - Q. 7 There are two points P and Q on a planar rigid body. The relative velocity between the two points (A) should always be along PQ (B) can be oriented along any direction (C) should always be perpendicular to PQ (D) should be along QP when the body undergoes pure translation Sol. 7 Option (C) is correct. GATE ME 2010 ONE MARK GATE ME 2010 ONE MARK Page 5 No part of this publication may be reproduced or distributed in any form or any means, electronic, mechanical, photocopying, or otherwise without the prior permission of the author. GATE SOLVED PAPER Mechanical Engineering 2010 Copyright © By NODIA & COMPANY Information contained in this book has been obtained by authors, from sources believes to be reliable. However, neither Nodia nor its authors guarantee the accuracy or completeness of any information herein, and Nodia nor its authors shall be responsible for any error, omissions, or damages arising out of use of this information. This book is published with the understanding that Nodia and its authors are supplying information but are not attempting to render engineering or other professional services. NODIA AND COMPANY B-8, Dhanshree Tower Ist, Central Spine, Vidyadhar Nagar, Jaipur 302039 Ph : +91 - 141 - 2101150 www.nodia.co.in email : enquiry@nodia.co.in GATE SOLVED PAPER - ME 2010 © www.nodia.co.in Q. 1 The parabolic arc yx = , x 12 ## is revolved around the x -axis. The volume of the solid of revolution is (A) /4 p (B) /2 p (C) / 34 p (D) / 32 p Sol. 1 Option (D) is correct. We know that the volume of a solid generated by revolution about x -axis bounded by the function () fx & limits between a to b is given by the equation. V ydx a b 2 p =# Given y x = & a 1 = , b 2 = Therefore, V () xdx 2 1 2 p =# xdx 1 2 p = # On integrating above equation, we get x 2 2 1 2 p = :D Substitute the limits, we get V 2 4 2 1 2 3 p p =- = :D Q. 2 The Blasius equation, d df f d df 2 0 3 3 2 2 h h += , is a (A) second order nonlinear ordinary differential equation (B) third order nonlinear ordinary differential equation (C) third order linear ordinary differential equation (D) mixed order nonlinear ordinary differential equation Sol. 2 Option (B) is correct. Given: d df f d df 2 3 3 2 2 h h + 0 = Order " It is determined by the order of the highest derivation present in it. So, It is third order equation but it is a nonlinear equation because in linear equation, the product of f with / dfd 22 h is not allow. Therefore, it is a third order non-linear ordinary differential equation. Q. 3 The value of the integral x dx 1 2 + 3 3 - # is (A) p - (B) /2 p - (C) /2 p (D) p Sol. 3 Option (D) is correct. Let I x dx 1 2 = + 3 3 - # I tan x 1 = 3 3 - - 6@ GATE ME 2010 ONE MARK GATE ME 2010 ONE MARK GATE ME 2010 ONE MARK © www.nodia.co.in GATE SOLVED PAPER - ME 2010 I [( ) ( )] tan tan 11 33 =+ - - -- I 22 pp p =-- = ak () () tan tan 11 qq -=- -- Q. 4 The modulus of the complex number i i 12 34 - + bl is (A) 5 (B) 5 (C) / 15 (D) 1/5 Sol. 4 Option (B) is correct. Let, z i i 12 34 = - + Divide & multiply z by the conjugate of () i 12 - to convert it in the form of abi + . So, z i i i i 12 34 12 12 # = - + + + () ( ) ()() i ii 12 34 12 22 = - ++ i ii 14 310 8 2 2 = - ++ () i 14 310 8 = -- +- i i 5 510 12 = -+ =- + z () () 12 5 22 =- + = aib a b 22 += + Q. 5 The function yx 23 =- (A) is continuous xR 6 ! and differentiable xR 6 ! (B) is continuous xR 6 ! and differentiable xR 6 ! except at / x 32 = (C) is continuous xR 6 ! and differentiable xR 6 ! except at / x 23 = (D) is continuous xR 6 ! except x 3 = and differentiable xR 6 ! Sol. 5 Option (C) is correct. () yfx = 23 0 (2 3 ) if if if xx x xx 3 2 3 2 3 2 < > = - = -- Z [ \ ] ] ] ] ] ] Checking the continuity of the function. at x 3 2 = , () Lf x limfh 3 2 h 0 =- " bl 2 lim h 23 3 h 0 =- - " bl lim h 223 h 0 =-+ " 0 = and () Rf x limfh 3 2 h 0 =+ " bl lim h 3 3 2 2 h 0 =+- " bl lim h 23 2 0 h 0 =+ -= " Since () lim Lfx h 0 " () lim Rfx h 0 = " So, function is continuous xR 6 ! GATE ME 2010 ONE MARK GATE ME 2010 ONE MARK © www.nodia.co.in GATE SOLVED PAPER - ME 2010 Now checking the differentiability : () Lf x l lim h fh f 3 2 3 2 h 0 = - -- " bb ll lim h h 23 3 2 0 h 0 = - -- - " bl lim lim h h h h 223 3 3 hh 00 = - -+ = - =- "" And () Rf x l lim h fh f 3 2 3 2 h 0 = +- " bb ll lim lim h h h h 3 3 2 20 23 2 hh 00 = +- - = +- "" bl 3 = Since Lf 3 2 l bl Rf 3 2 ! l bl , () fx is not differentiable at x 3 2 = . Q. 6 Mobility of a statically indeterminate structure is (A) 1 # - (B) 0 (C) 1 (D) 2 $ Sol. 6 Option (A) is correct. Given figure shows the six bar mechanism. We know movability or degree of freedom is 3( 1) 2 nl jh =- - - The mechanism shown in figure has six links and eight binary joints (because there are four ternary joints ,, & ABC D, i.e. , l 6 = j 8 = h 0 = So, n () 36 1 2 8 # =- - 1 =- Therefore, when n 1 =- or less, then there are redundant constraints in the chain, and it forms a statically indeterminate structure. So, From the Given options (A) satisfy the statically indeterminate structure n 1 # - Q. 7 There are two points P and Q on a planar rigid body. The relative velocity between the two points (A) should always be along PQ (B) can be oriented along any direction (C) should always be perpendicular to PQ (D) should be along QP when the body undergoes pure translation Sol. 7 Option (C) is correct. GATE ME 2010 ONE MARK GATE ME 2010 ONE MARK © www.nodia.co.in GATE SOLVED PAPER - ME 2010 Velocity of any point on a link with respect to another point (relative velocity) on the same link is always perpendicular to the line joining these points on the configuration (or space) diagram. v QP = Relative velocity between P & Q v QP vv PQ =- always perpendicular to PQ. Q. 8 The state of plane-stress at a point is given by 200 , MPa x s =- 100 MPa y s = 100 MPa xy t = . The maximum shear stress (in MPa) is (A) 111.8 (B) 150.1 (C) 180.3 (D) 223.6 Sol. 8 Option (C) is correct. Given : x s 200 MPa =- , y s 100 MPa = , xy t 100 MPa = We know that maximum shear stress is given by, max t () 2 1 4 xy xy 22 ss t =- + Substitute the values, we get max t () () 2 1 200 100 4 100 22 # =- - + 2 1 90000 40000 =+ . 180 27 = 180.3 MPa - Q. 9 Which of the following statements is INCORRECT ? (A) Grashof’s rule states that for a planar crank-rocker four bar mechanism, the sum of the shortest and longest link lengths cannot be less than the sum of the remaining two link lengths (B) Inversions of a mechanism are created by fixing different links one at a time (C) Geneva mechanism is an intermittent motion device (D) Gruebler’s criterion assumes mobility of a planar mechanism to be one Sol. 9 Option (A) is correct. According to Grashof’s law “For a four bar mechanism, the sum of the shortest and longest link lengths should not be greater than the sum of remaining two link GATE ME 2010 ONE MARK GATE ME 2010 ONE MARKRead More

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