# obtained using Simpson s rule with threepoint

Transcription

1 YEAR Q. 5. The etimate of # obtained uing Simon rule with threeoint function evaluation eceed the eact value by 5..5 (B).68. (D). Q. The annual reciitation data of a city i normally ditributed with mean and tandard deviation a mm and mm, reectively. The robability that the annual reciitation will be more than mm i < 5% (B) 5% 75% (D) % Q. The infinite erie... correond to!!! ec (B) e co (D) in YEAR Q. The error in d f ^ h for a continuou function etimated with = h =. uing the central difference formula d f^ f hh-f^-hh ^ h = i = h # -. The value of and f^ h are 9.78 and 5., reectively. The correonding error in the central difference etimate for h =. i aroimately. # - (B). # - 5. # - (D) 9. # - Q. 5 In an eeriment, oitive and negative value are equally liely to occur. The robability of obtaining at mot one negative value in five trial i / (B) / / (D) 6/ Q. 6 The eigen value of matri 9 5 > H are and 6.86 (B).8 and.5.7 and 6.86 (D) 6.86 and 9.5

2 Q. 7 For the arallelogram OPQR hown in the etch, OP = at bj and OR = ct dj. The area of the arallelogram i ad - bc (B) ac bd ad bc (D) ab - cd Q. 8 The olution of the ordinary differential equation y = for the boundary condition, y = 5 at = i y = e - (B) y = e - y =. 95e - (D) y = 6. 95e - Q. 9 A YEAR i quare matri which i neither ymmetric nor ew-ymmetric and 6 A@ T i it tranoe. The um and difference of thee matrice are defined a 6S@ = 6A@ 6A@ T and 6D@ = 6A@ -6A@ T, reectively. Which of the following tatement i TRUE? Both 6 S@ and 6 D@ are ymmetric (B) Both 6 S@ and 6 D@ are ew-ymmetric 6 S@ i ew-ymmetric and 6 D@ i ymmetric (D) 6 S@ i ymmetric and 6 D@ i ew-ymmetric Q. The quare root of a number N i to be obtained by alying the Newton Rahon iteration to the equation - N =. If i denote the iteration inde, the correct iterative inde, the correct iterative cheme will be N i = i b l (B) N i = i i c m N i = i c m (D) N i = i- b l i i i Q. There are two container, with one containing red and green ball and the other containing blue and green ball. One ball i drawn at random from each container. The robability that one of the ball i red and the other i blue will be /7 (B) 9/9 /9 (D) /7 YEAR Q. For an analytic function, f^ iyh= u^, yh iv^, yh, u i given by u = -y. The ereion for v, conidering K to be a contant i y - K (B) 6-6y K 6y- 6 K (D) 6 y K

3 Q. What hould be the value of l uch that the function defined below i continuou at = /? Zl co ] if! / f^h = [ - ] if = / \ (B) / (D) / Q. What i the value of the definite integral, (B) a/ a a #? a- (D) a Q. 5 If a and b are two arbitrary vector with magnitude a and b, reectively, a b # will be equal to ab-^a: bh (B) ab - a: b ab ^a: bh (D) ab a: b y Q. 6 The olution of the differential equation =, with the condition that y = at =, i y = (B) y = y = (D) y = YEAR in : D Q. 7 The lim i " / (B) / (D) Q. 8 Two coin are imultaneouly toed. The robability of two head imultaneouly aearing i /8 (B) /6 / (D) / Q. 9 The order and degree of the differential equation d y b y = l are reectively and (B) and and (D) and YEAR d y Q. The olution to the ordinary differential equation 6y - = i - y = c e c e (B) y = c e c e - y = c e c e (D) y = c e c e - -

4 i i Q. The invere of the matri > H i -i -i i i - > i - i H (B) i i - - > H i i i -i > i - ih (D) -i -i > H i i Q. The table below give value of a function F_ i obtained for value of at interval of F _ i The value of the integral of the function between the limit to uing Simon rule i.785 (B).56.6 (D) 7.5 Q. The artial differential equation that can be formed from z = a by ab ha the form (with = z and q = z ) y z = qy (B) z = q z = qy q (D) z = qy q Q. A arabolic cable i held between two uort at the ame level. The horizontal an between the uort i L. The ag at the mid-an i h. The equation of the arabola i y = h L, where i the horizontal coordinate and y i the vertical coordinate with the origin at the centre of the cable. The ereion for the total length of the cable i L L/ 6 h # (B) 6 h # L L L/ 6 h # (D) L # L/ 6 h L Q. 5 Given a function f_, yi = 6y -8- y 8 The otimal value of f_, yi i a minimum equal to / (B) i a maimum equal to / i a minimum equal to 8/ (D) i a maimum equal to 8/ YEAR 9 Q. 6 A quare matri B i ew-ymmetric if B T =- B (B) B T = B B -T = B (D) B - = B T Q. 7 For a calar function fyz (,,) = y z, the gradient at the oint P(,, - ) i i v 6 v j v (B) i v v j - v iv vj v (D) 56

5 Q. 8 The analytic function fz () = z - ha ingularity at z and - (B) and i and - i (D) i and -i YEAR 9 Q. 9 For a calar function fyz (,,) = y z, the directional derivative at the oint P(,, - ) in the direction of a vector iv- vj v i - 8 (B) (D) 8 Q. The value of the integral by z = ) i i i 5 co( z) # dz ( where C i a cloed curve given ( z-)( z-) C (B) i 5 (D) i Q. Solution of the differential equation y = rereent a family of ellie (B) circle arabola (D) hyerbola Q. Lalace tranform for the function f () = coh( a) i a a (B) - a - a a (D) a Q. In the olution of the following et of linear equation by Gau elimination uing artial ivoting 5 y z = ; y- z = ; and - y z =- ; the ivot for elimination of and y are and (B) and 5 and (D) 5 and - Q. The tandard normal robability function can be aroimated a F ( N ) = e( ) n n where N = tandard normal deviate. If mean and tandard deviation of annual reciitation are cm and 7 cm reectively, the robability that the annual reciitation will be between 9 cm and cm i 66.7% (B) 5.%.% (D) 6.7% YEAR 8 Q. 5 The roduct of matrice ( PQ) - P i P - (B) Q P Q P (D) PQP -

6 d y Q. 6 The general olution of y = i y = Pco Qin (B) y = Pco y = Pin (D) y = Pin YEAR 8 Q. 7 The equation h h z = can be tranformed to z h h = by ubtituting z t X X t t X z = (B) X = X = (D) Xt = z Q. 8 The value of ## (6 - -y) i.5 (B) 7..5 (D) 5. t z z Q. 9 Three value of and y are to be fitted in a traight line in the form y = a b by the method of leat quare. Given S = 6, Sy =, S = and S y = 6, the value of a and b are reectively and (B) and and (D) and Q. Solution of =- y at = and y = i - y =- (B) y = - y =- (D) y = Q. If robability denity function of random variable X i f () = for - # #, and = for any other value of then, the ercentage robability P b- # # l i.7 (B) - 6 and 5 and (D) and Q. The Eigen value of the matri 6 P@ = 5 > -5 H are 7 and 8 (B) 6 and 5 and (D) and Q. A eron on a tri ha a choice between rivate car and ublic tranort. The robability of uing a rivate car i.5. While uing the ublic tranort, further choice available are bu and metro, out of which the robability of commuting by a bu i.55. In uch a ituation, the robability (rounded u to two decimal) of uing a car, bu and metro, reectively would be.5,. and.5 (B).5,.5 and..5,.55 and. (D).5,.5 and.

7 Q. The following imultaneou equation y z = y z = y z = 6 will NOT have a unique olution for equal to (B) 5 6 (D) 7 Q. 5 The inner (dot) roduct of two vector P v and Q v i zero. The angle (degree) between the two vector i (B) 9 (D) YEAR 7 Q. 6 The minimum and the maimum eigen value of the matri 6, reectively. What i the other eigen value? 5 (B) (D) - R S S S T 5 V W W are - and W X Q. 7 The degree of the differential equation d = i (B) (D) Q. 8 The olution for the differential equation at = i y ln() y = e (B) ln() y = (D) y e = = y with the condition that y = = YEAR 7 Q. 9 For the value of a and b the following imultaneou equation have an infinite number of olution? y z = 5; y z = 9; y az = b,7 (B),8 8, (D) 7, Q. 5 A velocity vector i given a Vv = 5yiv y vj yz v. The divergence of thi velocity vector at (,, ) i 9 (B) (D) 5 Q. 5 A bo originally at 6c C cool down to c C in 5 minute when et in air at a temerature 5c C. What will be the temerature of the bo at the end of minute? 5.c C (B).5cC 8.7c C (D) 5cC

8 Q. 5 The following equation need to be numerically olved uing the Newton-Rahon method. - 9 = The iterative equation for thi uroe i ( indicate the iteration level) X X 9 = (B) X X = X X 9 X = X- X (D) X X = 9X Q. 5 Evaluate # in t t (B) (D) 8 Q. 5 Potential function f i given a f= - y. What will be the tream function ( Y ) with the condition Y= at = y =? y (B) y - y (D) y Q. 55 The invere of the # matri > 5 7 H i -7 > 5 - H (B) 7 > 5 H 7 - > -5 H (D) -7 - > -5 - H Q. 56 Given that one root of the equation - - = i 5, the other two root are and (B) and and (D) - and - Q. 57 If the tandard deviation of the ot eed of vehicle in a highway i 8.8 mh and the mean eed of the vehicle i mh, the coefficient of variation in eed i.57 (B) YEAR 6 Q. 58 Solution for the ytem defined by the et of equation y z = 8; - z = and y = 5 i = ; y = ; z = (B) = ; y = ; z = = ; y = ; z = (D) non-eitent Q. 59 The differential equation = 5. y i to be olved uing the bacward (imlicit) Euler method with the boundary condition y = at = and with the a te ize of. What would be the value of y at =?. (B).67. (D).

9 YEAR 6 Q. 6 R - V S W For a given matri A =- S - 6W, one of the eigenvalue i. S W The other two eigenvalue T are X, - 5 (B), - 5, 5 (D), 5 Q. 6 The directional derivative of fyz (,,) = y z at the oint P(,,) in the direction of the vector a = i- i (B) (D). Q. 6 A cla of firt year B. Tech. tudent i comoed of four batche A, B, C and D, each coniting of tudent. It i found that the eional mar of tudent in Enginnering Drawing in batch C havea mean of 6.6 and tandard deviation of.. The mean and tandard deviation of the mar for the entire cla are 5.5 and., reectively. It i decided by the coure intructor to normalize the mar of the tudent of all batche to have the ame mean and tandard deviation a that of the entire cla. Due to thi, the mar of a tudent in batch C are changed from 8.5 to 6. (B) (D) 9. Q. 6 A nd degree olynomial, f () ha value of, and 5 at =, and, reectively. The integral # f () i to be etimated by alying the traezoidal rule to thi data. What i the error ( defined a true value - aroimate value ) in the etimate? - (B) - (D) Q. 6 What i the area common to the circle r = a and r = aco q?.5 a (B).6 a.7 a (D).8 a Q. 65 Uing Cauchy integral theorem, the value of the integral (integration being taen in counterclocwie direction) # z - 6 dz i z- i c - i (B) - 6i 8 8-6i (D) 8 Q. 66 There are 5 calculator in a bo. Two of them are defective. Suoe 5 calculator are randomly iced for inection (i.e., each ha the ame chance of being elected), what i the robability that only one of the defective calculator will be included in the inection? (B) (D) 5

10 Q. 67 A herical nahthalene ball eoed to the atmohere loe volume at a rate roortional to it intantaneou urface are due to evaoration. If the initial diameter of the ball i m and the diameter reduce to cm after month, the ball comletely evaorate in 6 month (B) 9 month month (D) Infinite time Q. 68 The olution of the differential equation, y - =, given that at =, y = i - (B) - - (D) - YEAR 5 Q. 69 Conider the matrice X( ), Y( ) and P( ) T - T T The order of [ PXY ( ) P] will be ( # ) (B) ( # ) ( # ) (D) ( # ) # # #. Q. 7 Conider a non-homogeneou ytem of linear equation rereenting mathematically an over-determined ytem. Such a ytem will be conitent having a unique olution (B) conitent having a unique olution inconitent having a unique olution (D) inconitent having no olution Q. 7 Which one of the following i NOT true for comle number Z and Z? Z ZZ = r Z Z (B) Z Z # Z Z Z -Z # Z - Z (D) Z Z Z - Z = Z Z Q. 7 Which one of the following tatement i NOT true? The meaure of ewne i deendent uon the amount of dierion (B) In a ymmetric ditribution, the value of mean, mode and median are the ame In a oitively ewed ditribution : mean > median > mode (D) In a negatively ewed ditribution : mode > mean > median

11 YEAR 5 Q. 7 Conider the ytem of equation A( n# n) X( n# t) = l( n# ) where, l i a calar. Let ( l i, Xi) be an eigen-air of an eigen value and it correonding eigen vector for real matri A. Let be a ( n# n) unit matri. Which on of the following tatement i NOT correct? For a homogeneou n# n ytem of linear equation, ( A- l ) = having a nontrivial olution, the ran of ( A - l ) i le than n. m m (B) For matri A, m being a oitive integer, ( l i, Xi m ) will be the eigen-air for all i. T If A = A -, then l = for all i (D) If A T = A i, then l i i real for all i Q. 7 Tranformation to linear form by ubtituting v = y of the equation n ty () = qty () ; n> will be dv ( - nv ) = ( - nq ) (B) dv ( - nv ) = ( -nq ) dv ( nv ) = ( - nq ) (D) dv ( nv ) = ( nq ) Q. 75 A rail engine accelerate from it tationary oition for 8 econd and travel a ditance of 8 m. According to the Mean Value Theorem, the eedometer at a certain time during acceleration mut read eactly (B) 8 mh 75 mh (D) 6 mh d y Q. 76 The olution of y ; y( ), 7 = = a = in the range < < i given by - e co b in l (B) e co b - in l - e co b - in l - (D) e co b - in l Q. 77 Value of the integral # ( y - y ), where, c i the quare cut from the firt quadrant by c the line = and y = will be (Ue Green theorem to change the line integral into double integral) (B) (D) 5 Q. 78 Conider the liely alicability of Cauchy Integral Theorem to evaluate the following integral counter clocwie around the unit circle c. I = # ec zdz, c z being a comle variable. The value of I will be I = : ingularitie et = f (B) I = : ingularitie et =! n &, n =,,... I = / : ingularitie et = {! n; n =,,...} (D) None of above

12 Statement for Lined Q. 79 and 8 : Give a >, we wih to calculate it recirocal value /a by uing Newton Rahon Method for f () =. Q. 79 The Newton Rahon algorithm for the function will be X X X a = a (B) X X a X = a X = X = ax (D) X X a X = - Q. 8 For a = 7 and tarting with =., the firt two iteration will be.,.99 (B).,.9.,.6 (D).,.8 YEAR Q. 8 Real matrice [ A] #,[ B] #,[ C] # 5,[ D] 5# 5 and [ F] 5# are given. Matrice [B] and [E] are ymmetric. Following tatement are made with reect to thee matrice.. Matri roduct [ F] T T [ C] [ B][ C][ F] i a calar. T. Matri roduct [ D] [ F][ D] i alway ymmetric. With reference to above tatement, which of the following alie? Statement i true but i fale (B) Statement i fale but i true Both the tatement are true (D) Both the tatement are fale Q. 8 The ummation of erie S = i.5 (B) (D). Q. 8 The value of the function f ( ) = lim i " - 7 (B) (D) YEAR - Q. 8 The eigenvalue of the matri > - H are and (B) are - and are and 5 (D) cannot be determined Q. 85 The function f () = ha it maima at =- only (B) = only = only (D) both =- and =

13 Q. 86 Biotranformation of an organic comound having concentration ( ) can be modeled uing an ordinary differential equation =, where i the reaction rate contant, If = a at t =, the olution of the equation i = ae - t (B) = t a t = a( -e - ) (D) = a t Q. 87 A hydraulic tructure ha four gate which oerate indeendently. The robability of failure of each gate i.. Given that gate ha failed, the robability that both gate and will fail i. (B).. (D).8 YEAR R V S W Q. 88 Given Matri [ A] = S6 7W, the ran of the matri i S W T X (B) (D) Q. 89 A bo contain crew, of which are defective. Two crew are drawn at random with relacement. The robability that none of the two crew i defective will be % (B) 5% 9% (D) None of thee Q. 9 If P, Q and R are three oint having coordinate (, -, - ), (,, ), (, - ) in XYZ ace, then the ditance from oint P to lane OQR (O being the origin of the coordinate ytem) i given by (B) 5 7 (D) 9 YEAR TWO MARK Q. 9 If L define the Lalace Tranform of a function, L[ in( at )] will equal to a a (B) a - a a (D) a - Q. 9 The Fourier erie eanion of a ymmetric and even function, f () where f () = ( / ), - # # and = - ( / ), # # will be / ( co n) n = n (B) / ( co n n ) - n = / ( - in n) n (D) / ( in n n ) n = n =

14 YEAR Q. 9 A vector normal to i t t j - t i i t - t j - t (B) - it - tj t - i t t j t (D) i t t j - t Q. 9 The neceary condition to diagonalie a matri i that it eigen value hould be ditinct (B) it eigen vector hould be indeendent it eigen value hould be real (D) the matri i non-ingular Q. 95 lim YEAR in a - equal " - (B) / (D) YEAR R V S W Q. 96 The ran of the matri A = S 5W S 6 8W, i T X (B) (D) YEAR Q. 97 The characteritic root of the ytem decribed a, (B) -, = y, - j, j (D) None of the above Q. 98 For a ingular matri, at leat one eigen value would be at the origin (B) all eigen value would be at the origin no eigen value would be at the origin (D) None of the above =- are at **********

15 ANSWER KEY (D) (B) (D) (D) (B) (D) (D) (D) (B) (D) (B) (D) (B) (D) (B) (B) (B) (B) (D) (D) (D) (B) (B) (D) (B) (B) (D) (D) (B) (B) (D) (B) (D) (D) (B) (D) (D) (B) (B) (D) (B) (B) (D) (B) (B) (D) (D)