R. T. M. Nagpur University Mathematics-II MCQs PDF
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Rashtrasant Tukadoji Maharaj Nagpur University
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This document contains multiple-choice questions in mathematics, focusing on integral calculus and beta function for a second-semester BE. The questions cover various concepts and calculations in the respective modules.
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R. T. M. Nagpur University SUBJECT: Mathematics - II B. E. 2nd Semester Multiple Choice Questions Module - 1 - Integral Calculus - I (1) 7 The value of is...
R. T. M. Nagpur University SUBJECT: Mathematics - II B. E. 2nd Semester Multiple Choice Questions Module - 1 - Integral Calculus - I (1) 7 The value of is 2 𝜋 15 𝜋 (A) (B) 2 8 3 𝜋 5 𝜋 (C) (D) 4 2 Ans : B (2) The value of n 1 n is 1 𝜋 (A) (B) 𝑠𝑖𝑛𝑛𝜋 𝑐𝑜𝑠𝑛𝜋 𝜋 1 (C) (D) 𝑠𝑖𝑛𝑛𝜋 𝑐𝑜𝑠𝑛𝜋 Ans : C (3) Γ(n+1) = n! can be used when (A) n is any integer (B) n is a positive integer (C) n is a negative integer (D) n is any real number Ans : B (4) 1 What is the value of 𝛤 ? 2 𝜋 (A) 𝜋 (B) 2 𝜋 𝜋 (C) (D) 2 2 Ans : A (5) 9 What is the value of 𝛤 ? 4 (A) 5/4 ∗ 1/4 ∗ 𝛤(1/4) (B) 9/4 ∗ 5/4 ∗ 1/4 ∗ 𝛤(1/4) (C) 5/4 ∗ 1/4 ∗ 𝛤(5/4) (D) 1/4 ∗ 𝛤(1/4) Ans : A (6) The value of e z z1/ 2 dz is 0 (A) (B) / 2 (C) / 3 (D) / 4 Ans: B (7) 1 1 The value of x x is 2 2 (A) / sin x (B) / cos x (C) / sin x (D) / cos x Ans: B (8) dx e x x 3 / 4 dx is x 1 / 4 The value of e x 0 0 (A) (B) 2 (C) 2 (D) 3 Ans: C (9) The value of 2e z z 3 / 2 dz is 0 3 (A) (B) 2 (C) 2 (D) 3 Ans: A (10) Which of the following is true? 1 (A) (m, n) x m1 (1 x) n1 dx (B) (m, n) [ x m1 /(1 x) mn ]dx 0 0 (C) (m, n) [ x n1 /(1 x) mn ]dx (D) All of the above 0 Ans: D (11) If (n,3) 1 / 3 and n is a positive integer, then the value of n is (A) 4 (B) 3 (C) 2 (D) 1 Ans: D (12) (m, n) ? (A) (m 1, n) (B) (m 1, n 1) (C) (m, n 1) (m 1, n) (D) (m, n 1) (m 1, n) Ans: C (13) (m, n) ? m (A) (m, n) (B) m (m, n) mn (C) (m n) (m, n) (D) (m n) (m, n) Ans: A (14) 1 The value of t 2 (1 t )1/ 2 dt is 0 (A) 16/15 (B) 13/11 (C) 17/32 (D) 19/32 Ans: A (15) 1 The value of t 1/ 2 (1 t )1/ 2 dt is 0 (A) (B) 2 (C) / 2 (D) / 2 Ans: A (16) 1 The value of x3 / 2 (1 x)1/ 2 dx is 0 (A) / 2 (B) /16 (C) / 20 (D) / 2 Ans: B (17) 𝜋 𝜋 What is the value of 2 sin 𝜃 𝑑𝜃 + 2 cos 𝜃 𝑑𝜃? 0 0 3 3 Γ Γ 4 4 (A) 8 𝜋 1 (B) 4 𝜋 1 Γ 4 Γ 4 1 1 Γ Γ 4 4 (C) 8 𝜋 3 (D) 4 𝜋 3 Γ 4 Γ 4 Ans : A (18) The value of 𝛽(𝑚 + 1, 𝑛) is 1 𝑚 (A) 𝛽(𝑚, 𝑛) (B) 𝛽(𝑚, 𝑛) 𝑚 +𝑛 𝑚 +𝑛 𝑛 𝑚⎾ ⎾ 𝑛 (C) 𝛽(𝑚, 𝑛) (D) 𝑚 +𝑛 𝑚 +𝑛 ⎾ Ans: B (19) Which of the following function is not called the Euler’s integral of the first kind? 1 𝑚 −1 (A) 𝛽 𝑚, 𝑛 = 0 𝑥 1 − 𝑥 𝑛−1 𝑑𝑥 (𝑚 > 0, 𝑛 > 0) 𝜋 (B) 𝛽(𝑚, 𝑛) = 2 sin 𝜃 2𝑚 −1 cos 𝜃 2𝑛 −1 𝑑𝜃 0 ∞ 𝑦 𝑛 +1 (C) 𝛽(𝑚, 𝑛) = 0 1+𝑦 𝑚 +𝑛 𝑑𝑦 𝜋 (D) 𝛽(𝑚, 𝑛) = 2 0 2 sin 𝜃 2𝑚 −1 cos 𝜃 2𝑛−1 𝑑𝜃 Ans: B (20) Which of the following is not the definition of Beta function? 1 𝑚 −1 (A) 𝛽 𝑚, 𝑛 = 2 0 𝑥 1 − 𝑥 𝑛−1 𝑑𝑥 (𝑚 > 0, 𝑛 > 0) 𝜋 (B) 𝛽(𝑚, 𝑛) = 2 0 2 sin 𝜃 2𝑚 −1 cos 𝜃 2𝑛−1 𝑑𝜃 ∞ 𝑦 𝑛 +1 (C) 𝛽(𝑚, 𝑛) = 0 1+𝑦 𝑚 +𝑛 𝑑𝑦 1 𝑥 𝑚 −1 +𝑥 𝑛 −1 (D) 𝛽(𝑚, 𝑛) = 0 1+𝑥 𝑚 +𝑛 𝑑𝑥 Ans: A (21) 1 1 What is the value of 𝛽 , ? 2 2 (A) / 2 (B) (C) / 4 (D) / 2 Ans: B (22) What is the value of 𝛽(3,2)? (A) 1/4 (B) 1/6 (C) 1/12 (D) 1/16 Ans: C (23) 1 4 x3 The value of dx is 0 1 x (A) 11/35 (B) 128/35 (C) 12/25 (D) 21/25 Ans: B (24) 1 3 What is the value of 𝛽 , ? 4 4 (A) 𝜋 (B) 2𝜋 (C) 2𝜋 (D) 2𝜋 Ans: B (25) 9 What is the value of 𝛽 ,3 ? 2 (A) 16/1287 (B) 16/127 (C) 14/1287 (D) 14/127 Ans: A (26) 1 5 What is the value of 0 𝑥 1 − 𝑥 6 𝑑𝑥 (A) 1/42 (B) 1/496 (C) 1/5544 (D) 1/9842 Ans: C (27) 𝜋/2 The value of 0 𝑡𝑎𝑛𝜃𝑑𝜃 is 𝜋 𝜋 (A) (B) 4 2 𝜋 𝜋 (C) (D) 4 2 Ans : B (28) Lebnitz’s first rule is applied when limits of integration are _____________ of parameter (A) Dependent (B) Power (C) Independent (D) Product Ans : C (29) 1 𝑥 𝑎 −1 If 𝐹 𝑎 = 0 log 𝑥 𝑑𝑥 , then 𝐹 ′ (𝑎) is −1 1 (A) (B) 1+𝑎 2 1+𝑎 2 −1 1 (C) (D) 𝑎+1 𝑎+1 Ans : D (30) If f ( x, ) and f ( x, ) are continuous functions of x and , then d b d a f ( x, ) dx is b b (A) f ( x, ) dx (B) f ( x, ) dx a a b (C) f ( x, ) dx (D) None of these a Ans: C (31) If f ( x, ) and f ( x, ) are continuous functions of x and , then ( ) d d ( ) f ( x, ) dx is ( ) d d (A) f ( x, ) dx d f [ ( ), ] d f [ ( ), ] ( ) ( ) (B) f ( x, ) dx ( ) ( ) (C) f ( x, ) dx ( ) (D) None of these Ans: A (32) 1 xb d If F (b) dx , b 0 , then the value of F (b) is 0 log x da 1 b 1 b (A) (B) (C) (D) 1 b 1 b 1 b 1 b Ans: A (33) The root mean square value of f ( x) x(1 x) 0 x 1is (A) 3.7131 (B) 5.2225 (C) 0.1825 (D) 1.1325 Ans: C (34) If a rod of length ‘a’ is divided into two parts at random. Then the mean value of the sum of the squares on these two segments is (A) 2a2/3 (B) 3a2/2 (C) 2a3/3 (D) 3a3/2 Ans: A (35) The root mean square value of f ( x) over the range (a, b) is is given by b b f ( x)dx f ( x)dx (A) a (B) a ba ba b b f ( x) dx f ( x) dx 2 2 (C) a (D) a ba ba Ans : C (36) The mean value of f ( x) over the range (a, b) is given by b f ( x)dx (A) a ba b f ( x)dx (B) a ba b f ( x)dx (C) a ba b f ( x) dx 2 (D) a ba Ans: A (37) The mean square value of f ( x) over the range (a, b) is given by b f ( x)dx (A) a ba b f ( x)dx (B) a ba b f ( x) dx 2 (C) a ba b f ( x) dx 2 (D) a Ans: D ba (38) The root mean square root value of y = ex + 1 over the range (0,2). (A) 1.2241 (B) 2.2317 (C) 3.6108 (D) 4.5595 Ans: D (39) The root mean square value of y = log e x over the range (1, e) is 1 1 (A) (B) (C) 1 (D) Does not exist e -1 e -1 Ans: B (40) The mean value of y = A sin pt over the range (0, ) is p (A) 0.637 A (B) 0.481 A (C) 0.332 A (D) Does not exist Ans: A 3 3 1.Thecur vex- y =3axyi ssy mmet ri cbout A)X- axi s B)Y- axi s C)bot htheaxesD)aboutt hel i ney =-x Ans. :D 2.Thecur vex3- y3 =3xyi ssy mmet ri cbout A)X- axi s B)Y- axi s C)bot htheaxesD)aboutt hel i ney =-x Ans. :D 2 3.Thecur vey(2a- x)=x3hasasy mpt otepar all elt oy- axi sat A)x=2a B)y =2a C)x=0 D)y=0 Ans. :A 2 4.Thecur vey(2- x)=x3hasasy mpt otepar all elt oy- axi sat A)x=2 B)y =2a C)x=0 D)y=0 Ans. :A 2 2 5.Equat iont othet angentator igi nfort hecur ve3y=x( x-1)is A)X=a B)y=a C)x=0 D)y=0 Ans. :C 6.TheCar tesi anf orm oft hepar amet ri ccur vex=acos3ꝋ, y n3ꝋi =asi s A)x2+y 2 2 =a B)x3+y 3 3 =a C)x2/3+y 2/3 2/ =a 3 D)x3/2+y 3/2 =a3/2 Ans. :C 7.TheCar tesi anf orm oft hepar amet ri ccur vex=cos3ꝋ, y n3ꝋi =si s A)x2+y 2 =1 B)x3+y 3 =1 C)x2/3+y 2/3 =1 D)x3/2+y 3/2 =1 Ans. :C 8.Thecur ver=a( 1+cosꝋ)l i ewi thi naci rcl eofr adi us A)a B)2aC)1D)2 Ans. :B 9.Theequat ionr =2acosꝋr epr esent saci rcl ewhosecent reandr adi us ar e: A)( 2a, 0)anda B)( a,0)andaC)( 2a, 0)and2aD)( a,0)and2a Ans. :B 10. Theequat ionr =2si nꝋr epr esent saci rcl ewhosecent reand radi usare: A)( 2a, 0)anda B)( 0,1)and1C)( 2a, 0)and2aD)( 1,0)and1 Ans. :B 11. Thear eaincl udedbet weent hecur v =x2andt esy hest rai ghtl i ne y =3x+4is: A)121/ 6 B)125/ 6 C)123/ 6 D)131/ 6 Ans.B 12. Thewhol elengt hoft hecur vex2/3+y 2/3 =1i s: A)6uni ts B)8uni ts C)16uni ts D)18uni ts Ans.A 13. Thear eaoutsi det heci rcl er=2acosꝋandi nsi det hecar dioi dr =a( 1+cosꝋ)is: A)πa2 B)3a2/ 2 C)πa2/ 2 D)3πa2/ 2 Ans. :C 14. 2 Thear eaencl osedbyt hepar abol asy=4axandx2=4ayi s: A)( 16/ 3)a2 B)( 32/ 3)a2 C)( 31/ 3)a2 D)( 23/ 3)a2 Ans. :A 15. 2 Thear eaofoneoft hel oopsoft hecur vey=x2–x4 i s A)1/ 3 B)2/ 3 C)1/ 6 D)½ Ans. :B 16. 2/3 2/ Thear eaoft hecur vex +y 3=a2/3 i s: A)3πa2/ 8 B)πa2/ 8 D)3a2/ 8 C)3π/ 8 Ans. :A 17. Thear eaoft hecar dioi dr=a( 1-cosꝋ)i s: A)πa2 B)3a2/ 2 C)8a2/ 2 D)3πa2/ 2 Ans. :D 18. Theareaout sidet heci rcl er=2cosꝋandi nsi det hecar dioi dr=( 1+ cosꝋ)is: A)π B)3/ 2 C)π/ 2 D)3π/ 2 Ans. :C 19. Thecur ver=( 1+cosꝋ)i ssy mmet ri cabout : A)pol e B)I nit iall i neC)bot hAandB D)Noneoft heabov e Ans. :B 20. Lengt hoft hear coft hecur vey =f( x)bet weent hepoi ntswhoseabsci ssasa andbis A) B) C) D) Ans. :A 21 Thear eaencl osedbyt hecur vex=f (y) ,theY- axi sandt heabsci ssay =cand y=di sgiv enby A) B) C) D) Ans.B 22 Thear eaencl osedbyt hecur vey =f( x), theX- axi sandt heor dinat esx=aand x=bi sgiv enby A) B) C) D) Ans. :A 23 Vol umeofthesol idgenerat edbyrevol vi ngoft hear eaboundedbyt he cur vey =f( x)abouttheX-axi sisgi venby A)V =π B)V =π C)V =π D)V =π Ans.D 24 Vol umeofthesol idgenerat edbyrevolv ingofthear eaboundedbyt he cur vey =f( x)aboutthel ineparal l eltotheX-axi sisgivenby A)V =π B)V =π C)V =π D)V =π Ans.D 25 Ar eaboundedbyt hecur ver =f( ϴ)andt her adi ivect orsϴ=ϴ1andϴ=ϴ2 i s A) B) C) D) Ans.: B 26. 2 2 Thear eaoft hel oopoft hecur vey=x(a- x)i s A)a2/ 3 B)2a2/ 3 C) 4a2/ 3 D)8a2/ 15 Ans.:D 27. 2 Thear eabet weent hepar abol ay xa =4x- ndt hel i ney =xi s: A)9/ 2 B)7/ 2 C)5/ 2 D)3/ 2 Ans.A 28. 2 2 Thear eaencl osedbyt het wopar abol asy=4xandy=- 4(x- 2)i s: A)16/ 5 B)16/ 3 C)9/ 3 D)9/ 4 Ans.B AnsD 29` I fattheori gint her ear etwot angent s,whi char ereal anddi ff erentt hent he or igi niscal l ed A)Cusp B)conj ugat epoi nt C)Node D)Noneoft hese Ans. :C 30. Thecur ve i ssy mmet ri cabout A)Bot htheaxes B)aboutt hel i ney =x C)i nopposi tequadr ant D)al loft he above Ans. :D 31. Whichofthefollowingchar acteri sti cisnoti ncl udedi nthest udyofgener al pr ocedurefortracingtheal gebrai ccurve? A)Symmet ry B)RegionorExtent C)Orthogonal it y D)TangentstotheCur veatt heor i gin Ans. :C 32. Whichofthef oll owi ngi snotanexampl eforcur vesy mmet ri caboutyaxi s? a)x2=4ay b)x2=ay 2 c)y =4ax 2 d)x=2ay Ans. :C 33 Whichoft hef ol l owi nggr aphsr epr esentsy mmet ri caboutt heor igi n? 2 A)y=4ax B)x5+y5 =5a2x2y C)x2=4ay D)x2+y2 2 =a Ans. :D 34. Whatismeantbyquadr atur epr ocessi nmat hemat ics? A)Fi ndingar eaofpl anecurves B)fi ndi ngv olumeofplanecur v es C)Fi ndinglengthofplanecurves D)fi ndingslopeofplanecurves Ans.C 35. Whatisthevolumegenerat edwhenther egi onsur roundedbyy=√x, y=2 and y=0i srevol vedabouty–axi s? a)32/πcubicunit s b)32/5cubicunit s c)32π/5cubicunit s d)5π/32cubicunit s Ans.C 36. Thear eaoft hecur vex2/3+y 2/3 =1i s: A)3πa2/ 2 B)3π/ 8 C)4π/ 5 D)4πa2/ 5 AnsB 37. Thel engt hoft hecar dioi dr=( 1+cosꝋ)i s: A)4 B)3 C)8 D)3/ 2 Ans. :C 38. Thear eabet weent hecur vesr=2cosꝋandr=4cosꝋi s A)3π B)4π C)π D)3π/ 2 Ans. :A 39. Thear eabet weent hecur vesr=2( 1+cosꝋ)andr=( 1+cosꝋ)i s A)3π B)4π C)9π/ 2 D)3π/ 2 Ans.C 40. Thear eaencl osedbyt hepar abol asy =2 andx= i s: A)16/ 3 B)32/ 3C)6π D)23/ 3 Ans. :A M-II Module -III Multivariable Calculus (Integration) Question Bank Q 1) Using double integration, the area of the region bounded by parabolas y = x2 and x = y2 is a) 3 b) 1 c) 1/3 d) 0 Ans: c Q 2) If area A of the region bounded by curve 𝑟 = 𝑓 (𝜃) , 𝑟 = 𝑓 (𝜃) and 𝜃 = 𝑎 , 𝜃 = 𝑏 then which of the following is true? ( ) ( ) a)𝐴𝑟𝑒𝑎 = ∫ ∫ ( ) 𝑟 𝑑𝑟𝑑𝜃 b) 𝐴𝑟𝑒𝑎 = ∫ ( ) ∫ 𝑟 𝑑𝑟𝑑𝜃 ( ) c) 𝐴𝑟𝑒𝑎 = ∫ ∫ ( ) 𝑑𝑟𝑑𝜃 d) None of these Ans: a Q 3) To find the area outside the circle 𝑟 = 𝑎 𝑐𝑜𝑠 𝜃 and inside the circle 𝑟 = 2𝑎 𝑐𝑜𝑠 𝜃, the limits of 𝜃 will vary from a) 𝑎 𝑐𝑜𝑠 𝜃 𝑡𝑜 2𝑎 𝑐𝑜𝑠 𝜃 b) 0 𝑡𝑜 𝑎 𝑐𝑜𝑠 𝜃 c) 0 𝑡𝑜 𝜋 d) 0 𝑡𝑜 Ans: d Q 4) For finding area of plate in form of quadrant of ellipse + = 1then which of the following is correct? √ √ a) 𝐴𝑟𝑒𝑎 = ∫ ∫ 𝑑𝑦𝑑𝑥 b) 𝐴𝑟𝑒𝑎 = 2 ∫ ∫ 𝑑𝑦𝑑𝑥 √ c) 𝐴𝑟𝑒𝑎 = 4 ∫ ∫ 𝑑𝑦𝑑𝑥 d) None of these Ans: c Q 5) To find area lying between parabola + = 1 and the straight line 2x+3y = 6, the required region is a) R1 b) R2 c) R3 d) R4 Ans: c Q 6) To find area bounded by ellipse 𝑦 = 4𝑥 − 𝑥 and the line y=x, the strip for the limits is a) Parallel to X-axis b) Parallel to Y-axis c) Radial d) Any of the above Ans: b Q 7) To find the area outside the circle 𝑟 = 𝑎 𝑐𝑜𝑠 𝜃 and inside the circle 𝑟 = 2𝑎 𝑐𝑜𝑠 𝜃, the strip for the limits is a) only vertical strip b) only horizontal strip c) Radial strip d) both vertical strip and horizontal strip Ans: c Q 8) To find the volume of the region bounded by parabolas y = x 2 , x = y2 and z = 0 and z = 3 then which of the following is true? √ √ a)𝑉𝑜𝑙𝑢𝑚𝑒 = ∫ ∫ 𝑑𝑦𝑑𝑥 b) 𝑉𝑜𝑙𝑢𝑚𝑒 = ∫ ∫ 𝑑𝑦𝑑𝑥 √ c)𝑉𝑜𝑙𝑢𝑚𝑒 = ∫ ∫ 3 𝑑𝑦𝑑𝑥 d) None of these Ans: c Q 9) To find Center of Gravity of area parabola x = y2 and the line x + y = 2, the required region is a) R1 b) R2 c) R3 d) R1 + R2 Ans: b Q 10) By changing the order of integration, ∫ ∫ 𝑑𝑦𝑑𝑥 the limits of integration becomes a) x : 0 to a b) x : y to ∞ y : 0 to ∞ y : 0 to x c) x : 0 to y d) x : 0 to ∞ y : 0 to ∞ y : 0 to y Ans : c Q 11) ∫ ∫ 𝑑𝑦𝑑𝑥 = a) b) 𝑎 c) d) 𝑎 Ans: b Q 12) For evaluation of ∬ (𝑥 + 3𝑦 ) 𝑑𝑥𝑑𝑦 where A is te area of the rectangle 0 ≤ 𝑥 ≤ 3; 0 ≤ 𝑦 ≤ 1 a) First with respect to y then with respect to x b) First with respect to x then with respect to y c) Does not matter d) None of these Ans: c Q 13) ∫ ∫ 𝑑𝑥𝑑𝑦 = a) 1 b) 3 c) 1/3 d) Not define Ans: b Q 14) The integral ∫ ∫ 𝑥𝑦 𝑑𝑥𝑑𝑦 is solved by using change of order of integration. The strip for new limits is a) Vertical strip b) Horizontal strip c) Radial d) Any of the above Ans: b Q 15) For ∬ 𝑥𝑦𝑑𝑥𝑑𝑦, where R is region bounded by the circle 𝑥 + 𝑦 = 𝑎 , 𝑥 ≥ 0, 𝑦 ≥ 0 then the required region is a) R1 b) R2 c) R3 d) R4 Ans: a Q 16) Using double integration, to find Center of Gravity of area of the circle 𝑥 + 𝑦 = 𝑎 lying in the first quadrant and assumed density is K, which of the following is correct? ∫ ∫ ∫ ∫ a) 𝑥̅ = 𝐾 ,𝑦 = 𝐾 ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ b) 𝑥̅ = ,𝑦 = ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ c) 𝑥̅ = ,𝑦 = ∫ ∫ ∫ ∫ ∫ ∫ ∫ d) 𝑥̅ = ,𝑦 = ∫ ∫ ∫ ∫ Ans : c Q 17) After changing into polar coordinates, the integral ∫ ∫ 𝑥𝑦𝑑𝑦𝑑𝑥 changes to a) ∫ ∫ 𝑟 𝑠𝑖𝑛𝜃 𝜃 𝑑𝑥𝑑𝑦 b) ∫ ∫ 𝑟 𝑑𝑟𝑑𝜃 c) ∫ ∫ 𝑟 sin 𝜃 𝑐𝑜𝑠𝜃 𝑑𝑟𝑑𝜃 d) ∫ ∫ 𝑟 sin 𝜃 𝑐𝑜𝑠𝜃 𝑑𝑟𝑑𝜃 Ans: c ( ) Q 18) After changing into polar coordinates, the integral ∫ ∫ 𝑒 𝑑𝑥𝑑𝑦 changes to a) ∫ ∫ 𝑒 𝑑𝑟𝑑𝜃 b) ∫ ∫ 𝑒 𝑟𝑑𝑟𝑑𝜃 c) ∫ ∫ 𝑒 𝑟𝑑𝑟𝑑𝜃 d) ∫ ∫ 𝑒 𝑑𝑟𝑑𝜃 Ans: b Q 19) If 𝜌 = 𝑓(𝑟, 𝜃) at a point (𝑟, 𝜃) then Mass is a)∬ 𝑟 𝑑𝑟𝑑𝜃 b) ∬ 𝜌 𝑟 𝑑𝑟𝑑𝜃 c) ∬ 𝜌 𝑑𝑟𝑑𝜃 d) None of these Ans: b Q 20) After changing into polar coordinates, the integral ∫ ∫ 𝑒 ( ) 𝑑𝑥𝑑𝑦 then the limits of r a) 0 𝑡𝑜 ∞ b) 0 𝑡𝑜 𝑟 c) 0 𝑡𝑜 𝜋 d) None of these Ans: a Q 21) ∫ ∫ 𝑟 𝑑𝑟𝑑𝜃 = a) 𝑎 b) 𝑎 c) 𝑎 d) 𝑎 Ans: d Q 22) Applications of double integral for finding a) Mass b) Area c) Center of gravity d) all are correct Ans: d ( ) Q 23) After changing into polar coordinates, the integral ∫ ∫ 𝑒 𝑑𝑥𝑑𝑦 then the limits of 𝜃 a) 0 𝑡𝑜 b) 0 𝑡𝑜 c) 0 𝑡𝑜 𝜋 d) None of these Ans: b Q 24) For the integral, ∫ ∫ 𝑑𝑥𝑑𝑦 what is the order of integration after changing into polar coordinates? a) integrate with respect to 𝑦 first and then x. b) integrate with respect to 𝑥 first and then y c) integrate with respect to 𝑟 first and then 𝜃. d) integrate with respect to 𝜃 first and then r. Ans: c √ Q 25) The integral ∫ ∫ (𝑥 + 𝑦 ) 𝑑𝑥𝑑𝑦 is solved by changing into polar coordinates. The strip for new limits is a) Vertical strip b) Horizontal strip c) Radial d) Any of the above Ans: c Q 26) For evaluation of ∫ ∫ ∫ (𝑥)𝑑𝑥𝑑𝑦𝑑𝑧 a) First with respect to y, secondly with respect to z then with respect to x b) First with respect to x secondly with respect to y then with respect to z c) First with respect to z secondly with respect to y then with respect to x d) Does not matter Ans: d Q 27) For the integral, ∫ ∫ ∫ (𝑟)𝑑𝑟𝑑𝜃𝑑𝑧 what is the order of integration? a) integrate with respect to 𝜃 first then z and then r. b) integrate with respect to 𝜃 first then r and then z. c) integrate with respect to z first then r and then 𝜃. d) integrate with respect to r first then z and then 𝜃. Ans: c Q 28) For the integral, ∫ ∫ ∫ (𝑥 + 𝑦 + 𝑧)𝑑𝑥𝑑𝑦𝑑𝑧 what is the order of integration? a) integrate with respect to z first then y and then x. b) integrate with respect to x first then y and then z. c) integrate with respect to y first then z and then x. d) integrate with respect to y first then x and then z. Ans: d 1 1 1 x Q 29) For the integral, 0 y 0 ( x y z) dz dx dy which of the following is true? a) x: x = y to x = 1, y: y = 0 to y = 1, z: z = 0 to z = 1-x b) x: x = 0 to x = 1, y: y = y to y = 1, z: z = 0 to z = 1-x c) x: x = 0 to x = 1-x , y: y = y to y=1, z: z = 0 to z = 1 d) x: x = y to x = 1, y: y = 0 to y = 1-x, z: z = y to z = 1 Ans: a Q 30) For ∬ 𝑦 𝑑𝑥𝑑𝑦, where R is the region bounded by the parabolas y 2 4 x and x 2 4 y.Which of the following is true? y2 a) New limits can be y: y = 0 to y = 4 and x : x to x 2 y 4 x2 b) New limits can be x: x = 0 to x =4 and y : y 2 x to y 4 48 c) y dx dy R 5 d) all are correct Ans: d Q 31) Evaluate e over the triangle bounded by x = 0, y = 0 2 x 3 y dx dy , and x + y = 3 a) New limits are y: y = 0 to y = 3-x and 𝑥: 𝑥 = 0 to x 3 b) New limits are x: x=0 to x=3 and y : y 3 x to y 0 c) New limits are x: x=0 to x=3 and y: y=0 to y=3 d) all options are correct. Ans: a Q 32) By changing the order of integration∫ ∫ 𝑑𝑥𝑑𝑦 , the limits of integration becomes a) x: y to a b) x: y to a y: 0 to a y: 0 to x c) x: 0 to a d) x: 0 to a y: 0 to a y: 0 to x Ans: d Q 33) Using the change of order of integration, the order of evaluation of ∫ ∫ (𝑥 + 2𝑦)𝑑𝑥𝑑𝑦 is a) First with respect to y then with respect to x b) First with respect to x then with respect to y c) Does not matter d) None of these Ans : a Q 34) The integral ∫ ∫ 𝑑𝑥𝑑𝑦 is solved by using change of order of integration. The strip for new limits is a) Parallel to X-axis b) Parallel to Y-axis c) Radial d) Any of the above Ans: a ( ) Q 35) The integral ∫ ∫ 𝑒 𝑑𝑥𝑑𝑦 is changed into polar form, then the value of integral is a) π/2 b) π/3 c) 2π d) π/4 Ans: d a a x2 Q 36) After changing into polar coordinates the integral 0 y x 2 y 2 dx dy changes to a a 4 a sec r cos r cos dr d 2 a) 2 dx dy b) 0 y 0 0 4 a sec a a cos dr d r cos 2 c) d) 2 dr d 0 0 0 y Ans: b Q 37) To convert the given Cartesian coordinates to polar coordinates the substitutions are a) x r cos, y r sin , dx dy r dr d b) x r cos, y r sin , dx dy dr d c) x r cos, y r sin , dx dy r dr d d) x r sin, y r cos , dx dy r dr d Ans: a Q 38) To find the area outside the circle 𝑟 = 𝑎 𝑐𝑜𝑠 𝜃 and inside the circle 𝑟 = 2𝑎 𝑐𝑜𝑠 𝜃 the limits of r will vary from a) 𝑎 𝑐𝑜𝑠 𝜃 𝑡𝑜 2𝑎 𝑐𝑜𝑠 𝜃 b) 0 𝑡𝑜 𝑎 𝑐𝑜𝑠 𝜃 c) 0 𝑡𝑜 2𝑎 𝑐𝑜𝑠 𝜃 d) 0 𝑡𝑜 𝑟 Ans: d Q 39) For ∬ 𝑥𝑦𝑑𝑥𝑑𝑦, where R is the region bounded by X-axis x = 2a and 𝑥 = 4𝑎𝑦, then the required region is a) R1 b) R2 c) R3 d) R1 + R2 Ans: a Q 40) If the equation of circle is 𝑟 = 𝑎 𝑐𝑜𝑠 𝜃, then radius of this circle is a) a b) a/2 c) a/4 d) 2 Ans: b Subject: Mathematics-II (Question Bank MCQs) Module IV: Vector Calculus 1. a (b c ) b (c a ) c (a b ) (a) 5 (b) 0 (c) 2 (d) 1 Ans: (b) 2. iˆ (a iˆ) ˆj (a ˆj ) kˆ (a kˆ) (a) 2 (b) a (c) 2a (d) 2a Ans: (d) 3. a (b c ) (a) (a c )b (a b )c (b) (a b )c (c) (a b )c (d) (a b )c (a c )b Ans: (a) 4. (b c ) [(c a ) (a b )] (a) (a b )c (b) {(b a ) c } (c) {(a b ) c } (d) a (b c ) 2 2 Ans: (c) 5. (b c ) (a d ) (c a ) (b d ) (a b ) (c d ) (a) 1 (b) 3 (c) 2 (d) 0 Ans: (d) 6. ( A B ) (C D ) (a) 0 (b) ( A D ) ( B C ) (c) ( A C )( B D ) (d) ( A C )( B D ) ( A D ) ( B C ) Ans: (d) 7. (a b ) (c d ) (a) [b c d ]a [a c d ]b (b) [a c d ]b [b c d ]a (c) [b c d ]a [a c d ]b (d) [a c d ]b [b c d ]a Ans: (b) 8. (a b ) (c d ) (a) [a b d ]c [a b c ]d (b) [b c d ]a [a c d ]b (c) [a b c ]d [a b d ]c (d) [a b d ]c [a b c ]d Ans: (a) 9. If a (b c ) (a b ) c then (a) (a c ) b 0 (b) (a c ) b 0 (c) (a c ) b 0 (d) None of these Ans: (c) 10. (b c ) (a d ) (c a ) (b d ) (a b ) (c d ) (a) 0 (b) 2[a b c ]d (c) 2[a b c ]d 2 (d) None of these Ans: (b) 11. A particle moves along the curve x t 1, y t , z 2t 5 where t is the time, then the 3 2 component of velocity at t=1in the direction of iˆ ˆj 3kˆ is (a) 11 (b) 10 (c) 13 (d) 15 Ans: (a) 12. A particle moves along the curve r (t 3 4t )iˆ (t 2 4t ) ˆj (8t 2 3t 3 )kˆ where t is the time, then the tangential component of acceleration at t=2 is (a) 15 (b) 16 (c) 20 (d) 13 Ans: (b) 13. If t1 iˆ 2 ˆj 3kˆ and t 2 iˆ 2 ˆj 3kˆ be two tangent vectors to the curve, then angle between them is 7 3 3 7 (a) cos (b) cos (c) sin (d) sin 1 1 1 1 3 7 7 3 Ans: (b) d 14. If a t 2iˆ tˆj (2t 1)kˆ and b 2tiˆ ˆj tkˆ , then at t = 0 (a b ) dt (a) 2iˆ 2 ˆj (b) 2iˆ ˆj (c) iˆ 2 ˆj (d) 2i 2 ˆj ˆ Ans: (d) 15. A particle moves so that its position vector is given by r (cos wt )iˆ (sin wt ) ˆj where w is constant, then r v (a) w (b) constant vector (c) constant scalar (d) None of these Ans: (b) 16. If r xiˆ yˆj zkˆ then grad (r ) r r (a) (b) 0 (c) (d) r r r Ans: (a) 17. If r xiˆ yˆj zkˆ then grad (r ) n r r n 1 (a) (b) n r 2 r (c) (d) nr n 2 r r n2 r Ans: (d) 18. If r xiˆ yˆj zkˆ then div(r ) (a) 3 (b) 3 (c) 0 (d) 5 Ans: (b) 19. If r xiˆ yˆj zkˆ then curl (r ) r (a) (b) 1 (c) 0 (d) r r Ans: (c) 20. The directional derivative of is maximum in the direction of (a) (b) (c) (d) None of these Ans: (b) 21. The directional derivative of 4e at the point (1,1,1) in the direction towards the point 2 x y z (3,5,6) is 2 10 20 20 (a) (b) (c) (d) 9 9 9 9 Ans: (d) 22. The value of the constant a & b so that the surface ax byz (a 2) x will be orthogonal to the 2 surface 4 x y z 4 at the point (1,1,2) are 2 3 (a) 5, 1 (b) 0, 2 (c) 5/2, 1 (d) 1, 2 Ans: (c) 23. If divF 0 then F is (a) irrotational (b) rotational (c) solenoidal (d) None of these Ans: (c) 24. If vector field F is irrotational then (a) curlF 0 (b) curlF 1 (c) curlF 0 (d) curlF 1 Ans: (a) 25. Work done by the force F along the path C from point A to B where a scalar potential is given by (a) ( A) ( B ) 0 (b) ( B A) 0 (c) ( A B ) 0 (d) ( B ) ( A) 0 Ans: (d) 1 26. The directional derivative of in the direction of r where r xiˆ yˆj zkˆ is r 1 1 1 (a) (b) (c) 2 (d) r r r r Ans: (c) 27. The value of n for which vector field r r will be solenoidal where r xiˆ yˆj zkˆ is n (a) 3 (b) -3 (c) 1 (d) - 2 Ans: (b) 28. The value of constant a so that the vector V ( x 3 y )i ( y 2 z ) j ( x az )k is solenoidal is (a) -2 (b) -3 (c) -1 (d) 2 Ans: (a) 29. The divergence of V ( xyz)i (3x y ) j ( xz y z )k at the point (2,1,1). 2 2 2 (a) 15 (b) 13 (c) 16 (d) 14 Ans: (d) 30. The curl of V ( xyz)i (3x y ) j ( xz y z )k at the point (2,1,1). 2 2 2 (a) 2iˆ 2 ˆj 3kˆ (b) 2iˆ 3 ˆj 14kˆ (c) iˆ 2 ˆj kˆ (d) 2iˆ 2 ˆj 3kˆ Ans: (b) 31. curl grad -------, where is scalar point function. (a) 0 (b) 1 (c) 6 (d) 4 Ans: (a) 32. div curlA -------, where A is vector point function. (a) 1 (b) 0 (c) 2 (d) 5 Ans: (b) 33. div( A ) (a) div A grad A (b) div A grad A (c) div A grad A (d) None of these Ans: (a) 34. curl ( A ) (a) curl A grad A (b) curl A grad A (c) curl A grad A (d) None of these Ans: (c) 35. A vector field A ( x yz)i ( y zx) j ( z xy)k is 2 2 2 (a) rotational (b) conservative (c) solenoidal (d) None of these Ans: (b) 36. The value of constants a , b , c so that the vector F ( x 2 y az )iˆ (bx 3 y z ) ˆj (4 x cy 2 z )kˆ is irrotational are (a) 1, 3, 2 (b) 3, 2, -1 (c) 1, 0, -1 (d) 4, 2, -1 Ans: (d) 37. If A is irrotational where A ( y z )iˆ ( z x) ˆj ( x y )kˆ then its scalar potential is (a) xy yz zx (b) xy yz zx (c) xy yz zx (d) None of these Ans: (c) 38. If x y xz be the scalar potential for the conservative vector field F then the work done in 2 3 moving an object in this field from (1,2,1) to (3,1,4) is (a) 202 (b) 200 (c) 201 (d) 210 Ans: (a) dr 39. If F sin(a sin )iˆ a cos [1 cos(a sin )] ˆj and ( a sin )iˆ (a cos ) ˆj then the value d of F dr from 0 to 2 C is (a) (b) 2a 2 (c) a (d) a 2 Ans: (d) A dr from 0 to 2 dr 40. If A 4 8 cos 2 8 sin 2 then the value of is d C (a) 8 (b) 2 (c) (d) 4 Ans: (d) Multiple Choice Questions Module - 5 - Statistics 1. Which of the following is one of the normal equations of y a bx A) xy na b x 2 B) y na b x C) y a b x D) None of these A-B 2. Which of the following is one of the normal equations of y a bx A) xy a x b x 2 B) y na b x C) y a b x D) None of these A-A 3.For the following values of x and y the equation of the best fit straight line y a bx is------ x 1 2 3 4 6 8 y 2.4 3 3.6 4 5 6 A) y 1.976 0.506 x B) y 1.976 0.506 x C) y 1.976 0.506 x D) y 1.976 0.506 x A-D 4. Which of the following is one of the normal equations of y ax b A) xy a x 2 b x B) y na b x C) y a b x D) None of these A-A 5. Which of the following is one of the normal equations of y ax b A) xy na b x 2 B) y na b x C) y a b x D) y a x nb A-D 6. Which of the following is one of the normal equations of y mx c A) xy nm c x 2 B) y nm c x C) y m c x D) y m x nc A-D 7. Which of the following is one of the normal equations of y a bx cx 2 A) y na b x c x 2 B) y a x b x 2 c x 3 C) xy a x 2 b x 3 c x 4 D) None of these A-A 8. Which of the following is one of the normal equations of y a bx cx 2 A) y a b x c x 2 B) xy a x b x 2 c x 3 C) xy a x 2 b x 3 c x 4 D) None of these A-B 9. Which of the following is one of the normal equations of y a bx cx 2 A) xy na b x c x 2 B) y a x b x 2 c x 3 C) x 2 y a x 2 b x 3 c x 4 D) None of these A-C 10.For the following values of x and y the equation of the best fit parabola y a bx cx 2 is- x 0 2 5 10 y 4 7 6.4 -6 A) y 4.1 1.979 x 0.299 x 2 B) y 4.1 1.979 x 0.299 x 2 C) y 4.1 1.979 x 0.299 x 2 D) y 4.1 1.979 x 0.299 x 2 A-D 11. Which of the following is one of the normal equations of y ax 2 bx c A) y nc b x a x 2 B) xy a x b x 2 c x 3 C) x 2 y a x 2 b x 3 c x 4 D) None of these A-A 12. Which of the following is one of the normal equations of y ax 2 bx c A) y na b x c x 2 B) xy a x b x 2 c x 3 C) x 2 y c x 2 b x 3 a x 4 D) None of these A-C 13. Which of the following is one of the normal equations of y a bx 2 A) xy na b x 2 B) y na b x C) y na b x 2 D) None of these A-C 14. Which of the following is one of the normal equations of y a bx 2 A) xy na b x 2 B) x 2 y a x 2 b x 4 C) y a x b x 2 D) None of these A-B 15.For the following values of x and y the equation of the best fit parabola y a bx 2 is------ x 0 1 2 3 y 2 4 10 15 A) y 2.7 1.44 x 2 B) y 2.7 1.44 x 2 C) y 2.7 1.44 x 2 D) y 2.7 1.44 x 2 A-C 16. Which of the following is one of the normal equations of y ax 2 b A) xy na b x 2 B) y a x nb 2 C) y na b x 2 D) None of these A-B 17.Which of the following is one of the normal equations of y ax b A) log xy n log a b log x B) log xy n log a x log b C) log x log y log a log x b (log x) 2 D) None of these A-C 18.Which of the following is one of the normal equations of y ab x A) log y a log n log b x B) log y n log a log b x C) log y n log a log x b D) None of these A-B 19. For the following values of x and y the equation of the best fit curve y ab x is------ x 2 3 4 5 6 y 144 172.3 207.4 248.8 298.5 A) y 100(1.2) x B) y 100(1.2) x C) y 100(1.2) x D) y 100(1.2) x A-A 20.Which of the following is one of the normal equations of y ae b x A) log y n log a b log e x B) log y a log n b log e x C) log y n log a log x b log e D) None of these A-A 21.For the following values of x and y the equation of the best fit curve y ae bx is------ x 0 2 4 y 5.012 10 31.62 A) y 4.642e 0.46 x B) y 4.642e 0.46 x C) y 4.642e 0.46 x D) y 4.642e 0.46 x A-B 22.Two variables are said to be ------------if increase or decrease in one variable is accompanied by increase or decrease in the other variable. A)correlated B) unrelated C)related D) none of these. A-A 23. Karl Pearson’s defined the coefficient r = ------------ A) XY B) X Y 2 2 C) XY D) X Y 2 2 X.Y X.Y 2 2 X.Y 2 2 X.Y A-C 24. The value of coefficient of correlation always varies from ___________. A) 0 to 1 B) -1 to 0 C)-1 to 1 D) none of these. A-C 25. The equation of line of regression of y on x is ------------. A) x b0 b1 y B) y a0 a1 x C) y a0 a1 y D) y a0 a1 x 2. A-B 26. The equation of line of regression of y on x is useful to predict the value of ------------. A)y B) x C) both x and y D) None of these A-A 27. The equation of line of regression of x on y is useful to predict the value of ------------. A) y B) x C) both x and y D) None of these A-B 28. If r=0 then lines of regression are------------. A) parallel B)coincide C) perpendicular D) None of these A-C 29. If r=1 then lines of regression are ------------. A) different B)equal C) perpendicular D) None of these A-B 30. Point of intersection of lines of regression is---- A) y x B) x y C) x x D) y y A-B 31. The equation of line of regression of x on y is ------------. A) x b0 b1 y B) y a0 a1 x C) x b0 b1 x D) x b0 b1 y 2. A-A 32. The regression coefficient of y on x is given by a1 =----------. y y x y A) r 2 B) r 2 C) r D) r. x x y x A-D 33. The coefficient of correlation r in terms of regression coefficients is given by -------- A) r a1b1 B) r a1 b1 C) r a1b1 D) r a1b1 2 2 A-A 34. The coefficient of rank correlation r = -----------. d 6 d i 6 d i 6 d i 2 2 2 A) 1 B) 1 C) 1 D) 1 i. n (n 2 1) n (n 2 1) n (n 1) n (n 2 1) A-D 35. Two lines of regression are given by x 2 y 5 0 and 2 x 3 y 8 0 then the mean values ofx and y are ---------- A) 1, 2 B) 2, 1 C)-1, -2 D) -2,-1 A-A 36. If lines of regression are 5 y 8x 17 0 and 2 y 5x 14 0 and if y 16 then the 2 standard deviation of x is --------- A) 4 B) -4 C)-2 D) 2 A-D 37. If lines of regression are 5 y 8x 17 0 and 2 y 5x 14 0 then the coefficient of correlation between x and y is-------- A) 0 B) 0.8 C)0.99 D) None of these A-B 38. In rank correlation if all the d’s are zero then r = ------- A) 0 B) 1 C)-1 D) None of these A-B 39. If six values of X and Y are 2, 4, 5, 6, 8, 11 and 18, 12, 10, 8, 7, 5 respectively then sum of the differences of ranks of corresponding values of X and Y is A) 4 B) 3 C)2 D) 0 A-D 40. For the following values of X and Ythe rank correlation coefficient is r = -------- X 2 4 5 6 8 11 Y 18 12 10 8 7 5 A) 1 B) -1 C)0.5 D) 0.8 A-B MCQs Finite Differences Module 6 1. The shifting operator is denoted by ________. A) E B) nabla C) omega D) T Ans: A 2.Δ f (x) = A) f ( x + h) B) f ( x) − f ( x + h) (c) f ( x + h ) − f ( x) D) f ( x) − f ( x − h) AnsC 3. E ≡ A) 1 + Δ B) 1 − Δ C)1 + ∇ D) 1 − ∇ AnsA 4. If C is a constant then Δ C = A) C B) Δ C) 𝛥2 D) 0 Ans: D 5. If m and n are positive integers then 𝛥𝑚 𝛥𝑛 f (x) = A) 𝛥m +n f ( x) B) 𝛥𝑚 f ( x) C) 𝛥𝑛 f ( x) D) 𝛥m −n f ( x) AnsA 6. E f (x) = A) f ( x − h) B) f ( x) C) f ( x + h) D) f ( x + 2h) AnsC 7. For the given points ( 𝑥0 , 𝑦0 ) and (𝑥1 , 𝑦1 ) the Lagrange’s formula is 𝑥− 𝑥 1 𝑥− 𝑥 0 𝑥1− 𝑥 𝑥− 𝑥 0 A) y(x) = 𝑦0 + 𝑦1 B) y(x) = 𝑦0 + 𝑦 𝑥0 − 𝑥1 𝑥1 − 𝑥0 𝑥0 − 𝑥1 𝑥1 − 𝑥0 1 𝑥− 𝑥 1 𝑥− 𝑥 0 𝑥1 − 𝑥 𝑥− 𝑥 0 C) y(x) = 𝑦1 + 𝑦0 D) y(x) = 𝑦1 + 𝑦0 𝑥0 − 𝑥1 𝑥1 − 𝑥0 𝑥0 − 𝑥1 𝑥1 − 𝑥0 AnsA 8. If f (x) = 𝑥 2 + 2x + 2 and the interval of differencing is unity then Δf( x) is ? A) 2x – 3 B) 2x + 3 C) x + 3 D) x − 3 AnsB 9. The process of finding the values inside the interval (𝑥0 , 𝑥𝑛 ) is called A) Interpolation B) Extrapolation C) Iterative D) Polynomial equation Ans A 10. The Delta of power two is called the ____order difference operator. A) First B) second C) Third D) Fourth Ans B 11. For the given distributed data find the value of𝛥3 𝑦0 is? x 3.60 3.70 3.65 3.75 y 36.59 8 38.47 5 40.44 7 42.52 1 A) 0.095 B) 0.007 C) 1.872 D) 0.123 Ans B 12. Find Δ (x + cos x)? A) 1+2sin(x+1/2).sin1/2 B) 1 -2sin(x+1/2).sin1/2 C) 1 -2sin(x -1/2).sin1/2 D) 1+2sin(x -1/2).sin1/2 Ans B 13. If f(1) = 2, f(2) = 4 and f(4) = 16, what is the value of f(3)using Lagrange’s interpolation formula? 1 2 A) 8 (B) 8 (C) 8 D) 9 3 3 Ans C 14. In Simpson’s 1/3rd rule of integration is exact for all polynomials of degree not exceeding_________. A) 4. B) 1. C) 3. D) 2. Answer: D 15. In Simpson’s 3/8th rule which is applicable only when_____. A) n is multiple of 3 B) n is multiple of 6. C)n is multiple of 8. D) n is multiple of 24. Answer: A 16. In Simpson’s 1/3rd rule the number of intervals must be _____. A) Multiple of 3. B) Multiple of 6. C).Odd. D) Even Answer: D 17.The degree of y(x) in Trapezoidal Rule is _______. A)1. B)2. C)3. D)6. Answer: A 18. The degree of y(x) in Simpson’s (3/8)th is________. A)1. B) 2. C) 3. D) 6. Answer: C 19. In Simpson’s (1/3)rd Rule the number of intervals ______. A)odd. B)even. C) multiple of 3. D) multiple of 6. Answer: B 20. Interpolating polynomial is also known as______. A)smoothing function. B) interpolating function. C) collocation polynomial. D) interpolating formula. Answer: D 21. In Lagrange’s interpolation formula, the value of 𝑙0 (x) = _____. 𝑥1− 𝑥0 𝑥− 𝑥 1 𝑥 − 𝑥1 𝑥1− 𝑥0 A) B) C). D) 𝑥− 𝑥 0 𝑥0− 𝑥1 𝑥− 𝑥 0 𝑥2− 𝑥0 Answer: B 𝑥4 22. The Trapezoidal rule for = 𝑥0 𝑦 𝑑𝑥 ℎ ℎ A) { 𝑦0 +2(𝑦1 + 𝑦2 + 𝑦3 )+ 𝑦4 }. B) { 𝑦0 + 2(𝑦1 + 𝑦2 + 𝑦3 )+ 𝑦4 }. 2 3 ℎ ℎ C) { 𝑦0 + 2𝑦1 + 4( 𝑦2 + 𝑦3 )+ 𝑦4 }. D) { 𝑦0 + 𝑦1 + 𝑦2 + 𝑦3 +𝑦4 } 2 2 Answer: A 23. In deriving the trapezoidal formulae, the arc of the curve y=f(x) over each subinterval is replaced by its_____. A) Straight line. B) Ellipse. C) Chord D) Tangent line. Answer: C 24. In Simpson’s rule will give exact result, if the entire curve y=f(x) is itself a ____. A) Straight line. B) Chord. C) Parabola. D)Tangent line. Answer: C 25. Difference equation is used in : A) Discrete time analysis B) Continuous time analysis C) Digital analysis D) None of the mentioned Answer: A 26. Match the CORRECT pairs. Numerical Integration Scheme Order of Fitting Polynomial P. Simpson’s 3/8 Rule 1. First Q. Trapezoidal Rule 2. Second R. Simpson’s 1/3 Rule 3. Third A) P-2, Q-1, R-3 B) P-3, Q-2, R-1 C) P-1, Q-2, R-3 D) P-3, Q-1, R-2 Answer: D 27. The (n+1)th forward difference of nth degree of polynomial is ---- A) Zero B) a constant C) a variable D) None of these Answer: A 28. Order of the difference equation xn+2 - xn+1 + 2xn = n is ---- A) Zero B) 1 C) 2 D) 3 Answer: C 29. The interpolating function may be a straight line passing through the points. This is called the trapezoidal rule. A) TRUE B) FALSE C) Can be true or false D) Can not say Answer: A 30. The first forward difference of constant function is A) Constant B) 0 C) 1 D) None of these Answer: B 31. In the function y = f(x) , the independent variable x is called A) Entry B) Argument C) Intermediate D) interpolation Answer: B 32. The following function(s) can be used for interpolation: A) polynomial B) exponential C) trigonometric (D) all of the above Answer: D 33. Which of the following statement is true? A) Simpson’s 1/3rd rule can be applied when the range is divided into even number of subintervals B) Simpson’s 3/8th rule can be applied when the range is divided into number of subintervals, which must be a multiple of 3. C) Trapezoidal rule can be applied for any number of subintervals D) All of the above Answer: D 34. If ∅(E)𝑦𝑛 = F(n) and F(n) = 0, then solution of equation is given by A) Only PI B) Only CF C) CF + PI D) all of the above Answer: B 35. CF of Auxiliary Equation (A.E.) is( 𝑚2 - 5m + 6) = 0 A) 𝑐1(−2)𝑛 + 𝑐2 (−3)𝑛 B) 𝑐1 2𝑛 + 𝑐2 3𝑛 C) 𝑐1 (−2)𝑛 + 𝑐2 3𝑛 D) 𝑐1 2𝑛 + 𝑐2 (−3)𝑛 Answer: B 36. Find P.I. of difference equation: yn+2 - 3yn+1 + 2yn = 5𝑛 1 A) 12n B) 5n 5 1 C) 5n D) None of these 12 Answer: C 2 37. Find value of [ 1 - ∆ + ∆2 -----] [ 𝑛(2) + 𝑛(1) ] 3 1 A) [𝑛2 -2n + ] B) [𝑛2 -2n ] 3 1 1 C) [𝑛(2) -2𝑛(1) + ] D) [𝑛2 + ] 3 3 Answer: A 38. ∆3 (3𝑥 (2) ) = -- A) 0 B) 3 C) 2 D) 6 Answer: A 39. If ∆5 y = 0 then the number of entries are A) 6 B) 5 C) 4 D) 3 Answer: B 1 40. (𝑥 (3) ) = ? ∆3 𝑥 (6) 𝑥 (5) A) B) 120 20 𝑥 (2) 𝑥 (4) C) D) 2 4 Answer: A
