Model Question Paper Subject Code: MC0074 Subject Name: Statistical and Numerical methods using C++ Credits: 4Marks: 140 Part A (One mark questions) 1. Probability is the study of experiments, the outcomes of whose result are .. A) Certain B) Uncertain C) Countable D) Uncountable 2. Every point of a sample space is called as a .. A) Sample point B) element C) cluster D) limit point 3. A variable which takes a definite set of values with a definite probability associated with each value of the variable is called the .. A) Random set B) Random number C) Random variable D) Random value

4.Theexpectedvalueofthenumberofpointsthatwillbeobtainedinasinglethrowwithan ordinary die isA)27 B) 26

C) 25 D) 29 5. Poisson distribution is the limiting form of the binomial distribution A) (q+p)2n B) (q+p)n C) (q+2p)n D) (q-p)n 6. Mean of the Poisson distribution is .. A) m B) 2m C) 2m D) m2 7. The moment generating function about the origin is given by .A) M0(t) =((

+ + + ......! 2) (12ttB) M0(t) = ((

+ + + ......! 2) () ( 122ptptC) M0(t) =((

+ ......! 2) (12ptpt

D) M0(t) = ((

+ + + ......! 2) (12ptpt 8. M.G.F about mean will be given by m(t) = .. A) ) 1 ( 2 t e mte B) ) 2 ( t e mte C) ) 1 ( t e mte D) ) ( t e mte 9...isatechniqueusedforthemodelingandanalysisofnumericaldata consisting of values of a dependent variable and of independent variables.A) Complex analysis B) Real analysis C) Regression analysis D) Correlation analysis 10. In the value of the variate is assumed to be concentrated at the mid value of the interval.A) Class interval B) Frequency polygon C) Mode D) Median 11. A .. is a family of random variables {X(t) / teT}, defined on a given probability space, indexed by the parameter t, where t varies over an index set T. A) Random process B) Arrival process C) Stochastic process D) Markov process

12. A is defined to be discrete parameter independent process {Xn/n=1,2,.}whereX1,X2,..areindependent,identicallydistributed,nonnegative random variables. A) Renewal process B) Transaction process C) Modal process D) Unimodal process 13. t is an example of ..A) Approximate numbers B) Random numbers C) Exact numbers D) Natural numbers 14. .. are those that represent the numbers to a certain degree of accuracy A) Approximate numbers B) Natural numbers C) Real numbers D) Complex numbers. 15. A is an array of mn elements arranged in m rows and n columns. A) Vector B) Set C) Matrix D) Natural numbers 16. A square matrix in which all the elements are zeros is said to be a ..A) Skew matrix B) Skew symmetric matrix C) Orthogonal matrix D) Null matrix

17. Every algebraic equation of nth degree, where n is a positive integer, has only..A) n 1 roots B) n roots C) n + 1 roots D) 2n roots 18. If (a + ib) is a root of f(x) = 0, then the other root of the equation f(x) is ....A) ib B) a C) a ib D) b ia 19. .. play an important role in numerical techniques, where tabulated values of the functions are available.A) infinite differences B) normal differences C) Finite differences D) factorial differences 20. The methodofcomputingthevalueofy, fora givenvaluesofx,lyingoutside the tableof values of x is known as.A) interpolation B) factorization C) normalization D) extrapolation 21. The general method for deriving the numerical differentiation formulae is to differentiate the .. A) interpolation polynomial B) extrapolating polynomial C) trinomial D) differential polynomial.

22. The trapezoidal rule is .. A)] ) .... ( 2 [21 3 1 00n nxxy y y y yhdx yn+ + + + + =} B)] ) .... ( 2 [21 2 1 00n nxxy y y y yhdx yn+ + + + + =}

C)] ) .... ( 2 [31 2 1 00n nxxy y y y yhdx yn+ + + + + =} D)] ) .... ( 2 [21 2 1 10n nxxy y y y yhdx yn+ + + + + =} 23.A..isanequationwhichinvolvesindependentanddependentvariablesand the derivatives of the dependent variables.A) Quadratic equation B) Differential equation C) Transcendental equation D) Logarithmic equation 24. The equationx ydxdyxdxy d= + +22is a . order differential equation. A) firstB) second C) third D) zero 25. MATLAB is built around the MATLAB language called .. A) B-code B) Dual-code C) M-code D) Log-code

26. The values we get when we give the command >> ari = 1:5 is . A) ari = 1 B) ari = 1 2 3 4 5 C) ari = 1 5 D) ari = zero 27. . is the study of experiments the outcomes of whose result are uncertain. A) ProbabilityB) Distribution C) Trials D) Simulation 28. A of an experiment is the set of possible result of that experiment. A) sample space B) random space C) real space D) metric space 29.Theoutcomeswhichmakenecessarythehappeningofaneventinatrialarecalled A) unfavourable event B) favourable event C) retraceable eventD) traceable event 30. E(x) + E(y) =A) E(x + y) B) E(x y) C) E(xy) D) E(y x) 31) If C is a constant then V(CX) =

A) CV(X) B) C3V(X) C) C2V(X) D) C4V(X) 32) The joint probability density function satisfies the condition .. A)0 ) , ( =}}Rdxdy y x fB) =}}Rdxdy y x f ) , ( C)1 ) , ( =}}Rdxdy y x fD)1 ) , ( =}}Rdxdy y x f 33. Poisson distribution has .. parameter/s A) oneB) twoC) three D) four 34) The gamma function for (p>0) is defined as A)dx e x px p}= I01) (B)dx e x px p} = I01) (C)dx e x px p}= I0) (D)dx e x px p} += I01) ( 35. If X has a discrete distribution with density function f, then Mx(t) =

A) e+S xx tx f e ) ( B) eS xtx f e ) ( C) eS xtxx f e ) (D) e' 'S xtxx f e ) ( 36. The arithmetic mean of 129, 117, 112, 200, 172, 138, 183 is A) 160.14 B) 170.14 C) 140.14 D) 150.14 37. State true(T) or false(F) i.The probability of any event lies between 0 and 1. ii.If an event A is contained in event B, we should have P(A) > P(B) A)(i) T (ii) T B)(i) F (ii) F C)(i) T (ii) F D)(i) F(ii) T 38. State true(T) or false(F) If S is the sample space and E is any event in a random experiment then, (i) P( S ) = 0 (ii) 0 s P( E ) s 1 A)(i) T (ii) T B)(i) F (ii) F C)(i) T (ii) F D)(i) F (ii) T 39. State true(T) or false(F)

For a continuous random variable X, a probability density function is a function such that (i)f(x) > 0 (ii)1 ) ( =} dx x fE) (i) T (ii) T F)(i) F (ii) F G)(i) T (ii) F H)(i) F(ii) T 40. State true(T) or false(F) (i) In a perfect positive correlation, the coefficient of correlation is +1. (ii) Covariance is a measure of quadratic relationship between the random variables.E)(i) T (ii) T F)(i) F (ii) F G)(i) T (ii) F H)(i) F (ii) T Part B (Two mark questions) 41. The equation of the normal curve is . A) 22230oxxe y y=B) 2220oxxe y y =C) 210o= e y yx D) 2220oxxe y y= 42. State true(T) or false(F)

(i)IfthetotalfrequencyisN,thenthecorrespondingnormalcurveisgivenby 2222ot oxeNy=. (ii)Iforiginistakenat0success,theequationofthenormalcurveisgivenby 222) (2ot om xeNy= A)(i) T (ii) T B)(i) F (ii) F C)(i) T (ii) F D)(i) F (ii) T 43. In case of normal distribution the probability function with the mean at the origin is given byA) 6 2 2) (o tme t M =B) 22 2) (m tme t Mo=C) 2 2 2) (o tme t M =D) mtme t M22 2) (o= 44. State true(T) or false(F) (i) The point estimator u is an unbiased estimator for the parameter u ifu u = )( E . (ii) The sample variance S2 is unbiased estimator of the population variance o2. A)(i) T (ii) T B)(i) F (ii) F C)(i) T (ii) F D)(i) F (ii) T 45. The arithmetic mean in rupees from table given below is

A) Rs. 165 B) Rs. 265 C) Rs. 140 D) Rs. 175 46. State true(T) or false(F) (i) Simple bar diagram is used for two dimensional comparison. (ii)Piechartconsistsofacirclewhichisdividedintoanumberofsegmentsequaltothe number of components in the corresponding table.A)(i) T (ii) T B)(i) F (ii) F C)(i) T (ii) F D)(i) F (ii) T 47.State true(T) or false(F) A Stochastic process is said to be side-sense stationary if (i)(t) = E[X(t)] is dependent on t. (ii)R(0) = E[X3(t)] < A) (i) T(ii) T B) (i) T(ii) F C) (i) F(ii) T D) (i) F(ii) F 48. State true(T) or false(F) LetP(A)=p(h),P(B)=q(h)andP(C)=e(h).N(t)formsaPoissonprocess,providedthe following conditions are met(i)Events occurring in non overlapping intervals of time are mutually independent. Monthly100150200250300500 Number of Labourers3020151041

(ii)Probabilitiesp(h),q(h)ande(h)dependonlyonthelengthhoftheintervalandnoton the origin t. E)(i) T (ii) T F)(i) F (ii) F G)(i) T (ii) F H)(i) F (ii) T 49. State true(T) or false(F) (i) The number 57.396 is accurate to six significant digits. (ii) The number 67.576 has a precision of 0.001.A) (i) T(ii) T B) (i) T(ii) F C) (i) F(ii) T D) (i) F(ii) F 50. State true(T) or false(F) The following statements describe the notion of significant digits (ii)All non-zero digits are significant. (ii)All zeros occurring between non-zero digits are insignificant digits A) (i) T (ii) T B) (i) F (ii) F C) (i) T (ii) F D) (i) F (ii) T 51. State true(T) or false(F) (i) A square matrix is said to be an upper triangular matrix if aij = 0 for i > j(ii) A square matrix is said to be lower triangular matrix if aij = 0 for i < j.A) (i) T(ii) T B) (i) T(ii) F C) (i) F(ii) T D) (i) F(ii) F

52. State true(T) or false(F) (i) ((((

9 7 57 3 65 6 2 is a symmetric matrix. (ii) ((((

0 7 57 0 65 6 0is a skew symmetric matrix. A) (i) T (ii) T B) (i) F (ii) F C) (i) T (ii) F D) (i) F (ii) T 53. State true(T) or false(F) (i) In the method of false position we choose two points xn and xn1 such that f(xn) and f(xn1) are of opposite signs. (ii)Initialapproximationstotherootdoesnothaveanyconnectionwiththephysical considerations of the problem.A) (i) T(ii) T B) (i) T(ii) F C) (i) F(ii) T D) (i) F(ii) F 54.Bythemethodofiterationtherealrootoftheequationex=10xcorrecttofourdecimal places is A) 0.0824 B) 0.0713 C) 0.0813 D) 0.0913

55. State true(T) or false(F) (i) In the backward difference table, the rth difference of y is 11 1 V V = Viririry y y, r = n, (n1),..,r (ii) The inverse operator E-1 is defined as) ( ) (1h x f x f E =. A) (i) T(ii) T B) (i) T(ii) F C) (i) F(ii) T D) (i) F(ii) F 56. In the analysis of data with equal spacing h, 1+o22 =A) 2221||.|

\| + o B) 2231||.|

\| + o C) 2231||.|

\| o D) 2251||.|

\| + o 57. State true(T) or false(F) (i) The Simpsons one-third rule is ] ) .... ( 2 ) ... ( 2 [32 6 4 2 1 5 3 1 02n n nxxy y y y y y y y y yhdx yn+ + + + + + + + + + + = }

(ii)Numericaldifferentiationtechniquesmaybeusedtoobtainthederivativeofcontinuousas well as tabulated functions. A) (i) T(ii) T

B) (i) T(ii) F C) (i) F(ii) T D) (i) F(ii) F 58. Using Simpsons three-eighth rule the value of }+213) 1 ( dx x is A) 2.75 B) 3.75 C) 4.75 D) 5.75 59. State true(T) or false(F) (i)x xydxdysin3= +|.|

\|is a first order and third degree differential equation. (ii) Trapezoidal rule is a method to solve initial value problem. A) (i) T(ii) T B) (i) T(ii) F C) (i) F(ii) T D) (i) F(ii) F 60. The method of numerical solution by using Taylor series is of the type. A) Multi-step untruncatedB) Single-stepC) UntruncatedD) Single-step untruncated Part C (Four mark questions) 61. State true(T) or false(F) The key features of MATLAB are (i) High level language for technical computing (ii) Tools for building custom graphical user interfaces

A) (i) T(ii) T B) (i) T(ii) F C) (i) F(ii) T D) (i) F(ii) F Ans: A 62. State true(T) or false(F) (i) Identity plot can be used to produce a graph from two vectors x and y. (ii) NAG specializes in the provision of software for the solution of mathematical, statistical and data mining problems.A) (i) T(ii) T B) (i) T(ii) F C) (i) F(ii) T D) (i) F(ii) F Ans: C 63.Aboxcontaining5000ICchips,ofwhich1000aremanufacturedbycompanyXandthe restbycompanyY.TenpercentofthechipsmadebycompanyXand5percentofthechips made by company Y are defective. If a randomly chosen chip is found to be defective, then the probability that it came from company X is A) 0.01 B) 0.03 C) 0.02 D) 0.04 64. The expectation of the number of failures preceding the first success in an infinite series of independent trials with constant probability p of success in a trial isA) pp+11 B) 21p p

C) 21p p D) pp 1 65.Theprobabilitythattheaceofspadeswillbedrawnfromapackofwellshuffledcardsat least once in 104 consecutive trials using Poisson distribution is . A) 0.5321 B) 0.6942 C) 0.5941 D) 0.6942 66. If the independent variants xi(i=1,2,3,4,.,n) are normally distributed with common mean m and common variance o2, then variance isA) n 22o B) n 22o

C) 22no D) n2o 67. The value of 3 for the data given below is

A) 0.015 B) 0.215C) 0.075 D) 0.025 X012345678 F19265972522971

68. State whether true(T) or false(F) (i) The values assumed by the random variable X(t) are called states, and the set of all possible values forms the state space of the process. (ii)A stochastic process [X(t)] is said to be stationary if for n > 1, its nth order joint CDF satisfies the condition F(x;t) = F(x;t) A) (i) T (ii) T B) (i) F (ii) F C) (i) T (ii) F D) (i) F (ii) T 69. The value of101 102 correct to four significant digits is A) 0.05963 B) 0.04963 C) 0.06963 D) 0.08978 70. The inverse of the matrix ((((

=0 1 41 1 24 2 5A is A) ((((

1 13 613 16 46 0 1371 B) ((((

1 13 613 16 26 4 1371 C) ((((

1 13 613 16 46 4 1371

D) ((((

1 13 613 16 46 4 1371 71. The real root of the equation x3 x 1 = 0 using Newton-Raphson method, correct to four decimal places is . A) 1.5675 B) 1.3247 C) 1.6455 D) 1.4567 72.Thevalueoff(15),giventhefollowingtableofvalues,byNewtonsforwarddifference interpolation formula is

A) 55.8792 B) 54.8976 C) 56.8672 D) 57.9876 73. The approximate value ofdx x}20costby dividing the interval into six parts is A) 1.1873 B) 2.1873 C) 6.1873 D) 4.1873 74. From the Taylors series for y(x), if y(x) satisfies y' = x y2 and y(0) = 1.Find y(0.1) correct to four decimal places. A) 0.9138 x1020304050 y = f(x)46668193101

B) 0.8138 C) 1.1873 D) 0.9100 75. State whether true(T) or false(F) (i) MATLAB cannot be used as an interactive mathematical shell. (ii) MATLAB provides a simple way to define simple arrays. (iii) In MATLAB semicolon is compulsory.(iv) A matrix can be inverted if and only if its determinant is non-zero.A) (i) T (ii) T (iii) T (iv) T B) (i) T (ii) F (iii) T (iv) F C) (i) F (ii) T (iii) F (iv) T D) (i) F (ii) F (iii) F (iv) F

Answer Keys Part - APart - BPart - C Q. No.Ans. KeyQ. No.Ans. KeyQ. No.Ans. KeyQ. No.Ans. Key 1B21A41D61A 2A22B42A62C 3C23B43C63C 4A24B44A64D 5B25C45A65C 6A26B46D66D 7D27A47D67A 8C28A48A68C 9C29B49C69B 10B30A50C70C 11B31C51B71B 12A32C52A72C 13C33A53B73A 14A34B54D74A 15C35C55A75C 16D36D56A 17B37C57C 18C38D58C 19C39A59B 20D40C60D