Geeral Aptitude (GA) Set-8 Q. Q. 5 carry oe mark each. Q. The fisherme, the flood victims owed their lives, were rewarded by the govermet. (A) whom (B) to which (C) to whom (D) that Q.2 Some studets were ot ivolved i the strike. If the above statemet is true, which of the followig coclusios is/are logically ecessary?. Some who were ivolved i the strike were studets. 2. No studet was ivolved i the strike. 3. At least oe studet was ivolved i the strike. 4. Some who were ot ivolved i the strike were studets. (A) ad 2 (B) 3 (C) 4 (D) 2 ad 3 Q.3 The radius as well as the height of a circular coe icreases by 0%. The percetage icrease i its volume is. (A) 7. (B) 2.0 (C) 33. (D) 72.8 Q.4 Five umbers 0, 7, 5, 4 ad 2 are to be arraged i a sequece from left to right followig the directios give below:. No two odd or eve umbers are ext to each other. 2. The secod umber from the left is exactly half of the left-most umber. 3. The middle umber is exactly twice the right-most umber. Which is the secod umber from the right? (A) 2 (B) 4 (C) 7 (D) 0 Q.5 Util Ira came alog, Idia had ever bee i kabaddi. (A) defeated (B) defeatig (C) defeat (D) defeatist GA /3
Geeral Aptitude (GA) Set-8 Q. 6 Q. 0 carry two marks each. Q.6 Sice the last oe year, after a 25 basis poit reductio i repo rate by the Reserve Bak of Idia, bakig istitutios have bee makig a demad to reduce iterest rates o small savig schemes. Fially, the govermet aouced yesterday a reductio i iterest rates o small savig schemes to brig them o par with fixed deposit iterest rates. Which oe of the followig statemets ca be iferred from the give passage? (A) Wheever the Reserve Bak of Idia reduces the repo rate, the iterest rates o small savig schemes are also reduced (B) Iterest rates o small savig schemes are always maitaied o par with fixed deposit iterest rates (C) The govermet sometimes takes ito cosideratio the demads of bakig istitutios before reducig the iterest rates o small savig schemes (D) A reductio i iterest rates o small savig schemes follow oly after a reductio i repo rate by the Reserve Bak of Idia Q.7 I a coutry of 400 millio populatio, 70% ow mobile phoes. Amog the mobile phoe owers, oly 294 millio access the Iteret. Amog these Iteret users, oly half buy goods from e-commerce portals. What is the percetage of these buyers i the coutry? (A) 0.50 (B) 4.70 (C) 5.00 (D) 50.00 Q.8 The omeclature of Hidustai music has chaged over the ceturies. Sice the medieval period dhrupad styles were idetified as baais. Terms like gayaki ad baaj were used to refer to vocal ad istrumetal styles, respectively. With the istitutioalizatio of music educatio the term gharaa became acceptable. Gharaa origially referred to hereditary musicias from a particular lieage, icludig disciples ad grad disciples. Which oe of the followig pairigs is NOT correct? (A) dhrupad, baai (B) gayaki, vocal (C) baaj, istitutio (D) gharaa, lieage Q.9 Two trais started at 7AM from the same poit. The first trai travelled orth at a speed of 80km/h ad the secod trai travelled south at a speed of 00 km/h. The time at which they were 540 km apart is AM. (A) 9 (B) 0 (C) (D).30 GA 2/3
Geeral Aptitude (GA) Set-8 Q.0 I read somewhere that i aciet times the prestige of a kigdom depeded upo the umber of taxes that it was able to levy o its people. It was very much like the prestige of a head-huter i his ow commuity. Based o the paragraph above, the prestige of a head-huter depeded upo (A) the prestige of the kigdom (B) the prestige of the heads (C) the umber of taxes he could levy (D) the umber of heads he could gather END OF THE QUESTION PAPER GA 3/3
Q. Q. 25 carry oe mark each. Q. is equal to lim 2 + k 2 k= (A) e 3 (B) 5 6 (C) 3 4 (D) π 4 Q.2 Let F = (x y + z)(i +j ) be a vector field o R 3. The lie itegral F dr, where C is C the triagle with vertices (0,0,0), (5,0,0) ad (5,5,0) traversed i that order is (A) -25 (B) 25 (C) 50 (D) 5 Q.3 Let {,2,3,4} represet the outcomes of a radom experimet, ad P({}) = P({2}) = P({3}) = P({4}) = /4. Suppose that A = {,2}, A 2 = {2,3}, A 3 = {3,4}, ad A 4 = {,2,3}. The which of the followig statemets is true? (A) A ad A 2 are ot idepedet. (B) A 3 ad A 4 are idepedet. (C) A ad A 4 are ot idepedet. (D) A 2 ad A 4 are idepedet. Q.4 A fair die is rolled two times idepedetly. Give that the outcome o the first roll is, the expected value of the sum of the two outcomes is (A) 4 (B) 4.5 (C) 3 (D) 5.5 Q.5 The dimesio of the vector space of 7 7 real symmetric matrices with trace zero ad the sum of the off-diagoal elemets zero is (A) 47 (B) 28 (C) 27 (D) 26 Q.6 Let A be a 6 6 complex matrix with A 3 0 ad A 4 = 0. The the umber of Jorda blocks of A is (A) or 6 (B) 2 or 3 (C) 4 (D) 5 Q.7 Let X,, X be a radom sample from uiform distributio defied over (0, θ), where θ > 0 ad 2. Let X () = mi {X,, X } ad X () = max{x,, X }. The the covariace betwee X () ad X () / X () is (A) 0 (B) ( + )θ (C) θ (D) 2 ( + )θ ST /0
Q.8 Let X,, X be a radom sample draw from a populatio with probability desity fuctio f(x; θ) = θx θ, 0 x, θ > 0. The the maximum likelihood estimator of θ is (A) i= log e X i (B) i= log e X i (C) ( i= X i ) / (D) i= X i Q.9 Let Y i = β 0 + β x i + β 2 x 2i + ε i, for i =,,0, where x i s ad x 2i s are fixed covariates, ad ε i s are ucorrelated radom variables with mea 0 ad ukow variace σ 2. Here β 0, β ad β 2 are ukow parameters. Further, defie Y i = β 0 + β x i + β 2 x 2i, where (β 0, β, β 2) is the ubiased least squares estimator of (β 0, β, β 2 ). The a ubiased estimator of σ 2 is (A) 0 i= (Y i Y i ) 2 0 0 (C) i= (Y i Y i ) 2 8 (B) 0 i= (Y i Y i ) 2 0 7 (D) i= (Y i Y i ) 2 9 Q.0 For i =,2,3, let Y i = α + βx i + ε i, where x i s are fixed covariates, ad ε i s are idepedet ad idetically distributed stadard ormal radom variables. Here, α ad β are ukow parameters. Give the followig observatios, Y i 0.5 2.5 0.5 x i -2 the best liear ubiased estimate of α + β is equal to (A).5 (B) (C).8 (D) 2. Q. Cosider a discrete time Markov chai o the state space {,2,3} with oe-step trasitio 2 3 0.7 0.3 0 probability matrix. Which of the followig statemets is true? 2 [ 0 0.6 0.4] 3 0 0 (A) States, 3 are recurret ad state 2 is trasiet. (B) State 3 is recurret ad states, 2 are trasiet. (C) States, 2, 3 are recurret. (D) States, 2 are recurret ad state 3 is trasiet. Q.2 The miimal polyomial of the matrix [ 0 2 0 0 0 0 (A) (x )(x 2) (B) (x ) 2 (x 2) (C) (x )(x 2) 2 (D) (x ) 2 (x 2) 2 2 0 0 ] is 0 0 2 ST 2/0
Q.3 Let (X, X 2, X 3 ) be a trivariate ormal radom vector with mea vector ( 3,,4) ad 4 0 0 variace-covariace matrix [ 0 3 3]. Which of the followig statemets are true? (i) 0 3 4 X 2 ad X 3 are idepedet. (ii) X + X 3 ad X 2 are idepedet. (iii) (X 2, X 3 ) ad X are idepedet. (iv) 2 2 + X 3 ) ad X are idepedet. (A) (i) ad (iii) (B) (ii) ad (iii) (C) (i) ad (iv) (D) (iii) ad (iv) Q.4 A 2 3 factorial experimet with factors A, B ad C is arraged i two blocks of four plots each as follows: (Below () deotes the treatmet i which A, B ad C are at the lower level, ac deotes the treatmet i which A ad C are at the higher level ad B is at the lower level ad so o.) Block () ab ac bc Block 2 a b c abc The treatmet cotrast that is cofouded with the blocks is (A) BC (B) AC (C) AB (D) ABC Q.5 Cosider a fixed effects two-way aalysis of variace model Y ijk = μ + α i + β j + γ ij + ε ijk, where i =,, a; j =,, b ; k =,, r, ad ε ijk s are idepedet ad idetically distributed ormal radom variables with zero mea ad costat variace. The the degrees of freedom available to estimate the error variace is zero whe (A) a = (B) b = (C) r = (D) Noe of the above. Q.6 For k =,2,,0, let the probability desity fuctio of the radom variable X k be The E( 0 k= k X k ) is equal to x f Xk (x) = { e k k, x > 0 0, otherwise. Q.7 The probability desity fuctio of the radom vector (X, Y) is give by The the value of c is equal to.. c, 0 < x < y < f X,Y (x, y) = { 0, otherwise. ST 3/0
Q.8 Let { X } be a sequece of idepedet ad idetically distributed ormal radom variables with mea 4 ad variace. The lim P ( X i= i > 4.0006) is equal to Q.9 Let (X, X 2 ) be a radom vector followig bivariate ormal distributio with mea vector (0, 0), Variace(X ) = Variace(X 2 ) = ad correlatio coefficiet ρ, where ρ <. The P(X + X 2 > 0) is equal to Q.20 Let X,, X be a radom sample from ormal distributio with mea μ ad variace. Let Φ be the cumulative distributio fuctio of the stadard ormal distributio. Give Φ(.96) = 0.975, the miimum sample size required such that the legth of the 95% cofidece iterval for μ does NOT exceed 2 is Q.2 Let X be a radom variable with probability desity fuctio f(x; θ) = θe θx, where x 0 ad θ > 0. To test H 0 : θ = agaist H : θ >, the followig test is used: Reject H 0 if ad oly if X > log e 20. The the size of the test is Q.22 Let {X } 0 be a discrete time Markov chai o the state space {,2,3} with oe-step trasitio probability matrix 2 3 0.4 0.3 0.3 2 [ 0.5 0.2 0.3] 3 0.2 0.4 0.4 ad iitial distributio P(X 0 = ) = 0.5, P(X 0 = 2) = 0.2, P(X 0 = 3) = 0.3. The P(X = 2, X 2 = 3, X 3 = ) (rouded off to three decimal places) is equal to Q.23 Let f be a cotiuous ad positive real valued fuctio o [0, ]. The π f(si x) cos x dx is equal to 0 Q.24 A radom sample of size 00 is classified ito 0 class itervals coverig all the data poits. To test whether the data comes from a ormal populatio with ukow mea ad ukow variace, the chi-squared goodess of fit test is used. The degrees of freedom of the test statistic is equal to ST 4/0
Q.25 For i =,2,3,4, let Y i = α + βx i + ε i where x i s are fixed covariates ad ε i s are ucorrelated radom variables with mea 0 ad variace 3. Here, α ad β are ukow parameters. Give the followig observatios, Y i 2 2.5-0.5 x i 3 2-4 - the variace of the least squares estimator of β is equal to ST 5/0
Q. 26 Q. 55 carry two marks each. Q.26 Let a = ( )+, 0, ad b! = k=0 a k, 0. The, for x <, the series =0 b x coverges to (A) e x +x (B) e x x 2 e x (C) x (D) ( + x)e x Q.27 Let {X k } k be a sequece of idepedet ad idetically distributed Beroulli radom variables with success probability p (0,). The, as, coverges almost surely to (A) p (B) p (X k) k k= (C) p p (D) Q.28 Let X ad Y be two idepedet radom variables with χ m 2 ad χ 2 distributios, respectively, where m ad are positive itegers. The which of the followig statemets is true? (A) For m <, P(X > a) P(Y > a) for all a R. (B) For m >, P(X > a) P(Y > a) for all a R. (C) For m <, P(X > a) = P(Y > a) for all a R. (D) Noe of the above. Q.29 x z The matrix [ 0 2 y] is diagoalizable whe 0 0 (x, y, z) equals (A) (0,0,) (B) (,,0) (C) ( 2, 2, 2) (D) ( 2, 2, 2) Q.30 Suppose that P ad P 2 are two populatios with equal prior probabilities havig bivariate ormal distributios with mea vectors (2, 3) ad (, ), respectively. The variacecovariace matrix of both the distributios is the idetity matrix. Let z = (2.5, 2) ad z 2 = (2,.5) be two ew observatios. Accordig to Fisher s liear discrimiat rule, (A) z is assiged to P, ad z 2 is assiged to P 2. (B) z is assiged to P 2, ad z 2 is assiged to P. (C) z is assiged to P, ad z 2 is assiged to P. (D) z is assiged to P 2, ad z 2 is assiged to P 2. Q.3 Let X,, X be a radom sample from a populatio havig probability desity fuctio f X (x; θ) = 2 x, 0 < x < θ. The the method of momets estimator of θ is θ2 (A) 3 i= X i 2 (B) 3 i= X i 2 2 (C) i= X i (D) 3 i= X i (X i ) 2 ST 6/0
Q.32 Let X be a ormal radom variable havig mea θ ad variace, where θ 0. The X is (A) sufficiet but ot complete. (B) the maximum likelihood estimator of θ. (C) the uiformly miimum variace ubiased estimator of θ. (D) complete ad acillary. Q.33 Let {X } be a sequece of idepedet ad idetically distributed radom variables with mea θ ad variace θ, where θ > 0. The i= X i 2 is a cosistet estimator of (A) +θ (B) +θ θ (C) θ i= X i (D) θ +θ Q.34 Let X,, X 0 be a radom sample from a populatio with probability desity fuctio f(x; θ) = e x θ, < x <, < θ <. 2 The the maximum likelihood estimator of θ (A) does ot exist. (B) is ot uique. (C) is the sample mea. (D) is the smallest observatio. Q.35 Cosider the model Y i = β + ε i, where ε i s are idepedet ormal radom variables with zero mea ad kow variace σ i 2 > 0, for i =,,. The the best liear ubiased estimator of the ukow parameter β is (A) (Y 2 i= i/σ i ) i=(/σ 2 i ) (B) i= Y i (C) i= (Y i/σ i ) (D) i= (Y i/σ i ) i=(/σ i ) Q.36 Let (X, Y) be a bivariate radom vector with probability desity fuctio f (X,Y) (x, y) = { e y, 0 < x < y, 0, otherwise. The the regressio of Y o X is give by (A) X + (B) X 2 (C) Y 2 (D) Y + ST 7/0
Q.37 Cosider a discrete time Markov chai o the state space {, 2} with oe-step trasitio probability matrix 2 P = 0.2 0.8 [ 2 0.3 0.7 ]. The lim P is (A) [ 3 3 8 8 ] (B) [ 0 ] (C) [0 0 0 ] (D) [ 8 8 3 3 ] Q.38 Let (X, X 2 ) be a radom vector with variace-covariace matrix [ 4 0 ]. The two 0 2 pricipal compoets are (A) X ad X 2 (B) X ad X 2 (C) X ad X 2 (D) X + X 2 ad X 2 Q.39 Cosider the objects {,2,3,4} with the distace matrix 2 3 4 0 5 2 [ 0 2 3 ]. 3 2 0 4 4 5 3 4 0 Applyig the sigle-likage hierarchical procedure twice, the two clusters that result are (A) {2,3} ad {,4} (B) {,2,3} ad {4} (C) {,3,4} ad {2} (D) {2,3,4} ad {} Q.40 The maximum likelihood estimates of the mea vector ad the variace-covariace matrix of a bivariate ormal distributio based o the realizatio {( 2 ), (4 4 ), (4 ) } of a radom 3 sample of size 3, are give by (A) ( 3 ) ad [2 3 2/3 ] (B) (3 ) ad [2 3 3/2 ] (C) ( 3 3 3/2 ) ad [ 3 3/2 2/3 ] (D) (3 3 2/3 ) ad [ 3 2/3 ] Q.4 Cosider a fixed effects oe-way aalysis of variace model Y ij = μ + τ i + ε ij, for i =,, a, j =,, r, ad ε ij s are idepedet ad idetically distributed ormal radom variables with mea zero ad variace σ 2. Here, r ad a are positive itegers. Let Y i = r j= Y ij. The Y i is the least squares estimator for (A) μ + τ i 2 r (B) τ i (C) μ + τ i (D) μ ST 8/0
Q.42 Let A be a positive semi-defiite matrix with eigevalues λ λ, ad with α as the maximum diagoal etry. We ca fid a vector x such that x t x =, where t deotes the traspose, ad (A) x t A x > λ (B) x t A x < λ (C) λ x t A x λ (D) x t A x > α Q.43 Let X be a radom variable with uiform distributio o the iterval (,) ad Y = (X + ) 2. The the probability desity fuctio f(y) of Y, over the iterval (0,4), is (A) 3 y 6 (B) 4 y (C) 6 y (D) y Q.44 Let S be the solid whose base is the regio i the xy-plae bouded by the curves y = x 2 ad y = 8 x 2, ad whose cross-sectios perpedicular to the x-axis are squares. The the volume of S (rouded off to two decimal places) is Q.45 Cosider the triomial distributio with the probability mass fuctio 7! P(X = x, Y = y) = x!y!(7 x y)! (0.6)x (0.2) y (0.2) 7 x y, x 0, y 0, ad x + y 7. The E(Y X = 3) is equal to Q.46 Let Y i = α + βx i + ε i, where i =, 2, 3, 4, x i s are fixed covariates ad ε i s are idepedet ad idetically distributed stadard ormal radom variables. Here, α ad β are ukow parameters. Let Φ be the cumulative distributio fuctio of the stadard ormal distributio ad Φ(.96) = 0.975. Give the followig observatios, Y i 3-2.5 5-5 x i -2 3-2 the legth (rouded off to two decimal places) of the shortest 95% cofidece iterval for β based o its least squares estimator is equal to Q.47 Cosider a discrete time Markov chai o the state space {,2,3} with oe-step trasitio 2 3 0 0.2 0.8 probability matrix. The the period of the Markov chai is 2 [ 0.5 0 0.5] 3 0.6 0.4 0 Q.48 Suppose customers arrive at a ATM facility accordig to a Poisso process with rate 5 customers per hour. The probability (rouded off to two decimal places) that o customer arrives at the ATM facility from :00 pm to :8 pm is ST 9/0
Q.49 Let X be a radom variable with characteristic fuctio φ X ( ) such that φ X (2 π) =. Let Z deote the set of itegers. The P(X Z) is equal to Q.50 Let X be a radom sample of size from uiform distributio over (θ, θ 2 ), where θ >. To test H 0 : θ = 2 agaist H : θ = 3, reject H 0 if ad oly if X > 3.5. Let α ad β be the size ad the power, respectively, of this test. The α + β (rouded off to two decimal places) is equal to Q.5 Let Y i = β 0 + β x i + ε i, i =,,, where x i s are fixed covariates, ad ε i s are ucorrelated radom variables with mea zero ad costat variace. Suppose that β 0 ad β are the least squares estimators of the ukow parameters β 0 ad β, respectively. If x i = 0, the the correlatio betwee β 0 ad β is equal to i= Q.52 Let f: R R be defied by f(x) = (3x 2 f(h)+f( h) 8 + 4) cos x. The lim h 0 h 2 is equal to Q.53 The maximum value of (x ) 2 + (y 2) 2 subject to the costrait x 2 + y 2 45 is equal to Q.54 Let X,, X 0 be idepedet ad idetically distributed ormal radom variables with mea 0 ad variace 2. The E ( X 2 X 2 + +X 0 2 ) is equal to Q.55 Let I be the 4 4 idetity matrix ad v = (, 2, 3, 4) t, where t deotes the traspose. The the determiat of I + vv t is equal to END OF THE QUESTION PAPER ST 0/0