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IIT JAM 2013 : Computer Applications

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Z013-CA QUESTION BOOKLET CODE Test Paper Code: CA Time: 3 Hours o Maximum Marks: 100 INSTRUCTIONS A. Geueral: 1. 2. 3. 4. 5. 6. 7. 8. This Booklet is your Question Paper. It contains 20 pages and has 100 questions. The Question Booklet Code is printed on the right-hand top comer of this page. The Question Booklet contains blank spaces for your rough work. No additional sheets will be provided for rough work. Clip board, log tables, slide rule, cellular phoue aud electrouic gadgets in any form are NOT allowed. Non Programmable Calculator is allowed. Write your Name and Registration Number in the space provided at the bottom. All answers are to be marked only on the machine gradable Objective Response Sheet (ORS) provided along with this booldet, as per the instructions therein. The Question Booldet along with the Objective Response Sheet (ORS) must be handed over to the Invigilator before leaving the examination hall. Refer to Special InstructionlUseful Data on reverse of this sheet. B. Filling-in the ORS : 9. 10. 11. Write your Registration Number in the boxes provided on the ORS and darken the appropriate bubble under each digit of your Registration Number using a black ink ball point pen. Ensure that the code on the Question Booklet and the code on the ORS are the same. If the codes do not match, report to the Invigilator immediately. On the ORS, write your Name, Registration Nwnber, Name of the Test Centre and put your signature in the appropriate box with ball-point pen. Do not write these anywhere else. C. Marking of Answers on the ORS : 12. 13. 14. Each question has 4 choices for its answer: (A), (B), (C) and (D). Only ONE of them is the correct answer. On the right-hand-side of ORS, for each question number, darken with a black ink ball point pen ONLY one bubble corresponding to what you consider to be the most appropriate answer, from among the four choices. There will be negative maI:king for wrong answers. MARKING SCHEME: (a) For each correct answer, you will be awarded 1 (One) mark. (b) For each wrong answer, you will be awarded-V3 (Negative V3) mark. (c) Multiple answers to a question will be treated as a wrong answer. (d) For each un-attempted question, you will be awarded 0 (Zero) mark. i I Name Registration Number ~ Special Instruetionsl Useful Data R denotes the set of real numbers In x denotes log, x f'denotes the first derivative of the function I I" denotes the second derivative of the function I !, = : denotes the partial derivative of the function I with respect to x VI denotes the gradient of the function I P(X = 11) denotes the probability of X = 11 x' denotes the complement of the Boolean variable x LPP denotes Linear Programming Problem max I denotes the maximum of function min I denotes the minimum of fimction .f f x T denotes the transpose of vector x For all C programs and segments assume that all the standard library functions are accessible ~ Q.I If the function f:R-7[0,2] is defined by f(x)=lcosxj+jsinxl,then (A) (C) Q.2 Q.5 f is differentiable on R (D) the minimum value of f is I is onto all three negative real roots one negative and two positive real roots one positive and two negative real roots one real and two complex roots (B) (D) exactly one zero exactly three zeros exactly two zeros no zero If (2.001)x (3.999)' is approximated using first order Taylor polynomial of two variables, then the approximate value is (A) 31.999 (B) 32.001 (C) 32.000 (D) 31.891 af af X4 + 4 If f(x,Y) = x 3 tan- l ( Y + Y, then the value of x-+ Y - is x) x-y ax i!Y (A) Q.6 (B) The function f(x) = xsinx+ cosx-#- has (A) (C) QA is one-one If the functionf(x) = x 3 + ax' + 6x -I has a critical point at x = -2, then f(x) = 0 has (A) (B) (C) (D) Q.3 f f f If f(x,y) (B) i x3 _ (C) 3f = I, f y (0,0) = I f x (0,0) = 1, 1" (0,0) = 0 (D) 4f y3 = I x I + I y I if o 2f (x,y) '" (0,0) if (x,y) = (0,0) then (A) (B) (C) Q.7 !, (0,0) = 0, 1" (0,0) = 0 f x (0,0) = I, f y (0,0) = -I (D) fx (0,0) If f and g are two differentiable functions such that 1'(x) = g(x) and g'(x) = - f(x) with f(O) = 0 and g(O) = I, then f2 (x) + g2 (x) is (A) Q.8 4 (B) I (C) f(x) + g(x) (D) f(x) - g(x) (D) 4 The area of the region bounded by the lines I x I + I y I = I is (A) 1 (B) 2 (C) CA-1I20 3 ~ Q.9 If a function f then I(x) is defined for all positive real numbers is such that r(x') (A) (B) 2 x Yo (C) Q.IO 3 71 4 -x l '+- 4 71 3 -xl' +- (D) with f(l) = I, I Yo 5 -x-+- 7 7 7 7 J The general solution of the differential equation d ;' - dx (A) y = c1e ' + c, cos x + c J sin x (C) Q.1I = x' y = c1e +c,xe +cJe x x 7 6 +2 7 6 d' d --? + 2:. - Y = 0 is d;c dx x (B) (D) -x y=c1e- +c,sinx+cJcosx y=c1e- ' +c,sin(x+c J ) Consider the table x -I o I f(x) -2 -I o Using Lagrange interpolation the value of f(0.6) is (A) Q.l:l 0.4 -(.JiS +3) (B) (C) 1.0 2 3 (C) 7 Consider the iterative scheme (A) Q.14 (B) - 0.4 (D) 0.32 If the non-zero root of the equation sin x - x' = 0 is approximated using the first two terms in the Taylor's series expansion of sin x , then the approximate root is (A) Q.13 - 0.32 X"+l =--3 -x n with 6 (B) Xo 62 63 Ji5-3 (D) = 0, then the value of X 4 (C) 7 0.8 (D) is 30 31 Consider the table I : I ~ I : I : I : I Then the value of /';J y , where /'; is the forward difference operator, is (A) -3 (B) 3 (C) CA-2/20 2 (D) -2 ~ 100 Q.15 Let [x] denote the largest integer less than or equal to x. The value of the integral f[x]dx, o computed by the composite trapezoidal rule with step size I, is (A) 4950 (B) 5000 (C) 9900 The integral 5050 1 1 Q.16 (D) f j(x) dx f f(x)dx '" f(a) + fe-a), a is approximated by the formula E [0,1]. If -1 -1 the approximation is exact for all polynomials of degree at most 3, then the value of a is I I I I (C) (B) (A) (0) .fi 2 3 ..J3 Q.l7 The surface area ofthe surface x 2 + y2 = 2 - z, for z ~ 0, is (A) Q.18 (B) 13ff - (C) 2 I 13ff (C) - - (0) 2ff (D) 2 (0) 3 4 II The area bounded by y = xl x and y2 = x is (A) Q.19 104ff 3 -- 2 (B) - 3 3 2 1 3 The area ofa loop of the curve r = 4cos 2 BsinB is (A) ff 3 (B) ff (C) - 8 ff 2 ff 15 Q.20 The integral f f j(x,y)dxdy is equivalent to )'=0 x=)'2 1.Jx (A) 1 (B) f f j(x,y)dydx .r-=O )'=;r2 x=o y=O 4 .Jx (C) (0) f f !(x,y)dydx ..=0 y=x~ Q.21 1 f f j(x,y)dydx 4.fi f f j(x,y)dydx x=Oy=O The volume of the region bounded by the surfaces z = 4 _~X2 + y2 andz=~x2+y2 (A) 2ff 3 (B) (C) 8ff 3 4ff 3 CA-3/20 (0) 16ff 3 IS ~ Q.22 (A) Q.23 0 dx- (B) 1 x x 3 x (B) cte.! 3 x (D) 1 'x c1e x + c.,xe x +-x-e 2 cte +c 2 xe +x e (C) - I The integral f (D) 7 dx c1e + c,xe +-x e 1 2 o. Then yeO) 2:; _2 dy + Y = eX is dx x lim y(x) = x --7 00 dx 2 The general solution of d X 4 dy + 4y = ge-x satisfying (C) (A) Q.24 d2~ - Let y(x) be a solution of +c 2 xe x +x 2 e x I-x f cos(x+ y)eY-Xdydx is equivalent to x=-l y=l+x I (A) "2 1 1 1 f fe'cosudvdu (B) "2 u=-1 v=-u 1 (C) "2 1 1 fe'cosudvdu u=-Iv=-I I 1 (D) f fe' cosudvdu "2 1 1 f fe'cosudvdu u=-l u=-I v=u Q.25 1 f u 2 ~- Consider the following C function int fun(int n){ int b=0; while(n! =0){ b = b*10+n%10; n=n/l0;} return b;} The value of fun(7830) is (A) 7830 (B) Q.26 783 (C) 387 (D) 1000 (D) 42750 What does the following C program segment display when executed? int i; long int k=0; for(i=1;i<=S0;i++) printf( "%d", k); (A) 42075 k+=(i*i); (B) (C) 42925 CA-4/20 42950 is ~ Q.27 What does the following C program segment display when executed? int arr[] = {1,S,7,8,9}; int k=0,i; for(i=0;i<S;++i) k=k+*(arr+i); printf( "%d", k); (A) Q.28 29 (B) (C) 21 9 (D) 30 Consider the following C program. What does it display when executed? #include<stdio.h> static int b=0; int fun2(int); int main(){ int a; a=fun2(6); printf("(%d,%d)",a,b); return 0;} int fun2(int n){ if (n==l I I n==2) return 1; else{ b+=2; return fun2(n-1) +fun2(n-2);}} (A) Q.29 (8,14) (B) (8,12) (C) (12,8) (D) (14,16) (C) 3333 (D) 3222 The output of the following C code segment is int *p,*q, a=3, b=2; p=&a; q=&b; p=q; b=*q; printf("%d %d %d %d",a,b,*p,*q); (A) Q.30 3232 (B) 2222 The value of the variable k after the execution of the following C code segment is int i=0,j=0,k=0; int m=10,n=20; for(i=0;i<mji++){ j=i; while (j <=n){ ++k; j++; }} (A) 210 (B) 200 (C) CA-S/20 165 (D) 220 ~ In general, for a computer which of the following represents the memories in increasing order of their capacities Q.43 (A) (B) (C) (D) Q.44 Register < RAM < Cache < Hard Disk RAM < Cache < Hard Disk < Register Register < Cache < RAM < Hard Disk Cache < RAM < Hard Disk < Register The circuit given below Input1 output >0;--- Input2 L..._ is equivalent to 2-input (A) (C) (B) (D) NAND gate NOR gate AND gate OR gate The output of the circuit given below is Q.45 x y output x y (A) 0 (B) y (C) CA-8/20 x y (D) x ~ QA6 The truth table of a Boolean functionj(x,y,z) is given below y 0 0 I I 0 0 I Thenfix,y,z) is equal to (A) x y (B) QA7 z 0 I 0 I 0 I 0 f(x,y,z) 0 0 I I 0 0 I 1 x 0 0 0 0 I I I I 1 1 (C) x Let the Boolean function f(il' i 2 , , i lO ) = i, .i 2 y i lO be realized using two input AND gates only. Then the minimum number of two input AND gates required is (A) 5 (B) 10 (C) 8 QA8 x+ y (D) (D) 9 The output of the following four-bit even parity generator is x X' four-bit even parity generator y .. y' (A) QA9 (B) (C) 1 0 (D) x' +y' Suppose two fair dice are rolled. The probability of obtaining a sum of 6 or 7 is (A) Q.50 x+y output ~ 36 (B) ~ (C) 36 ~ 6 (D) be a random variable following Poisson distribution. If P(X P(X 2: 2) is equal to Let X (A) - OAln 004 (B) 0.6 - 0.4ln 004 0.6 (D) 0.6 + OAln 004 (C) CA-9/20 E. 36 = 0) = 004, then ~ Q.51 It is claimed that the circuit in the dotted box given below converts an S-R flip-flop to another type of flip-flop. Input I S + : Q Q - b Q I R . Q Clock Which of the following is true ? (A) S-R flip-flop is converted to T flip-flop (B) S-R flip-flop is converted to D flip-flop (C) Claim is false because S=I and R=I is not allowed in S-R flip-flop (D) Claim is false because s=o and R=O is not allowed in S-R flip-flop Q.52 Suppose two fair dice are rolled. The probability that one face is 4 given that the faces show different numbers is 5 I I I (A) (B) (C) (D) 6 3 6 2 Q.53 A box contains ten screws out of which four are defective. Six screws are drawn one by one at random, without replacement. The probability that the sixth screw drawn is the last defective one, IS 32 5 I 2 (A) (B) (C) (D) 729 21 21 729 Q.54 Suppose (A) Q.55 2 a =2f +} + 4k, b= -4f + 3k , ~ = 3f - 2} . The value of (B) -2 (C) a b+b,c+c a -4 The volume of the parallelepiped obtained by three edge vectors }-k 8 4 i + 2} + k, - i + }, is (A) (D) (B) 2 (C) CA-IO/20 1 IS (D) 4 and ~ Q.56 The equation of the plane passing through P(O,O,I), Q(l,I,O) and R(O,2,O) is (A) (B) x + y + 2z = 1 (C) Q.57 2x + y + z = I x+2y+z=1 (D) x+y+2z=2 If the vectors 2t + a} + 4fc, S} +!3 k, and - 20t - 4} + Sk are mutually orthogonal, then (a,!3) is (A) Q.5S 2..[5 4.J5 (B) 2 (B) (-2,-4) (D) (-2,4) (C) ..[5 (D) s..[5 4 (C) S (D) 16 242 bytes (B) 2 25 bytes (C) 240 bytes (D) 243 bytes The Boolean expression (x + y + z) (x'+y + z) (x + y'+z) (x + Y + z') is equivalent to (A) (x+ y) (x+z) (y+z) (B) (x'+y')' (x'+z') (y'+z') (C) (x+ y') (x+z') (y+z') (D) Q.62 (C) Four terabytes is equal to (A) Q.61 (2,4) The number of Boolean functions j(x,y) satisfying j(x,y) = j(x',y') is (A) Q.60 (B) The area of the triangle with vertices P(O,O,I), Q(I,2,3) and R(O,4,l) is (A) Q.59 (2,-4) (x'+y) (x'+z) (y'+z) Let S = {1,2,3,4,5,6,7,8,9,IO}. The number of subsets of S each having exactly one odd integer IS (A) 6 2' (B) 2 10 (C) CA-ll/20 2' (D) 5.2 5 ~ Q.63 In a class, each student takes at least one and at most two electives out of three electives namely, DMS, DS and ADA. The table below gives enrollment infonnation of the students in the above courses: Course DMS DS ADA DMS andDS DS and ADA ADAandDMS Number of students 90 70 90 30 30 40 Then the total number of students in the class is (A) 160 (B) 150 Q.64 (D) 250 an abelian group under matrix multiplication a non-abelian group under matrix multipiication an abelian group under matrix addition a non-abelian group under matrix addition Let G be the group of all 2 x 2 real matrices (: multiplication. Let HI = { (A) (B) (C) (D) Q.66 90 b Thesets={(: -a ) a'bER} is (A) (B) (C) (D) Q.65 (C) G~) b E R} and H 2 = { ~) with ad - be G ~) * 0, under matrix e E R}. Then HI is a subgroup of G but Hz is not a subgroup of G HI is not a subgroup of G but Hz is a subgroup of G neither HI nor Hz is a subgroup of G both HI and Hz are subgroups of G TheLPP min 4x+5y subject to x +Y (A) (C) ~ 10, ~ 2x + 5Y :s; 30, x 0, y~O has no optimal solution has exactly one optimal solution CA-12/20 (B) (D) has more than one optimal solution has no feasible solution ~ Q.67 Let a and fJ be two positive real numbers. If the number of optimal solutions of the LPP max ax+fJy subject to 2x + 3y ;:0: x + Y :<:; 10, 5, x ;:0: 0, y;:O: 0 is infinite, then which of the following is possible? (A) ~) a=3,fJ=2 (C) Q.68 a=2,fJ=3 a=2,fJ=2 ~) a=l,fJ=4 (C) 1 The number offeasible solutions of the LPP min x+ y subject to 2x + 3y ;:0: x + Y :<:; 1, 3, x ;:0: 0, y;:O: 0 IS (A) Q.69 ~) infinite 0 (D) 4 (D) 6 The number of extreme points of the set offeasible solutions of the LPP min x+5y subject to x +Y ;:0: 3x + 5 y :<:; 30, x:<:; 8, y;:o: 0 1, IS (A) Q.70 (B) 3 (C) 4 5 Which of the following set of vectors in R 3 forms a linearly independent set? (A) (B) {(1,2,3), (1,2,0), (0,0,2)} (C) Q.71 {(1,1,0), (0,1,1), (1,0,1)} {(1,1,1), (1,1,0), (0,0,1)} (D) {(O,O,O), (0,0,1), (0,1,0)} Which of the following is a subspace ofR3? = O} (A) {(xl'x"x,) E R'I5x l -3x, +2x, (B) {(xl'x"x,) E R' 15xl -3x, + 2x, = I} (C) {(xl'x"x,) E R' ~) {(xl'x"x,) E R' I x, I Xl +x, = 1,x, = O} =1} CA-13120 ~ Q.85 If A = (aij) is a 4 x 4 matrix such that aij = 2;+j, I ~ i, j ~ 4, then determinant of A is (A) 2' (B) (C) 2' 2 (D) 0 Q.86 If A is a square matrix such that x T Ax> 0 for all nonzero 11 x I column vectors x, then the system Ax = 0 has (A) exactly one solution (B) infinitely many solution (C) a nonzero solution (D) no solution Q.87 The first Indian to win the Academy Award (Oscar) is (A) A.R. Rahman (B) Satyajit Ray (C) Aamir Khan (D) Bhanu Athaiya Q.88 The Bharat Ratna in the year 2008 was awarded to (A) Lata Mangeshkar (B) (D) (C) Bismillah Khan Bhimsen Joshi Amartya Sen Q.89 The Man of the Series in ICC Cricket World Cup 2011 was (A) M.S. Dhoni (B) Yuvraj Singh (C) Sachin Tendulkar (D) Gautam Gambhir Q.90 The painting "Mona Lisa" is housed in (A) da Vinci Art Gallery (C) Albert Museum (B) (D) Louvre Museum Metropolitan Museum of Art Q.91 Who among the following was one of the Nine Gems (Navaratnas) in the court of Chandragupta II ? (A) Tansen (B) Kalidasa (C) Valmiki (D) Aryabhatta Q.92 Which one of the following is the oldest Indian Institute of Technology ? (A) Indian Institute of Technology Delhi (B) Indian Institute of Technology Kanpur (C) Indian Institute of Technology Bombay (D) Indian Institute of Technology Kharagpur Q.93 The Government of India has declared the year 2012 as the national year of (A) Chemistry (B) Physics (C) Mathematics (D) CA-16/20 Medicine ~ Q.94 The next element in the sequence 1, - 3,7, -IS,... is (A) Q.95 -27 (B) 27 (C) 31 (D) - 31 Consider the following two lists List I List II 1: Apple Inc. 2: Microsoft Corp. 3: Infosys Ltd. 4: Airtel The correct match is (A) 1-7 Q,2 -7 R,3 -7 P,4 -7 S (C) P: 0: R: S: N.R. Narayana Murthy Steve Jobs Bill Gates Sunil Bharti Mittal (B) CD) 1-7 S,2 -7 P,3 -7 R,4 -7 Q 1-7 S,2 -7 R,3 -7 P,4 ~ Q 1-7 Q,2 -7 P,3 -7 R,4 -7 S Q.96 Suppose a country has coins of denominations 1, 4, and 5. The minimum number of coins required to make the amount 13 is (A) 2 (B) 3 (C) 4 (D) 5 Q.97 Let F be a field ofprirne order p. Let G = {q(x) I q(x) is a polynomial of degree at most Il over F}. Then G is a field of order (A) nP (B) p" (C) p"+! (D) (Il + 1)' 5 0 5 2 Q.98 The number of linearly independent eigenvectors of A = I 0 4 2 0 0 ,0 (A) Q.99 4 (B) 3 (C) The minimum value taken by the function I(x) (A) 0 (B) -1 0 2 1 3 2 0 I IS 1, (D) 1 (D) 2 =_1x_I__ 1 is 1+lxl (C) 1 Q.100 The values of k for which the following system oflinear equations has non-zero solutions x + y + z = 0, 2x + fry + 3z = 0, 3x + 5y + kz = 0 are (B) 2 and 4 (A) 1 and 4 (C) 3 and 5 (D) 2 and 5 CA-17/20 OU8T-V;) JI.lOA\ q~ no.l .I0J aallds Jj.lOA\ OZl6l-V:J qJl no.l .I0J aauds OZlOZ-V:J Y(.I0M q~ no.l .IoJ ;laRds

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