MATHS TEST

IIT-JAM IIT-JAM 2016

MOCK TEST-I TEST-I MATHEMATICS (MA)

Time: 3 hour

Paper code: MA Maximum Marks: 100

General Instructions : Instructions : (i) All questions All questions are compulsory.

a r d n e n i a J

(ii) The question question paper paper consists consists of 60 of 60 questions divided into three sections A, B and C. Section A comprises of 30 30 single correct Mutiple correct Mutiple Choice Questions (MCQ ( MCQ)) carry one carry one or or two two marks each, each, Section Section B comprises comprises of of 10 multi correct Mutiple Mutiple Select Select Questi Questions ons (MSQ) MSQ) carry two marks each marks each and Section C Section C comprises of 20 2 0 Numerical Answer Type Questions (NA ( NAT T) carry one carry one or two marks each. marks each. (iii) Section-A (MCQ) Section-A (MCQ) contains a total of of 30 Multiple Choice Questions (MCQ) carrying one or two marks eac marks each. h. Each MCQ type question question has four choices choices out of which only one choice is the correct answer. answer .

(iv) Section-B (MSQ) Section-B (MSQ) contains a total of 10 10 Multiple Select Questions(MSQ) carrying Questions(MSQ) carrying two two marks eac marks each. h. Each Each MSQ type question question is simila similarr to MCQ but with a differe difference nce that there may ma y be one or more more than than one choice choice(s) (s) that are correc correctt out of the four given given choic choices. es. The candidate gets full gets full credit if he/she selects all the correct answers only and no wrong answers. answers. (v) Section-C (NAT) contains a total of of 20 Numerica Numericall Answe Answerr Type Type (NAT) (NAT) questions carrying one or two marks each. each. For these NAT type questions, questions, the answer is a signed real number. number. No choices will be shown for these type of questions.

h s e n i D

(vi) In all sections, sections, questions questions not attempted will result in zero in zero mark. mark.

(vii) (vii) In Section In Section A (MCQ), wrong answer will result in negative marks. For all 1 mark questions, 1/3 marks marks will be deducted for each wrong answer.For all 2 marks questions, 2/3 marks will marks will be deducted for each wrong answer.

(viii) (viii) In Section In Section B (MSQ),there (MSQ) ,there is no is no negative and negative and no partial marking partial marking provisions. (ix) In Section In Section C (NAT) there (NAT) there is no is no negative marking. negative marking. (x) Non-programmable calculators are calculators are permitted. Sharing of calculators is N is NOT OT allowed. allowed. Special Instructions / Usefull Data

N R

f  , f 

∂g ∂g ∂g , , ∂x ∂y ∂z f x , f y , f z log i, j, j, k [T : B, B]

: The set of all positive positive Integers Integers : The set of all Real Real numbers numbers : First and Second Second derivatives respectively respectively of a real real function f : Partial Partial derivatives derivatives of g with resp respect to x, y and z resp respec ectively tively

: Partial Partial derivatives derivatives of f with resp respect ect to x, y and z respec espectively tively : The logarit logarithm hm to the base base e : Standard Standard unit orthogon orthogonal al vectors vectors : The matrix of linear linear transfor transformatio mation n T w.r.t. w.r.t. the basis basis B

Prepared by: Jainendra Singh & Dinesh Khatri

Mathematics

I IT JAM 2016

Page 2 of 13

SECTION A (MCQ) Question Question numbers numbers 1 to 30 are are objective objective type single corre correct questions questions carry carry 1 or 2 mark each. each. For 1 all 1 mark questions, /3 marks will be deduct deducted ed for each wrong wrong answer. For all 2 marks questions, 2/ 3 will be deducted for each wrong answer.

a r d n e n i a J

1. Given Given that y that y = x = x is solution of the differential equation the second linearly independent solution is (a). x2 . x2

+ 1.

(c). x2

− 1.

(b).

(d). x3 .

( x2 (x

−

d2 y 1) 2 dx

dy − 2x dx + 2y = 0. Then,

[2]

d2 x dy 2. Let W ( W (x) denote the wronskian of the differential equation + x + x2 y = 0 and given 2 dx dx that W that W (0) (0) = 1, then the wronskian is given by (a). e−x 2 (b). e−x .

(c). e− (d). e

x

2

x

/2 .

/2 .

2

h s e n i D

[1]

3. The solut solution ion of y y  = y = y 2 , y (0) = 1 exists exists for all (a). x

∈ (−∞, 1) (b). x ∈ [0, [0, a], a > 1. (c). x ∈ (−∞, ∞). (d). x ∈ [1, [1, a], a > 1.

dy 4. Let S denote denote the set of all constant solutions of the differential equation = dx then the set S set S is

[2]

 y(y − 1)(y 1)(y − 2)(y 2)(y − 3),

(a). Empty Empty

(b). Uncountable Uncountable & infinite (c). Countable Countable & infinite

(d). (d). finite finite [1]


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5. Consider Consider the differential differential equation d2 y 1 dy + + (1 2 dx x dx

− x12 )y = 0, x ∈ (0, (0, 10]

and the wronskian at x at x = 2 is 1. Then the minimum value of the wronskian is

a r  d n e n i   a J h s  e n i   D    (a). 5

(b). 10 (c). 15

(d). 20

[1]

6. Suppose Suppose that f ( f (x) is differentiable for all x [0, [0, 1] and that f (0) f (0) = 0. Define Define the sequen sequence ce 1 < an > by the rule a n = n = n f . Then Then lim lim an equals n→∞ n

∈

(a).

∞

(b). 0

(c). f  (0)

(d).

−f (0)

√ n1+ 1 + √ n1+ 2 + · · · + √ 12n

7. lim

n→∞

[1]

equals

(a). 0

(b). 1 (c).

(d).

∞ 1 2

[1]

nxn 8. The set set of all value valuess of x for which, the series converges absolutely, is 1)(2x + 1)n n=1 (n + 1)(2x

∞

(a). (

(b).

R

(c). (

(d).

−∞, −1) ∪ (1, (1, ∞)

−∞, −1) ∪ −31 , ∞ −∞, − 13 ∪

1 , 3

∞

[2]


Mathematics

I IT JAM 2016

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 xy). Then 9. Let f Let f ((x, y ) = (xy).

(a). f x and f and f y do not exist at (0, (0 , 0)

(b). f x (0, (0, 0) = 1 (c). f y (0, (0, 0) = 0 (d). f is (0, 0). f is differentiable at (0,

x

10. Let f Let f ((x) =



g(t) dt. dt . Then f Then f  ( 1) equals

−

1/x

(a). 2g 2g( 1)

−

(b).

−2g(−1) 1 (c). g(−1) 2 1 (d). − g (−1) 2 11. The infim infimum um of the set A =



a r d n e  n   i a J

2( 1)n+1 + ( 1)

−

(a).

−5 11 (b). − 2

−

h s e n i D  (c). 5 11 (d). 2

(n+1) 2

n

2+

3 : n : n n

∈N

[2]

[2]

is

[2]

12. Which Which of the following following function is uniformly continuou continuouss on (0, (0, 1) ? (a). (a). sin 1

1 x

(b). e /

x

1 x 1 (d). x sin x (c). ex cos

13. The radius of convergenc convergencee of power power series

∞

n=1

(a). 0


[2] 3−n x3n is

Mathematics

I IT JAM 2016

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(b). 3 (c). 3 (d).

1/3

∞ [2]

14. 14.

  √ lim lim sin π n + n equals 2

n→∞

(a). 0 (b). 1 (c).

−1

(d). π

  √

(ln2)2 −y 2

ln 2

15. The integral integral

0

(a).

√ x +y

e

2

2

0

π 2 (2ln2 π 2 (2ln2

− 2) (b). − 1) (c). π (2ln2 − 1) (d). π (2ln2 − 2) 5

16. The value value of integral integral

5

 

(a). 7 sin5 (b). 2 1 sin sin 25 (c). 2 1 cos25 (d). 2

y

[1]

dxdy equals dxdy equals

[2]

sin(x sin(x2 )dx dy is

h s e n i D 0

a r d n e n i  a J

− −

[2]

17. The direction direction in which which the function f function f ((x, y) = x 2 y + exy sin y, decreases y, decreases most rapidly at P at P 0 (1, (1, 0) is (a). j

(b).

− j

(c). i

(d). i + j [2]


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I IT JAM 2016

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(3, (3,5,0)

18. The value value of the integral integral



(yzdx + xzdy + xzdy + xydz) xydz ) is

(1, (1,1,2)

(a). (b).

− 12 1 2

(c). 2 (d).

−2

a r d n e n i a J

[2]

19. The circulation circulation of the field F = x = x 2 i + 2x 2 x j + z 2 k around the curve C :the :the ellipse 4x 4 x2 + y 2 = 4 in the x the x y plane, counterclockwise (using Stoke’s theorem) is

−

(a).

−4π2

(b). 4π 4π 2 (c).

−4π

(d). 4π 4π

[2]

F = x j + (z 2 1)k 20. If F = xii y j+ 1)k and S and S is is the surface of the cylinder bounded by z = z = 0, z = 1, x2 + y2 = 4, then

 −

−

F.n dS equals dS equals

S

(a). π (b). π 2 (c). 3π 3π

h s e n i D (d). 4π 4π

[1]

21. D4 is dihedral group given as e,a,a2 , a3 ,ab,a2 b, a3 b where a4 = e, b2 = e, ba = ba = a a 3 b. Let N be the subgroup < subgroup < a2 >= e, a2 . then

{

{

}

}

D4 is cyclic. N D4 (b). N is N is normal subgroup of D D 4 and is not cyclic. N (c). N is N is not a normal subgroup of D D 4 . D4 (d). N is N is normal subgroup of D D 4 and is isomorphic to Z 4 N (a). N is N is normal subgroup of D D 4 and

22. Let φ : Z 4

(Z 4 × Z 6 ) × Z 6 → Z 4 × Z 3 by φ(x, y) = (x + 2y, 2 y, y ) is homomorphism then is {(0, (0, 0), 0), (2, (2, 3)}

isomorphic to (a). Z 12 12


[1]

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Page 7 of 13

(b). Z 2

× Z 2 × Z 3 (c). Z 2 × Z 6

(d). (d). None of above. above. [1]

a r d n e n i a J

23. Suppose Suppose that that G is non-abelian group of order p 3 (where p is prime) and Z ( Z (G) = e (where e (where e is identity of G G and Z ( Z (G) is centre of G) G ) then Z (G) (where Z (G) represents order of Z) may be

|

(a). p (b). p2 or p or p but not p not p 3 (c). p3 or p or p but not p not p 2 (d). (d). any any of p, p , p 2 , p 3

|

|

24. Let α Let α and a nd β β belongs belongs to S to S n (Set of all permutations of 1, 2, 3,

{ {

(a). βαβ −1 is even permutation. (b). βαβ −1 is odd permutation.

|



[2]

·· · · · , n }). Then

(c). βαβ −1 is even permutation when α is even permutation. (d). βαβ −1 is even permutation when β is β is even permutation.

25. Let Z Let Z n = 0, 1, 2, 3,

{

[1]

· · · , n − 1} represents cyclic group under addition modulo n. n . Then

h s e n i D (a). Z 40 40 have 4 elements of order 10.

(b). (b). Elemen Elementt 36 is a generat generator or of Z Z 40 40 . (c). The order of element 4 is 20.

(d). Z 40 40 is cyclic group with order of element 28 is 20. [1]

26. Let V Let V = P ( P (t) be the vector space of all real polynomials, then which of the following is not a subspace W subspace W of V V ? (a). W consists W consists of all polynomials with degree atmost 6.

(b). W consists W consists of all polynomials with degree

≥6 and the zero polynomial

(c). W consists W consists of all polynomials with only even powers of t of t..

(d). W consists W consists of all polynomials with only odd powers of t of t..


[1]

Mathematics

I IT JAM 2016

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27. Let A Let A be any square matrix of order n with det with det((A) = 2 then det(adj(adj det (adj(adjA A)) equals 2 (a). 2n −2n 2 (b). 2n −2n+2

(c). 2n−1 2

(d). 2(n−1)

28. Let A be an m where b where b is an m an m

a r d n e n i a J

[2]

× n matrix where m < n. Consid Consider er the system system of linear linear equation equationss Ax = b, × 1 column vector and b = 0. Which of the following is always true ?

(a). The system system of equations has no solution. solution.

(b). The system of equations equations has solution if and only if it has infinitely many many solutions. solutions. (c). The system of equations equations has a unique unique solution.

(d). The system of equations equations has atleast one solution.

29. Let S denote S denote the set of unit vectors in Then (a). V is V is always a subspace of R3

3 R

[2]

and W W a vector subspace of R3 , let V = W

∩ ∩ S .

(b). V is V is a subspace of R3 iff W has W has dimension 1 (c). V is V is a subspace of R3 iff W has W has dimension 3 (d). V is V is never a subspace of R3 .

h s e n i D  30. Let T Let T : P 2

[2]

→ P 2 be b e linear transformation transformati on P 2 , space of all polynomials of degree ≤ 2, defined  on on P x( x (x − 1) by T by T (( p( p(x)) = p( p (x + 1). Given B = 1, x, an ordered basis for P 2 , then the matrix 2

[T : B, B , B ] is

0 (a). 0 01 (b). 0 01 (c). 0 01 (d). 1

 1 /  /  0 /  /  / / 

1 1 0

1

1 1 0

1

1 1 0

0 1 1 0 /2

1

1 1 1

2 2

2 2 2

2 1/ 2

0 0 1/ 2

[2]


Mathematics

I IT JAM 2016

Page 9 of 13

SECTION B (MSQ) Quest Question ion numbers numbers 31 to 40 are are multis multisele elect ct questi questions ons carry carry 2 marks marks each. ach. No negat negative ive and and no partial marking.

31. Consider Consider the sequences sequences < < an > and a nd < < bn > given by an+1 =

an + bn 2

Then (a). < an > and < bn > are monotone (b). (b). lim lim an = lim bn = n→∞

n→∞

a r d n e n i a J 

and bn+1 =

√ a b

1 1

2a n b n , 0 < b1 < a1 , f or n an + bn

∈ N.

(c). both the sequences sequences < < an > and a nd < < bn > decrease

(d). the sequence sequence < < an > decreases and < and < bn > increases

32. Let f : R2

→R

be given given by f by f ((x, y ) = x + y , for (x, y)

|| ||

(a). f is f is continuous at (0,0) (b). f x (0, (0, 0) = f y (0, (0, 0) = 1 (c). both f both f x (0, (0, 0) and f and f y (0, (0, 0) do not exist (d). f is f is not differentiable at (0, (0 , 0)

h s e n i D

33. Let C Let C be be a simple closed curve in the xy-plane. Let I = Then

∈ R2. Then

F.dr , dr , where F where F =

C

[2]

[2]

−yi + x j . x2 + y 2

(a). I = 0

(b). I = 2π

(c). Stoke’s Stoke’s theorem theorem can not be applied applied

(d). curl F = 0

34. Let A and a nd B B are two matrices of m of m have exactly same solutions if

[2]

× n. Then homogeneous equations AX = = 0 and B and B X = 0

(a). det( det(A) = det( det (B )

(b). rank( rank (A) = rank( rank (B )

(c). A and B are row equivalent

(d). A = B = B [2]


Mathematics

I IT JAM 2016

Page 10 of 13

x

35. A differentiable differentiable function satisfies satisfies f ( f (x) =

 0

hold good?

−e (b). f ( f (x) has a maximum value 1 − e−1

{f ( f (t)cos t − cos( cos(t − x)}dt.Which dt.Which of the following

(a). f ( f (x) has minimum value 1

π (c). f  ( ) = e 2  (d). f (0) = 1

a r d n e n i a J

[2]

36. A curve curve y y = f = f ((x) passes through (1, (1, 1) and tangent at P at P ((x, y) cuts the x-axis and y-axis at A and B and B respectively such that BP B P : AP = AP = 3 : 1, then (a). equation equation of curve is xy is xy 

− 3y = 0

(b). (b). normal normal at (1, (1, 1) is x is x + 3y 3y = 4 1 (c). curve curve passes through through 2, 8 (d). equation equation of curve is xy is xy  + 3y 3 y = 0

 

37. Which Which of the following following function is uniformly continuou continuouss on [0, [0, (a). (a). sin( sin(x2 ) (b). ex (c). sin(sin sin(sin x)

h s e n i D 2)

sin(x (d). esin(x

[2]

∞)?

[2]

38. Let G = U (32) U (32) = 1, 3, 5, 7, 9, 11, 11, 13, 13, 15, 15, 17, 17, 19, 19, 21, 21, 23, 23, 25, 25, 27, 27, 29, 29, 31 is group with operation multiplic multiplication ation mod 32. And let H = U 16 U (32) 2) and and x 1 mod 16 . Then Then 16 (32) = x : x U (3 which of the followings is/are true ?

{

(a). (a). Order Order of

{

∈

}

≡

}

G is 8 H

G is abelian group H G (c). Order of is 4 H G (d). is isomorphic to U (16) U (16) H (b).


[2]

Mathematics

I IT JAM 2016

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39. Which Which of the followings followings is/are true for quotient quotient groups ? (a). A quotient quotient group of abelian group is abelian. (b). A quotient quotient group of cyclic group is cyclic. cyclic. Z 60 60 (c). Order of = 15 < 15 15 > >

a r d n e n i a J

(d). (d). Order Order of elemen elementt 14 + < 8 < 8 > > in quotient group

1

1

1

x2

1 y

   − 40. The integral integral dzdydx equals dzdydx equals −   −  − (a). dydzdx −   √ −  − (b). dydxdz √ − −   −  √ 1 x2

1

1

1 z

x2

0

1

1 z

1

1 z 1 z

0

0

1 z

(c).

x2 y

dxdydz

0

0

0

8 (d). 17

Z 24 24 is 4. < 8 > 8 >

[2]

[2]

SECTION C (NAT)

Quest Question ion numbers numbers 41 to 60 are are Numeri Numeric cal type type questi questions ons carry carry 1 or 2 marks marks each. ach. No negat negative ive markings for Wrong answer.

h s  e n i D

41. Suppose Suppose A and B are closed sets in R and let f : A B R be uniformly continuous on A and B and B . Must f Must f be be uniformly continuous on A on A B ? (Select 1 for yes and 0 for no)

∪

∪ →

[1]

42. Find the interval interval of convergenc convergencee of the power series

∞

n(2n (2n

n=1

43. Must Must the set cos n : n : n

{

− 1)x 1)x2n [2]

∈ N} be dense in [ −1, 1]. (Select 1 for yes and 0 for no)

[1]

44. Find the rational rational number number a a such that the following system of linear equations has no solution: x + 2y 2y

− 3z = 4

3x

− y + 5z 5 z = 2 4x + y + (a ( a2 − 14)z 14)z = a = a + 2 [2]


Mathematics

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45. Find Find the value value of a such that the vectors (1, (1 , 2, 3), 3), ( 1, 0, 2) and (1, (1, 6, a) in dep endent.

−

3 R

are linearly [1]

46. Let V Let V = R4 (R) be a vector space over the field R and let W let W 4 a = b b + c, c = b = b + d . Find the dimension of W. R : a =

⊂ ⊂ V defined V defined by W by W = {(a,b,c,d) a,b,c,d) ∈

}

[2]

47. Find the local minimum minimum value of the function y =

a r d n e n i a J

 3−x

3 + 2x 2x

−

x2

, x < 0 , x 0

≥

48. Find the absolute maximum maximum value value of the function function f ( f (x, y) = x2 + y 2 on the closed triangular plate bounded by the lines x = 0, y = 0, y + 2x 2 x = 2 in the first octant. t an xn − 1 − tan , x0 = 1 converges. sec2 xn √ such that dy + ιy = where ι = = −1 suc that φ (0) = 2. Find φ Find φ((π ). ιy = x x where ι φ(0)

49. Find the limit at which which the sequence sequence 50. Let φ Let φ be the solution of

xn+1 = x = x n

dx

51. Consider Consider the initial value problem problem y

− y − 2y = 0,

[2] [2] [2] [1]

y (0) = α, = α, y  (0) = 2. 2.

Then, find the value of α of α so that the solution approaches to zero as t

→ ∞.

[2]

52. Find the maximum maximum value of the solution solution of the initial value value problem 2y 

− 3y + y = 0,

h s e  n i   D 53. Evaluate Evaluate by Stoke’s Stoke’s theorem rectangle rectangle 0



(sin zdx

C

1 y (0) = 2, 2, y  (0) = . 2

[1]

− cos xdy + xdy + sin ydz) ydz ) where C is C is the boundary of the

≤ x ≤ π, 0 ≤ y ≤ 1, z = 3.

[2]

54. Let φ Let φ be the potential function for the field F = (z cos xz) xz )i + ey j + (x (x cos xz) xz )k.

with φ with φ(0 (0,, 0, 0) = 0. Find φ Find φ(0 (0,, 0, 1).

55. Apply Green’s Green’s theorem to evaluate evaluate

[2]

(3ydx (3ydx + 2xdy) xdy) where C where C : : the boundary of 0

C

y

≤ sin x.

≤ x ≤ π, 0 ≤ [2]

56. Find the volume volume of the region in the first octant bounded by the coordinate planes, planes, the plane πx y = 1 x and the surface z surface z = cos , 0 x 1. 2

[2]

57. How many many generators generators Z 20 20 have ?

[1]

−

≤ ≤

∈ ∈ S 7 and suppose β suppose β 4 = (2 1 4 3 5 6 7). 7). Find Find β β ..

58. Let permutatio permutation n β


[1]

Mathematics

I IT JAM 2016

Page 13 of 13

59. How many many homomorphism homomorphism are there from Z from Z 20 onto Z 8 ? 20 onto Z 1

60. Evaluate Evaluate

1

  0

2 ,y2 )

max(x emax(x

[1]

dydx, dydx, where

0

2

2

max(x max(x , y ) =

h s e n i D

x

2

y2

, if x 2 y2 , if x 2 < y2

≥

a r d n e n i a J


[2]

Answer Key (For Mock Test Paper mathematics)

1. (b)

2. (c)

3. (a)

4. (d)

5. (a)

6. (c)

7. (c)

8. (c)

9. (c)

10. (a)

11. (b) 21. (b) 31. (a, b, d)

12. (d) 22. (a) 32. (a, c, d)

13. (c) 23. (a) 33. (a, c, d)

14. (b) 24. (c) 34. (c, d)

16. (d) 26. (b) 36. (c, d)

17. (b) 27. (d) 37. (c)

18. (d) 28. (b) 38. (a, b, d)

19. (d) 29. (d) 39. (a,b,c,d) (a,b,c,d)

20. (d) 30. (b) 40. (a, b, d)

41. 0

42. (-1, 1)

43. 1

44. -4

15. (b) 25. (a) 35. (a, b, c) 45. 13

46. 2

47. 3

48. 4

49.

50.

51. -2

52.

53. 2

54. 0

55. -2

56.

57. 8

58.

59. 0

60. e-1

(2, 4, 5, 7, 1, 3, 6)

MATHS TEST

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