Maths Test 39456p
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IIT-JAM 2016
MOCK TEST-I
Paper code: MA
MATHEMATICS (MA)
Time: 3 hour
Maximum Marks: 100
General Instructions :
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(i) All questions are compulsory.
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(ii) The question paper consists of 60 questions divided into three sections A, B and C. Section A comprises of 30 single correct Mutiple Choice Questions (MCQ) carry one or two marks each, Section B comprises of 10 multi correct Mutiple Select Questions (MSQ) carry two marks each and Section C comprises of 20 Numerical Answer Type Questions (NAT) carry one or two marks each. (iii) Section-A (MCQ) contains a total of 30 Multiple Choice Questions (MCQ) carrying one or two marks each. Each MCQ type question has four choices out of which only one choice is the correct answer.
Ja
in
(iv) Section-B (MSQ) contains a total of 10 Multiple Select Questions(MSQ) carrying two marks each. Each MSQ type question is similar to MCQ but with a difference that there may be one or more than one choice(s) that are correct out of the four given choices. The candidate gets full credit if he/she selects all the correct answers only and no wrong answers. (v) Section-C (NAT) contains a total of 20 Numerical Answer Type (NAT) questions carrying one or two marks each. For these NAT type questions, the answer is a signed real number. No choices will be shown for these type of questions. (vi) In all sections, questions not attempted will result in zero mark.
es h
(vii) In Section A (MCQ), wrong answer will result in negative marks. For all 1 mark questions, 1/3 marks will be deducted for each wrong answer.For all 2 marks questions, 2/3 marks will be deducted for each wrong answer. (viii) In Section B (MSQ),there is no negative and no partial marking provisions. (ix) In Section C (NAT) there is no negative marking.
in
(x) Non-programmable calculators are permitted. Sharing of calculators is NOT allowed.
D
N R f 0 , f 00 ∂g ∂g ∂g , , ∂x ∂y ∂z fx , fy , fz log i, j, k [T : B, B]
Special Instructions / Usefull Data : The set of all positive Integers : The set of all Real numbers : First and Second derivatives respectively of a real function f : Partial derivatives of g with respect to x, y and z respectively : Partial derivatives of f with respect to x, y and z respectively : The logarithm to the base e : Standard unit orthogonal vectors : The matrix of linear transformation T w.r.t. the basis B
Prepared by: Jainendra Singh & Dinesh Khatri
Mathematics
IIT JAM 2016
Page 2 of 13
SECTION A (MCQ) Question numbers 1 to 30 are objective type single correct questions carry 1 or 2 mark each. For all 1 mark questions, 1/3 marks will be deducted for each wrong answer. For all 2 marks questions, 2/ will be deducted for each wrong answer. 3
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d2 y dy 1. Given that y = x is solution of the differential equation (x2 − 1) 2 − 2x + 2y = 0. Then, dx dx the second linearly independent solution is (a). x2 . (b). x2 + 1.
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(c). x2 − 1. (d). x3 .
2. Let W (x) denote the wronskian of the differential equation
d2 x dy +x + x2 y = 0 and given 2 dx dx
in
that W (0) = 1, then the wronskian is given by
[2]
(a). e−x 2
(b). e−x . 2
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(c). e−x /2 . 2
(d). ex /2 .
[1]
es h
3. The solution of y 0 = y 2 , y(0) = 1 exists for all (a). x ∈ (−∞, 1)
(b). x ∈ [0, a], a > 1. (c). x ∈ (−∞, ∞).
in
(d). x ∈ [1, a], a > 1.
[2]
D
p dy 4. Let S denote the set of all constant solutions of the differential equation = y(y − 1)(y − 2)(y − 3), dx then the set S is (a). Empty
(b). Uncountable & infinite (c). Countable & infinite (d). finite [1]
Prepared by: Jainendra Singh & Dinesh Khatri
Mathematics
IIT JAM 2016
Page 3 of 13
5. Consider the differential equation d2 y 1 dy 1 + + (1 − 2 )y = 0, x ∈ (0, 10] dx2 x dx x and the wronskian at x = 2 is 1. Then the minimum value of the wronskian is (a). 5
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(b). 10 (c). 15
[1]
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(d). 20
6. Suppose that f (x) is differentiable for all x ∈ [0, 1] and that f (0) = 0. Define the sequence 1 < an > by the rule an = n f . Then lim an equals n→∞ n (a). ∞ (b). 0
in
(c). f 0 (0)
7. lim
n→∞
√
[1]
Ja
(d). −f 0 (0)
1 1 1 +√ + ··· + √ n+1 n+2 2n
(a). 0
equals
(c). ∞ 1 (d). 2
es h
(b). 1
in
8. The set of all values of x for which, the series
[1] nxn converges absolutely, is n n=1 (n + 1)(2x + 1) ∞ P
(a). (−∞, −1) ∪ (1, ∞)
D
(b). R
−1 (c). (−∞, −1) ∪ ,∞ 3 1 1 (d). −∞, − ∪ ,∞ 3 3 [2]
Prepared by: Jainendra Singh & Dinesh Khatri
Mathematics
IIT JAM 2016
Page 4 of 13
p (xy). Then
9. Let f (x, y) =
(a). fx and fy do not exist at (0, 0) (b). fx (0, 0) = 1 (c). fy (0, 0) = 0
Zx 10. Let f (x) =
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(d). f is differentiable at (0, 0).
g(t) dt. Then f 0 (−1) equals
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1/x
(a). 2g(−1)
n(n+1) 2
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(a). −5 11 (b). − 2 (c). 5 11 (d). 2
2(−1)n+1 + (−1)
3 2+ :n∈N n
[2]
is
Ja
11. The infimum of the set A =
in
(b). −2g(−1) 1 (c). g(−1) 2 1 (d). − g(−1) 2
[2]
[2]
12. Which of the following function is uniformly continuous on (0, 1) ? 1 x
in
(a). sin 1
(b). e /x
1 x 1 (d). x sin x
D
(c). ex cos
13. The radius of convergence of power series
[2] ∞ X n=1
(a). 0
Prepared by: Jainendra Singh & Dinesh Khatri
3−n x3n is
Mathematics
IIT JAM 2016
Page 5 of 13
(b). 3 (c). 31/3 (d). ∞ [2] √ 14. lim sin π n2 + n equals
ra
n→∞
(a). 0 (b). 1 (d). π
Z
ln 2 Z
√
e
15. The integral 0
(b).
π 2 (2 ln 2 π 2 (2 ln 2
x2 +y 2
dxdy equals
0
− 2)
[1]
in
(a).
√
(ln 2)2 −y 2
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(c). −1
− 1)
(c). π(2 ln 2 − 1)
Z
5 Z 5
16. The value of integral 0
y
Ja
(d). π(2 ln 2 − 2)
[2]
sin(x2 )dx dy is
in
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(a). 7 sin 5 (b). 2 1 − sin 25 (c). 2 1 − cos 25 (d). 2
[2]
D
17. The direction in which the function f (x, y) = x2 y + exy sin y, decreases most rapidly at P0 (1, 0) is (a). j
(b). −j (c). i (d). i + j [2]
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Mathematics
IIT JAM 2016
Z
Page 6 of 13
(3,5,0)
18. The value of the integral
(yzdx + xzdy + xydz) is (1,1,2)
(a). − 21 (b).
1 2
(c). 2
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(d). −2 [2]
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19. The circulation of the field F = x2 i + 2xj + z 2 k around the curve C :the ellipse 4x2 + y 2 = 4 in the x − y plane, counterclockwise (using Stoke’s theorem) is (a). −4π 2 (b). 4π 2 (c). −4π
[2]
in
(d). 4π
2 −1)k and S is the surface of the cylinder bounded by z = 0, z = 1, x2 +y 2 = 20. If F = xi−yj+(z Z
4, then
F.n dS equals
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S
(a). π (b). π 2 (c). 3π
[1]
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(d). 4π
21. D4 is dihedral group given as {e, a, a2 , a3 , ab, a2 b, a3 b} where a4 = e, b2 = e, ba = a3 b. Let N be the subgroup < a2 >= {e, a2 }. then D4 is cyclic. N D4 (b). N is normal subgroup of D4 and is not cyclic. N (c). N is not a normal subgroup of D4 . D4 (d). N is normal subgroup of D4 and is isomorphic to Z4 N
D
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(a). N is normal subgroup of D4 and
22. Let φ : Z4 × Z6 → Z4 × Z3 by φ(x, y) = (x + 2y, y) is homomorphism then isomorphic to (a). Z12
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[1] (Z4 × Z6 ) is {(0, 0), (2, 3)}
Mathematics
IIT JAM 2016
Page 7 of 13
(b). Z2 × Z2 × Z3 (c). Z2 × Z6 (d). None of above. [1]
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23. Suppose that G is non-abelian group of order p3 (where p is prime) and Z(G) 6= e (where e is identity of G and Z(G) is centre of G) then |Z(G)| (where |Z(G)| represents order of Z) may be (a). p (b). p2 or p but not p3
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(c). p3 or p but not p2 (d). any of p, p2 , p3
[2]
24. Let α and β belongs to Sn (Set of all permutations of {1, 2, 3, · · · , n }). Then
in
(a). βαβ −1 is even permutation. (b). βαβ −1 is odd permutation.
(c). βαβ −1 is even permutation when α is even permutation.
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(d). βαβ −1 is even permutation when β is even permutation.
[1]
25. Let Zn = {0, 1, 2, 3, · · · , n − 1} represents cyclic group under addition modulo n. Then (a). Z40 have 4 elements of order 10.
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(b). Element 36 is a generator of Z40 . (c). The order of element 4 is 20.
(d). Z40 is cyclic group with order of element 28 is 20. [1]
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26. Let V = P (t) be the vector space of all real polynomials, then which of the following is not a subspace W of V ? (a). W consists of all polynomials with degree atmost 6.
D
(b). W consists of all polynomials with degree ≥6 and the zero polynomial (c). W consists of all polynomials with only even powers of t.
(d). W consists of all polynomials with only odd powers of t. [1]
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Page 8 of 13
27. Let A be any square matrix of order n with det(A) = 2 then det(adj(adjA)) equals 2 −2n
(a). 2n
2 −2n+2
(b). 2n
(c). 2n−1 2
(d). 2(n−1)
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[2] 28. Let A be an m × n matrix where m < n. Consider the system of linear equations Ax = b, where b is an m × 1 column vector and b 6= 0. Which of the following is always true ? (a). The system of equations has no solution. (c). The system of equations has a unique solution. (d). The system of equations has atleast one solution.
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(b). The system of equations has solution if and only if it has infinitely many solutions.
[2]
in
29. Let S denote the set of unit vectors in R3 and W a vector subspace of R3 , let V = W ∩ S. Then (a). V is always a subspace of R3
(b). V is a subspace of R3 iff W has dimension 1 (d). V is never a subspace of R3 .
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(c). V is a subspace of R3 iff W has dimension 3
[2]
D
in
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30. Let T : P2 → P2 be linear transformation on P2 , space of all polynomials of degree ≤ 2, defined x(x − 1) an ordered basis for P2 , then the matrix by T (p(x)) = p(x + 1). Given B = 1, x, 2 [T : B, B] is 0 1 1 (a). 0 1 1/2 0 0 1/2 1 1 0 (b). 0 1 1/2 0 0 1/2 1 1 1/2 (c). 0 1 1/2 0 0 1/2 1 0 0 0 (d). 1 1 1 1 0 /2 /2 [2]
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IIT JAM 2016
Page 9 of 13
SECTION B (MSQ) Question numbers 31 to 40 are multiselect questions carry 2 marks each. No negative and no partial marking. 31. Consider the sequences < an > and < bn > given by an + bn 2
and
bn+1 =
2an bn , 0 < b1 < a1 , f or n ∈ N. an + bn
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an+1 = Then
(a). < an > and < bn > are monotone √ (b). lim an = lim bn = a1 b1 n→∞
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n→∞
(c). both the sequences < an > and < bn > decrease (d). the sequence < an > decreases and < bn > increases
be given by f (x, y) = |x| + |y|,
(a). f is continuous at (0,0) (b). fx (0, 0) = fy (0, 0) = 1 (d). f is not differentiable at (0, 0)
Ja
(c). both fx (0, 0) and fy (0, 0) do not exist
for (x, y) ∈ R2 . Then
in
32. Let f : R2 → R
[2] Z
33. Let C be a simple closed curve in the xy-plane. Let I =
(a). I = 0 (b). I = 2π
F.dr , where F = C
−yi + xj . x2 + y 2
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Then
[2]
(c). Stoke’s theorem can not be applied
in
(d). curl F = 0
[2]
D
34. Let A and B are two matrices of m × n. Then homogeneous equations AX = 0 and BX = 0 have exactly same solutions if (a). det(A) = det(B)
(b). rank(A) = rank(B) (c). A and B are row equivalent
(d). A = B [2]
Prepared by: Jainendra Singh & Dinesh Khatri
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IIT JAM 2016
Z
Page 10 of 13
x
{f (t) cos t − cos(t − x)}dt.Which of the following
35. A differentiable function satisfies f (x) = 0
hold good? (a). f (x) has minimum value 1 − e
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(b). f (x) has a maximum value 1 − e−1 π (c). f 00 ( ) = e 2 0 (d). f (0) = 1
[2]
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36. A curve y = f (x) es through (1, 1) and tangent at P (x, y) cuts the x-axis and y-axis at A and B respectively such that BP : AP = 3 : 1, then (a). equation of curve is xy 0 − 3y = 0
in
(b). normal at (1, 1) is x + 3y = 4 1 (c). curve es through 2, 8 0 (d). equation of curve is xy + 3y = 0
[2]
Ja
37. Which of the following function is uniformly continuous on [0, ∞)? (a). sin(x2 ) (b). ex (c). sin(sin x) 2)
es h
(d). esin(x
[2]
38. Let G = U (32) = {1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31} is group with operation multiplication mod 32. And let H = U16 (32) = {x : x ∈ U (32) and x ≡ 1 mod 16}. Then which of the followings is/are true ? G is 8 H
in
(a). Order of
G is abelian group H G (c). Order of is 4 H G (d). is isomorphic to U (16) H
D
(b).
[2]
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IIT JAM 2016
Page 11 of 13
39. Which of the followings is/are true for quotient groups ? (a). A quotient group of abelian group is abelian.
(d). Order of element 14 + < 8 > in quotient group
Z
1
1Z
Z
1−y
dzdydx equals
40. The integral x2
1
Z
1−x2
Z
0
1−z
dydzdx
(a). −1
Z (b). 0
Z
1
0 Z √
x2 1−z
Z
1−z
√ − 1−z x2 √ 1 Z 1−z Z y
(c). 0
in
0
8 17
Ja
(d).
dydxdz dxdydz
0
[2]
en d
−1
Z
Z24 is 4. <8>
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(b). A quotient group of cyclic group is cyclic. Z60 = 15 (c). Order of < 15 >
[2]
SECTION C (NAT) Question numbers 41 to 60 are Numerical type questions carry 1 or 2 marks each. No negative markings for Wrong answer.
es h
41. Suppose A and B are closed sets in R and let f : A ∪ B → R be uniformly continuous on A and B . Must f be uniformly continuous on A ∪ B ? (Select 1 for yes and 0 for no)
[1]
in
42. Find the interval of convergence of the power series ∞ X
n(2n − 1)x2n
n=1
[2]
D
43. Must the set {cos n : n ∈ N} be dense in [−1, 1]. (Select 1 for yes and 0 for no)
[1]
44. Find the rational number a such that the following system of linear equations has no solution: x + 2y − 3z = 4 3x − y + 5z = 2 4x + y + (a2 − 14)z = a + 2 [2]
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45. Find the value of a such that the vectors (1, 2, 3), (−1, 0, 2) and (1, 6, a) in R3 are linearly dependent.
[1]
46. Let V = R4 (R) be a vector space over the field R and let W ⊂ V defined by W = {(a, b, c, d) ∈ R4 : a = b + c, c = b + d}. Find the dimension of W.
[2]
ra
47. Find the local minimum value of the function 3−x , x<0 y= 3 + 2x − x2 , x ≥ 0
xn+1 = xn −
49. Find the limit at which the sequence 50. Let φ be the solution of
dy + ιy = x dx
where ι =
√
en d
48. Find the absolute maximum value of the function f (x, y) = x2 + y 2 on the closed triangular plate bounded by the lines x = 0, y = 0, y + 2x = 2 in the first octant. tan xn − 1 , x0 = 1 sec2 xn −1
such that φ(0) = 2. Find φ(π).
in
51. Consider the initial value problem
converges.
[2] [2] [2] [1]
y 00 − y 0 − 2y = 0, y(0) = α, y 0 (0) = 2. [2]
Ja
Then, find the value of α so that the solution approaches to zero as t → ∞. 52. Find the maximum value of the solution of the initial value problem 1 2y 00 − 3y 0 + y = 0, y(0) = 2, y 0 (0) = . 2
es h
[1] Z
(sin zdx − cos xdy + sin ydz) where C is the boundary of the
53. Evaluate by Stoke’s theorem
C
rectangle 0 ≤ x ≤ π, 0 ≤ y ≤ 1, z = 3.
[2]
54. Let φ be the potential function for the field
in
F = (z cos xz)i + ey j + (x cos xz)k.
with φ(0, 0, 0) = 0. Find φ(0, 0, 1).
[2] I (3ydx+2xdy) where C : the boundary of 0 ≤ x ≤ π, 0 ≤
D
55. Apply Green’s theorem to evaluate y ≤ sin x.
C
[2]
56. Find the volume of the region in the πxfirst octant bounded by the coordinate planes, the plane , 0 ≤ x ≤ 1. y = 1 − x and the surface z = cos 2
[2]
57. How many generators Z20 have ?
[1]
58. Let permutation β ∈ S7 and suppose β 4 = (2 1 4 3 5 6 7). Find β.
[1]
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59. How many homomorphism are there from Z20 onto Z8 ? Z 1Z 1 2 2 emax(x ,y ) dydx, where 60. Evaluate 0
[1]
0 2
2
x2 , if y 2 , if
x2 ≥ y 2 x2 < y 2
D
in
es h
Ja
in
en d
ra
max(x , y ) =
Prepared by: Jainendra Singh & Dinesh Khatri
[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)
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. (2, 4, 5, 7, 1, 3, 6)
59. 0
60. e-1
MOCK TEST-I
Paper code: MA
MATHEMATICS (MA)
Time: 3 hour
Maximum Marks: 100
General Instructions :
ra
(i) All questions are compulsory.
en d
(ii) The question paper consists of 60 questions divided into three sections A, B and C. Section A comprises of 30 single correct Mutiple Choice Questions (MCQ) carry one or two marks each, Section B comprises of 10 multi correct Mutiple Select Questions (MSQ) carry two marks each and Section C comprises of 20 Numerical Answer Type Questions (NAT) carry one or two marks each. (iii) Section-A (MCQ) contains a total of 30 Multiple Choice Questions (MCQ) carrying one or two marks each. Each MCQ type question has four choices out of which only one choice is the correct answer.
Ja
in
(iv) Section-B (MSQ) contains a total of 10 Multiple Select Questions(MSQ) carrying two marks each. Each MSQ type question is similar to MCQ but with a difference that there may be one or more than one choice(s) that are correct out of the four given choices. The candidate gets full credit if he/she selects all the correct answers only and no wrong answers. (v) Section-C (NAT) contains a total of 20 Numerical Answer Type (NAT) questions carrying one or two marks each. For these NAT type questions, the answer is a signed real number. No choices will be shown for these type of questions. (vi) In all sections, questions not attempted will result in zero mark.
es h
(vii) In Section A (MCQ), wrong answer will result in negative marks. For all 1 mark questions, 1/3 marks will be deducted for each wrong answer.For all 2 marks questions, 2/3 marks will be deducted for each wrong answer. (viii) In Section B (MSQ),there is no negative and no partial marking provisions. (ix) In Section C (NAT) there is no negative marking.
in
(x) Non-programmable calculators are permitted. Sharing of calculators is NOT allowed.
D
N R f 0 , f 00 ∂g ∂g ∂g , , ∂x ∂y ∂z fx , fy , fz log i, j, k [T : B, B]
Special Instructions / Usefull Data : The set of all positive Integers : The set of all Real numbers : First and Second derivatives respectively of a real function f : Partial derivatives of g with respect to x, y and z respectively : Partial derivatives of f with respect to x, y and z respectively : The logarithm to the base e : Standard unit orthogonal vectors : The matrix of linear transformation T w.r.t. the basis B
Prepared by: Jainendra Singh & Dinesh Khatri
Mathematics
IIT JAM 2016
Page 2 of 13
SECTION A (MCQ) Question numbers 1 to 30 are objective type single correct questions carry 1 or 2 mark each. For all 1 mark questions, 1/3 marks will be deducted for each wrong answer. For all 2 marks questions, 2/ will be deducted for each wrong answer. 3
ra
d2 y dy 1. Given that y = x is solution of the differential equation (x2 − 1) 2 − 2x + 2y = 0. Then, dx dx the second linearly independent solution is (a). x2 . (b). x2 + 1.
en d
(c). x2 − 1. (d). x3 .
2. Let W (x) denote the wronskian of the differential equation
d2 x dy +x + x2 y = 0 and given 2 dx dx
in
that W (0) = 1, then the wronskian is given by
[2]
(a). e−x 2
(b). e−x . 2
Ja
(c). e−x /2 . 2
(d). ex /2 .
[1]
es h
3. The solution of y 0 = y 2 , y(0) = 1 exists for all (a). x ∈ (−∞, 1)
(b). x ∈ [0, a], a > 1. (c). x ∈ (−∞, ∞).
in
(d). x ∈ [1, a], a > 1.
[2]
D
p dy 4. Let S denote the set of all constant solutions of the differential equation = y(y − 1)(y − 2)(y − 3), dx then the set S is (a). Empty
(b). Uncountable & infinite (c). Countable & infinite (d). finite [1]
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5. Consider the differential equation d2 y 1 dy 1 + + (1 − 2 )y = 0, x ∈ (0, 10] dx2 x dx x and the wronskian at x = 2 is 1. Then the minimum value of the wronskian is (a). 5
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(b). 10 (c). 15
[1]
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(d). 20
6. Suppose that f (x) is differentiable for all x ∈ [0, 1] and that f (0) = 0. Define the sequence 1 < an > by the rule an = n f . Then lim an equals n→∞ n (a). ∞ (b). 0
in
(c). f 0 (0)
7. lim
n→∞
√
[1]
Ja
(d). −f 0 (0)
1 1 1 +√ + ··· + √ n+1 n+2 2n
(a). 0
equals
(c). ∞ 1 (d). 2
es h
(b). 1
in
8. The set of all values of x for which, the series
[1] nxn converges absolutely, is n n=1 (n + 1)(2x + 1) ∞ P
(a). (−∞, −1) ∪ (1, ∞)
D
(b). R
−1 (c). (−∞, −1) ∪ ,∞ 3 1 1 (d). −∞, − ∪ ,∞ 3 3 [2]
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Page 4 of 13
p (xy). Then
9. Let f (x, y) =
(a). fx and fy do not exist at (0, 0) (b). fx (0, 0) = 1 (c). fy (0, 0) = 0
Zx 10. Let f (x) =
ra
(d). f is differentiable at (0, 0).
g(t) dt. Then f 0 (−1) equals
en d
1/x
(a). 2g(−1)
n(n+1) 2
es h
(a). −5 11 (b). − 2 (c). 5 11 (d). 2
2(−1)n+1 + (−1)
3 2+ :n∈N n
[2]
is
Ja
11. The infimum of the set A =
in
(b). −2g(−1) 1 (c). g(−1) 2 1 (d). − g(−1) 2
[2]
[2]
12. Which of the following function is uniformly continuous on (0, 1) ? 1 x
in
(a). sin 1
(b). e /x
1 x 1 (d). x sin x
D
(c). ex cos
13. The radius of convergence of power series
[2] ∞ X n=1
(a). 0
Prepared by: Jainendra Singh & Dinesh Khatri
3−n x3n is
Mathematics
IIT JAM 2016
Page 5 of 13
(b). 3 (c). 31/3 (d). ∞ [2] √ 14. lim sin π n2 + n equals
ra
n→∞
(a). 0 (b). 1 (d). π
Z
ln 2 Z
√
e
15. The integral 0
(b).
π 2 (2 ln 2 π 2 (2 ln 2
x2 +y 2
dxdy equals
0
− 2)
[1]
in
(a).
√
(ln 2)2 −y 2
en d
(c). −1
− 1)
(c). π(2 ln 2 − 1)
Z
5 Z 5
16. The value of integral 0
y
Ja
(d). π(2 ln 2 − 2)
[2]
sin(x2 )dx dy is
in
es h
(a). 7 sin 5 (b). 2 1 − sin 25 (c). 2 1 − cos 25 (d). 2
[2]
D
17. The direction in which the function f (x, y) = x2 y + exy sin y, decreases most rapidly at P0 (1, 0) is (a). j
(b). −j (c). i (d). i + j [2]
Prepared by: Jainendra Singh & Dinesh Khatri
Mathematics
IIT JAM 2016
Z
Page 6 of 13
(3,5,0)
18. The value of the integral
(yzdx + xzdy + xydz) is (1,1,2)
(a). − 21 (b).
1 2
(c). 2
ra
(d). −2 [2]
en d
19. The circulation of the field F = x2 i + 2xj + z 2 k around the curve C :the ellipse 4x2 + y 2 = 4 in the x − y plane, counterclockwise (using Stoke’s theorem) is (a). −4π 2 (b). 4π 2 (c). −4π
[2]
in
(d). 4π
2 −1)k and S is the surface of the cylinder bounded by z = 0, z = 1, x2 +y 2 = 20. If F = xi−yj+(z Z
4, then
F.n dS equals
Ja
S
(a). π (b). π 2 (c). 3π
[1]
es h
(d). 4π
21. D4 is dihedral group given as {e, a, a2 , a3 , ab, a2 b, a3 b} where a4 = e, b2 = e, ba = a3 b. Let N be the subgroup < a2 >= {e, a2 }. then D4 is cyclic. N D4 (b). N is normal subgroup of D4 and is not cyclic. N (c). N is not a normal subgroup of D4 . D4 (d). N is normal subgroup of D4 and is isomorphic to Z4 N
D
in
(a). N is normal subgroup of D4 and
22. Let φ : Z4 × Z6 → Z4 × Z3 by φ(x, y) = (x + 2y, y) is homomorphism then isomorphic to (a). Z12
Prepared by: Jainendra Singh & Dinesh Khatri
[1] (Z4 × Z6 ) is {(0, 0), (2, 3)}
Mathematics
IIT JAM 2016
Page 7 of 13
(b). Z2 × Z2 × Z3 (c). Z2 × Z6 (d). None of above. [1]
ra
23. Suppose that G is non-abelian group of order p3 (where p is prime) and Z(G) 6= e (where e is identity of G and Z(G) is centre of G) then |Z(G)| (where |Z(G)| represents order of Z) may be (a). p (b). p2 or p but not p3
en d
(c). p3 or p but not p2 (d). any of p, p2 , p3
[2]
24. Let α and β belongs to Sn (Set of all permutations of {1, 2, 3, · · · , n }). Then
in
(a). βαβ −1 is even permutation. (b). βαβ −1 is odd permutation.
(c). βαβ −1 is even permutation when α is even permutation.
Ja
(d). βαβ −1 is even permutation when β is even permutation.
[1]
25. Let Zn = {0, 1, 2, 3, · · · , n − 1} represents cyclic group under addition modulo n. Then (a). Z40 have 4 elements of order 10.
es h
(b). Element 36 is a generator of Z40 . (c). The order of element 4 is 20.
(d). Z40 is cyclic group with order of element 28 is 20. [1]
in
26. Let V = P (t) be the vector space of all real polynomials, then which of the following is not a subspace W of V ? (a). W consists of all polynomials with degree atmost 6.
D
(b). W consists of all polynomials with degree ≥6 and the zero polynomial (c). W consists of all polynomials with only even powers of t.
(d). W consists of all polynomials with only odd powers of t. [1]
Prepared by: Jainendra Singh & Dinesh Khatri
Mathematics
IIT JAM 2016
Page 8 of 13
27. Let A be any square matrix of order n with det(A) = 2 then det(adj(adjA)) equals 2 −2n
(a). 2n
2 −2n+2
(b). 2n
(c). 2n−1 2
(d). 2(n−1)
ra
[2] 28. Let A be an m × n matrix where m < n. Consider the system of linear equations Ax = b, where b is an m × 1 column vector and b 6= 0. Which of the following is always true ? (a). The system of equations has no solution. (c). The system of equations has a unique solution. (d). The system of equations has atleast one solution.
en d
(b). The system of equations has solution if and only if it has infinitely many solutions.
[2]
in
29. Let S denote the set of unit vectors in R3 and W a vector subspace of R3 , let V = W ∩ S. Then (a). V is always a subspace of R3
(b). V is a subspace of R3 iff W has dimension 1 (d). V is never a subspace of R3 .
Ja
(c). V is a subspace of R3 iff W has dimension 3
[2]
D
in
es h
30. Let T : P2 → P2 be linear transformation on P2 , space of all polynomials of degree ≤ 2, defined x(x − 1) an ordered basis for P2 , then the matrix by T (p(x)) = p(x + 1). Given B = 1, x, 2 [T : B, B] is 0 1 1 (a). 0 1 1/2 0 0 1/2 1 1 0 (b). 0 1 1/2 0 0 1/2 1 1 1/2 (c). 0 1 1/2 0 0 1/2 1 0 0 0 (d). 1 1 1 1 0 /2 /2 [2]
Prepared by: Jainendra Singh & Dinesh Khatri
Mathematics
IIT JAM 2016
Page 9 of 13
SECTION B (MSQ) Question numbers 31 to 40 are multiselect questions carry 2 marks each. No negative and no partial marking. 31. Consider the sequences < an > and < bn > given by an + bn 2
and
bn+1 =
2an bn , 0 < b1 < a1 , f or n ∈ N. an + bn
ra
an+1 = Then
(a). < an > and < bn > are monotone √ (b). lim an = lim bn = a1 b1 n→∞
en d
n→∞
(c). both the sequences < an > and < bn > decrease (d). the sequence < an > decreases and < bn > increases
be given by f (x, y) = |x| + |y|,
(a). f is continuous at (0,0) (b). fx (0, 0) = fy (0, 0) = 1 (d). f is not differentiable at (0, 0)
Ja
(c). both fx (0, 0) and fy (0, 0) do not exist
for (x, y) ∈ R2 . Then
in
32. Let f : R2 → R
[2] Z
33. Let C be a simple closed curve in the xy-plane. Let I =
(a). I = 0 (b). I = 2π
F.dr , where F = C
−yi + xj . x2 + y 2
es h
Then
[2]
(c). Stoke’s theorem can not be applied
in
(d). curl F = 0
[2]
D
34. Let A and B are two matrices of m × n. Then homogeneous equations AX = 0 and BX = 0 have exactly same solutions if (a). det(A) = det(B)
(b). rank(A) = rank(B) (c). A and B are row equivalent
(d). A = B [2]
Prepared by: Jainendra Singh & Dinesh Khatri
Mathematics
IIT JAM 2016
Z
Page 10 of 13
x
{f (t) cos t − cos(t − x)}dt.Which of the following
35. A differentiable function satisfies f (x) = 0
hold good? (a). f (x) has minimum value 1 − e
ra
(b). f (x) has a maximum value 1 − e−1 π (c). f 00 ( ) = e 2 0 (d). f (0) = 1
[2]
en d
36. A curve y = f (x) es through (1, 1) and tangent at P (x, y) cuts the x-axis and y-axis at A and B respectively such that BP : AP = 3 : 1, then (a). equation of curve is xy 0 − 3y = 0
in
(b). normal at (1, 1) is x + 3y = 4 1 (c). curve es through 2, 8 0 (d). equation of curve is xy + 3y = 0
[2]
Ja
37. Which of the following function is uniformly continuous on [0, ∞)? (a). sin(x2 ) (b). ex (c). sin(sin x) 2)
es h
(d). esin(x
[2]
38. Let G = U (32) = {1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31} is group with operation multiplication mod 32. And let H = U16 (32) = {x : x ∈ U (32) and x ≡ 1 mod 16}. Then which of the followings is/are true ? G is 8 H
in
(a). Order of
G is abelian group H G (c). Order of is 4 H G (d). is isomorphic to U (16) H
D
(b).
[2]
Prepared by: Jainendra Singh & Dinesh Khatri
Mathematics
IIT JAM 2016
Page 11 of 13
39. Which of the followings is/are true for quotient groups ? (a). A quotient group of abelian group is abelian.
(d). Order of element 14 + < 8 > in quotient group
Z
1
1Z
Z
1−y
dzdydx equals
40. The integral x2
1
Z
1−x2
Z
0
1−z
dydzdx
(a). −1
Z (b). 0
Z
1
0 Z √
x2 1−z
Z
1−z
√ − 1−z x2 √ 1 Z 1−z Z y
(c). 0
in
0
8 17
Ja
(d).
dydxdz dxdydz
0
[2]
en d
−1
Z
Z24 is 4. <8>
ra
(b). A quotient group of cyclic group is cyclic. Z60 = 15 (c). Order of < 15 >
[2]
SECTION C (NAT) Question numbers 41 to 60 are Numerical type questions carry 1 or 2 marks each. No negative markings for Wrong answer.
es h
41. Suppose A and B are closed sets in R and let f : A ∪ B → R be uniformly continuous on A and B . Must f be uniformly continuous on A ∪ B ? (Select 1 for yes and 0 for no)
[1]
in
42. Find the interval of convergence of the power series ∞ X
n(2n − 1)x2n
n=1
[2]
D
43. Must the set {cos n : n ∈ N} be dense in [−1, 1]. (Select 1 for yes and 0 for no)
[1]
44. Find the rational number a such that the following system of linear equations has no solution: x + 2y − 3z = 4 3x − y + 5z = 2 4x + y + (a2 − 14)z = a + 2 [2]
Prepared by: Jainendra Singh & Dinesh Khatri
Mathematics
IIT JAM 2016
Page 12 of 13
45. Find the value of a such that the vectors (1, 2, 3), (−1, 0, 2) and (1, 6, a) in R3 are linearly dependent.
[1]
46. Let V = R4 (R) be a vector space over the field R and let W ⊂ V defined by W = {(a, b, c, d) ∈ R4 : a = b + c, c = b + d}. Find the dimension of W.
[2]
ra
47. Find the local minimum value of the function 3−x , x<0 y= 3 + 2x − x2 , x ≥ 0
xn+1 = xn −
49. Find the limit at which the sequence 50. Let φ be the solution of
dy + ιy = x dx
where ι =
√
en d
48. Find the absolute maximum value of the function f (x, y) = x2 + y 2 on the closed triangular plate bounded by the lines x = 0, y = 0, y + 2x = 2 in the first octant. tan xn − 1 , x0 = 1 sec2 xn −1
such that φ(0) = 2. Find φ(π).
in
51. Consider the initial value problem
converges.
[2] [2] [2] [1]
y 00 − y 0 − 2y = 0, y(0) = α, y 0 (0) = 2. [2]
Ja
Then, find the value of α so that the solution approaches to zero as t → ∞. 52. Find the maximum value of the solution of the initial value problem 1 2y 00 − 3y 0 + y = 0, y(0) = 2, y 0 (0) = . 2
es h
[1] Z
(sin zdx − cos xdy + sin ydz) where C is the boundary of the
53. Evaluate by Stoke’s theorem
C
rectangle 0 ≤ x ≤ π, 0 ≤ y ≤ 1, z = 3.
[2]
54. Let φ be the potential function for the field
in
F = (z cos xz)i + ey j + (x cos xz)k.
with φ(0, 0, 0) = 0. Find φ(0, 0, 1).
[2] I (3ydx+2xdy) where C : the boundary of 0 ≤ x ≤ π, 0 ≤
D
55. Apply Green’s theorem to evaluate y ≤ sin x.
C
[2]
56. Find the volume of the region in the πxfirst octant bounded by the coordinate planes, the plane , 0 ≤ x ≤ 1. y = 1 − x and the surface z = cos 2
[2]
57. How many generators Z20 have ?
[1]
58. Let permutation β ∈ S7 and suppose β 4 = (2 1 4 3 5 6 7). Find β.
[1]
Prepared by: Jainendra Singh & Dinesh Khatri
Mathematics
IIT JAM 2016
Page 13 of 13
59. How many homomorphism are there from Z20 onto Z8 ? Z 1Z 1 2 2 emax(x ,y ) dydx, where 60. Evaluate 0
[1]
0 2
2
x2 , if y 2 , if
x2 ≥ y 2 x2 < y 2
D
in
es h
Ja
in
en d
ra
max(x , y ) =
Prepared by: Jainendra Singh & Dinesh Khatri
[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)
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. (2, 4, 5, 7, 1, 3, 6)
59. 0
60. e-1
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