Dear all readers. It is Praveen Chhikara. Here is a JAM Mathematics mock test, prepared

according to the exams conducted in the last 3-4 years. Due to lack of time, I could not arrange

the questions them section-wise. A key has been prepared is okay, although I prepared it very

speedily. I hope, it is likely to help you a lot.

Pr

avee

n Chh

ikara

JAM Mathematics Mock Test

Date: Jan 27, 2019 -by Praveen Chhikara ..

1. The greatest value of α for which the power series∑

[2−(−1)n

3]nxn con-

verges for |x| < α, is ............

2. For A ⊆ R, which of the following statements is FALSE?

(a) The closure A of A is equal to the intersection of all closed setswhich contain A.

(b) The interior Ao of A is equal to the union of all open sets whichare contained in A.

(c) Ao = (Ac)c.

(d) (A)o = (Ao).

3. Which of the following series is(are) convergent?

(a)∑∞

n=11+2+3+...+n

2n.

(b) 1 + 12

+ 16

+ 112

+ 136

+ 172

+ . . ..

(c)∑∞

n=1π+tan−1 nn√n+n+1

.

(d)∑∞

n=1

(−1)n sin( 7nπ5

)

n7π3

.

4. Let A,B ⊆ R be nonempty sets.Define d(A,B) = inf{|x − y| : x ∈ A, y ∈ B}. Pick out the validstatement.

(a) If A and B are closed sets with A ∩B = ∅, then d(A,B) > 0.

(b) If A and B are closed sets with d(A,B) = 0, then A ∩B 6= ∅.(c) If A is closed and B is compact, where A ∩B = ∅,

then, d(A,B) = 0 MAY happen.

(d) If A is closed and B is compact, where A∩B = ∅, then, d(A,B) >0.

5. The supremum of the set { ex−e−xex+e−x

: x ∈ R} is ............

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6. The limit of the sequence { 1n

sin(log(n) + 72π)} is ..............

7. Let A ⊆ R be a nonempty set. Which of the following is a WRONGstatement?

(a) If A is a closed set, and (xn) is a Cauchy sequence in A, then (xn)must converge in A.

(b) If (xn) is a Cauchy sequence in N, then (xn) must be eventuallyconstant.

(c) If (xn) is a sequence in A such that (xn) → x, the x may not bean element of A′.

(d) If A is not a closed set, then there exists a sequence (xn) in A suchthat (xn)→ x, where x /∈ A.

8. The series∑ (−1)n

n!is

(a) not convergent.

(b) absolutely convergent.

(c) conditionally convergent.

(d) not absolutely convergent.

9. The set of all limit points of the set {n+ 12n

: n ∈ N} is

(a) N.

(b) N ∪ { 12n

: n ∈ N}.(c) the empty set ∅.(d) { 1

2n: n ∈ N}.

10. Which of the following set is not connected ?

(a) (0, 1].

(b) {x ∈ R : x2 ∈ (0, 1)}.(c) {x ∈ R : cosx < 2}.(d) {x ∈ R : 0 < 1

x< 2}.

11. The surface area of the portion of the paraboloid z = 1 + x2 + y2 thatlies below the plane z = 5 is

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(a)∫ π0

∫ 1

0

√(4r2 + 1).rdrdθ.

(b)∫ 2π

0

∫ 1

0

√(4r2 + 1).rdrdθ.

(c)∫ π0

∫ 2

0

√(4r2 + 1).rdrdθ.

(d)∫ 2π

0

∫ 2

0

√(4r2 + 1).rdrdθ.

12. Let S be the portion of circle (x − a)2 + (y − a)2 = a2 in the firstquadrant. Then

∫∫S

dxdy√(2a−x)

equals

(a) 43

√2a3/2.

(b) 83

√2a3/2.

(c) 23

√2a3/2.

(d)√23a3/2.

13. The limit lim(x,y,x)→(0,0,0)xyz

(x2+y2+z2)1/2is equal to ............

14. Consider the function

f(x, y) =

x sin( 1

x) + y sin( 1

y) if x 6= 0, y 6= 0

x sin( 1x) if x 6= 0, y = 0

y sin( 1y) if x = 0, y 6= 0

0 if x = 0, y = 0.

Then

(a) the partial derivatives of f exist at (0, 0).

(b) the function f is continuous at (0, 0).

(c) the function f is differentiable at (0, 0).

(d) the function f is not differentiable at (0, 0).

15. The volume of the solid that is below the surface z = 3x+ 2y over theregion R on the plane z = 0 bounded by the curves x = 0, y = 0 andx+ 2y = 4 is

(a) 325

.

(b) 645

.

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(c) 643

.

(d) 323

.

16. Suppose f(x, y) = 2x3 + 6xy2 − 3y3 − 150x. Then

(a) (5, 0) is a point of local maximum.

(b) (3, 4) is neither a point of local maximum nor of local minimum.

(c) (−5, 0) is neither a point of local maximum nor of local minimum.

(d) (−3,−4) is a point of local minimum.

17.The smallest subgroup of (a) containing a6 and a12 is(a) (a3).(b) (a48).(c) (a6).(d) (a24).18. Which of the following is(are) true ?(a) R∗ has no proper subgroup of finite index.(b) R∗ has exactly one proper subgroup of finite index.(c) C∗ has no proper subgroup of finite index.(d) C∗ has exactly one proper subgroup of finite index.19.Which of the following statements is true?(a) There exists a group homomorphism from Z24 onto Z6 ⊕ Z2.(b) There does not exist a group homomorphism from Z28 onto Z6.(c) There does not exist a group homomorphism from Z24 onto Z8.(d) There exists a group homomorphism φ : Z36 → Z20 such thatφ(1) = 2.20.The set of all n× n real orthogonal matrices with determinant 1 formsa group wrt matrix multiplication, denoted by SO(n,R) Which of thefollowing is true ?(a) SO(2,R) is isomorphic to {z ∈ C : |z| = 1}.(b) SO(2,R) is not isomorphic to {z ∈ C : |z| = 1}.(c) SO(2,R) is not isomorphic to the group of all rotations of the planeabout the origin.(d) SO(2,R) is not isomorphic to R/Z.

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21The number of the elements of order 5 in S7 equals .............22

Let G be the group with 101 elements. Suppose a belongs to G. Thenumber of solutions of x2 = a in G is equal to ...............23The last two digits of 131010 are .........24Suppose S is the part of cylinder z = 1 − x2, 0 ≤ x ≤ 1,−2 ≤ y ≤ 2

and, C is the boundary curve of the surface S and if−→F = yi+ yj + zk,

then |∮C

−→F .dR| equals......

25.Let S be the sphere x2 + y2 + z2 = 4 and,

−→F = 7xi − zk. Then

|∫ ∫

S=−→F .d−→S | equals

(a) 32π(b) 64π(c) 128π(d) 16π .26.Suppose S is the portion of the plane 2x − 2y + z = 1 lying in the

first octant. The flux∫ ∫

S

−→F .d−→S through S, where the normal points

upwards, is(a) 1/2.(b) 1/4.(c) 1/6.(d) 1/8.27.Consider the vector field

−→F = (2x cos y−2z3)i+(3+2yez−x2 sin y)j+

(y2ez − 6xz2)k. Then

(a) there does not exist a function f(x, y, z) such that−→F = 5f .

(b) if C1 and C2 are any two smooth curves from (1, 0, 1) to (0, 1, 0),

then∫C1

−→F .d−→r =

∫C2

−→F .d−→r .

(c) curl−→F = 0.

(d) if C is any closed smooth curve , then∮c

−→F .d−→r must be zero.

28.If the value of

∮c−y3dx+xy2dy

(x2+y2)2, where C is the ellipse x2 + 4y2 = 4, is nπ,

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then n is equal to ..............29.For the function f(x, y) = x2y , if the gradient5f at (3, 2) is (m,n),then m+ n is equal to ,,,,,,,,,,,30.Assuming continuity of all partial derivatives involved, which of thefollowing is FALSE?(a) curl grad r4 = 0.(b) div(5ϕ×5ψ) = 0.(c) curl(ϕ grad ϕ) 6= 0.

(d) div curl−→F = 0.

31.Which of the following statements is true ?(a) If A is an n× n matrix over R, then the dimension ofspan{I, A,A2...} is equal to n.(b) There exists a 5 × 5 matrix A over R such that the vector spaceM5(R) of all 5× 5 matrices over R is spanned by {I, A,A2..., A25}.(c) If A is a 5 × 5 real symmetric matrix with eigenvalues 1, 1, 1, 1, 1,then the dimension of span{I, A,A2...} is 5.(d) If A is a 5 × 5 real symmetric matrix with eigenvalue 1, 1, 1, 1, 1,then the dimension of span{I, A,A2...} is 1.32.If the system

x− 3y + 2z = 5

2x− 5y − 3z = 9

−x− y + kz = 0

has a unique solution, then k is anything other than ..........33.Let V be the vector space of all continuous functions over [0, 1]. Sup-pose T : V −→ V is a linear transformation defined by T (f(x)) =∫ 1

0f(t) sin(x− t)dt, for all f ∈ V , and for all x ∈ [0, 1]. Then

(a) the nullspace of T is the zero space.

(b) the nullspace of T is {f ∈ V :∫ 1

0f(t) cos(t)dt =

∫ 1

0f(t) sin(t)dt =

0}(c) the rangespace of T is V itself.(d) the rangespace of T is span {sinx, cosx}.34.

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Let T : R2 → R3 be a linear transformation such that T (1, 1) =(2, 1, 8), T (0,−1) = (1, 1,−5). The matrix representation of T wrt thestandard bases is

(a)

1 13 −12 5

.(b)

1 12 53 −1

.(c)

1 12 −13 5

.(d)noneofthese.

35.The dimension of the smallest subspace of R4 that contains (1, 2, 5, 1), (1, 0,−1, 2),(0, 2, 6,−1) and (1, 2, 5, 1) is .............36.Let A be a 6 × 6 matrix of rank 5 such that det(A − ωI) 6= 0, for

ω = e±2πi

3 . The rank of A2 + A+ I is ..............37.Suppose α, β, γ are the distinct roots of x3 = 1 , then the value of

∣∣∣∣∣∣α β γβ γ αγ α β

∣∣∣∣∣∣ is ...............

38.Consider the matrix

A =

1 5 −75 2 4−7 4 −3

.Then(a) there exists no nonsingular matrix P such that P−1AP is a diagonal

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matrix.(b) there exists a nonsingular matrix P such that P−1AP is a diagonal matrix.(c) there exist more than one nonsingular matrices P such that P−1AP is adiagonal matrix.(d) there exists an orthogonal matrix P such that P−1AP is a diagonalmatrix.39.Let V be the vector space of all sequences {an} in F which has only a finitelymany nonzero terms an wrt. the operations {an + bn} = {an} + {bn} and{can} = c{an} for c ∈ F. If S spans V , then(a) S may be finite.(b) S is uncountable.(c) S is at least countable infinite.(d) none of the above can be concluded.40.Suppose V is a 2-dimensional space, and T : V → V is a linear operator suchthat T 2 = 0, but T 6= 0. If v ∈ V such that Tv 6= 0 , then {v, Tv}(a) is linearly dependent(b) spans V(c) is not a basis for V(d) does not span V . 41.Consider f : R→ R defined by

f(x) = x2 − 2x, x ∈ Q,

f(x) = 3x− 6, x ∈ Qc.

(a) The function is continuous at x = 2.(b) The function is differentiable at x = 2.(c) The function is continuous at x = 3.(d) The function is differentiable at x = 3.42.Let f : R→ R be a monotone function. Then f(a) has at most finitely many discontinuities.(b) has at most countable infinitely many discontinuities.(c) may have uncountable number of discontinuities.(d) none of these.43.

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Which of the following statements is true ?(a) There exists a continuous function from [0, 1] onto R.(b) There does not exist a continuous function from R onto (0, 1).(c) There exists a continuous function f : R→ R which takes on every valuethrice.(d) There exists a continuous function f : R→ R which takes on every valueexactly twice.44.The limit limx→0

−2ln(cosx)x2

is equal to ...............45.Which of the following is always true ?(a) If f(x) and g(x) are continuous at x = a, then f(x)

g(x)is continuous at x = a.

(b) If f(x) + g(x) is continuous at x = a and f′(a) = 0, then g(x) is contin-

uous at x = a.(c) If f(x) + g(x) is differentiable at x = a, then f(x) and g(x) are bothdifferentiable at x = a.(d) There exist functions f(x) and g(x) such that f(x), f(x) + g(x) are bothcontinuous at x = a, but g(x) is discontinuous at x = a.46.Let f(x) =

∫ x0et sin tdt . The function increases in

(a) (0, π2).

(b) (0, 2π).(c) (−π

2, 0).

(d) (−2π, 0).47.The number of the values of x where the function f(x) = cos x + cos(

√2x)

attains its maximum is(a) 0.(b) 1.(c) 2.(d) infinitely many.48.Let f be a continuously differentiable function defined on the interval [0, 3]and assume that f(0) = 1, f(1) = 2 and f(3) = 2. Then(a) There exists a point c ∈ [0, 3] such that f(c) = c.(b) There exists a point c ∈ [0, 3] such that f

′(c) = 1/3.

(c) There exists a point c ∈ [0, 3] such that f′(c) = 1.

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(b) There exists a point c ∈ [0, 3] such that f′(c) = 1/4.

49.If [.] represents the greatest integer function, then

∫ 3

0[x]dx is equal to ............

50.The function f(x) = |x| + |x + 1| + |x + 2| is not differentiable at n points,where n is.................51.Consider the function f : [0, 1]→ R, defined by limm→∞limn→∞[1+cos2m(n!πx)].Then(a) f is only continuous at rational points.(b) f is only continuous at irrational points.(c) f is continuous in [0, 1].(d) f is discontinuous everywhere.52.The coefficient of x3 in the Taylor series expansion of f(x) =

√1 + x about

the point a = 0 is α16

, where α is .............

53.An integrating factor of (3xy + y2)dx+ (x2 + xy)dy = 0 is(a) x.(b) x2.(c) x3.(d) x4.54.The initial-value problem y

′=√y, y(0) = 0 has

(a) no solution.(b) a unique solution.(c) countably infinite solutions.(d) uncountably many solutions.55.Suppose a, b, c > 0 and λ ≥ 0. The solutions of ay

′+ by = ce−λx

(a) tend to 0 as x→∞ ,if λ = 0.(b) tend to c

bas x→∞ ,if λ = 0.

(c) tend to 0 as x→∞ ,if λ > 0.(d) tend to c

bas x→∞ ,if λ > 0.

56.Consider the initial-problem y

′+ 2y = f(x), where f(x) = 1 if 0 ≤ x ≤ 1,

and f(x) = 0 if 1 < x ≤ 2, y(0) = 0. Then f(3/2) equals

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(a) 1e2

sinh(1).(b) 1

e2cosh(1).

(c) 1e

sinh(1).(d) 1

ecosh(1).

57.The general solution of x2y

′′ − 2xy′+ 2y = 0, x < 0

(a) C1x+ C2x2.

(b) C1x− C2x2.

(c) −C1x+ C2x2.

(d) −C1x− C2x2.

58.The value of n for which each member of the family y = xn

6+ c intersects the

hyperbola xy = 2 orthogonally is .............59.Consider the differential equation y

′′+ 2y

′+ 5y = f(x), x ≥ 0 where f :

[0,∞)→ R is a bounded and continuous function.(a) There exists an unbounded solution of the given differential equation.(b) Each solution of the given differential equation is bounded(c) There exists a discontinuous solution of the given differential equation(d) Each solution of the given differential equation is continuous.60.If the solution y(x) of the initial-value problem y

′′+ 2y

′+ 5y = 0, y(0) =

0, y′(0) = 2 at x = 1 is enx sin(mx), then m+ n is............

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