institute for ias/ ifos/ csir/ gate ... - maths optional

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I ns t i t u t e f or I A S / I F oS / C S I R / G AT E E x a m i na t i on s

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About the CSIR-UGC (NET) / GATEAbout the CSIRCSIR-NET Exam is mandatory for candidates aspiring to teach in various degree colleges/ universitiesin all over India. CSIR-UGC conducts JRF/NET exam twice a year in the month of June andDecember. The exam will be conducted in different subjects like Mathematical Sciences, PhysicalSciences & Life Sciences etc.

Recent changes in CSIR-UGC:v Observing that “the courts should not venture into academic field, Delhi High Court has upheld the

mandatory requirement of clearing the NET or SLET for appointment to the post of Lecturer.v The University Grants Commission (UGC) framed the Rule & Regulations – 2009 in July. Which

says that NET or SLET is mandatory for appointment of Lecturers.v Atleast 55% marks or equivalent grade is required in master degree for NET qualification.v Atleast one Professor in each Dept. in P.G. College is a requirement.v The new regulations have also created an additional post senior professor. Accordingly, the new

hierarchy in ascending order is assistant professor, associate professor, professor and senior professor.v One post of a professor in each department of the postgraduate college, and of 10% posts in an

undergraduate college shall be of those from professors only.

About the GATEGraduate Aptitude Test in Engineering (GATE) is an all India examination that primarily tests a comprehensiveunderstanding of various undergraduate subjects in Engineering and Technology. The GATE score of acandidate reflects a relative performance level in a particular paper in the exam across several years. Thescore is used for admissions to post-graduate engineering programmes (eg. M.E., M.Tech, direct Ph.D.) inIndian higher education institutes with financial assistance provided by MHRD and other Governmentagencies. The score may also be used by Public sector units for employment screening purposes.

SAMPLE QUESTIONS1. Let w1, w2, w3 be subspaces of a vector space

V such that 3 1w w⊂ . Then which one of thefollowing is correct ?(a) 1 2 3 2 1 3( )w w w w w w∩ + = + ∩

(b) 1 2 3 3 1 2( ) ( )w w w w w w∩ + = + ∩

(c) 1 2 3 2 1 3( )w w w w w w∪ + = + ∪(d) None of the above

2. Let 1 2 3, andξ ξ ξ be vectors of a vector spaceV over the field F. If r and s are arbitraryelements of F and the set { }1 2 1 2 3, ,r sξ ξ ξ ξ ξ+ + is

linearly dependent, then { }1 2 3, ,ξ ξ ξ is:(a) linearly dependent set(b) a null set(c) linearly independent set(d) None of the above

3. Consider the following vectorspaces over the reals:1. The set of all complex numbers with

usual operations2. The set of all polynomials with real

coefficients of degree 3≤3. The set of all 2 , , 4 ,x t y t z t t= = − = ∈¡4. The set of all 3×3 matrices having real

entries with usual operationsThe correct sequence of these vector spacesin decreasing order of their dimensions is:(a) 1, 2, 3, 4 (b) 2, 1, 4, 3(c) 4, 2, 1, 3 (d) 4, 3, 2, 1

4. Consider the following assertions:1. Rank (ST) = Rank S = Rank T2. Rank (ST) = Rank S, if T is non singular3. Rank (ST) = Rank T, if T is non singularwhere S, T : V V→ are linear transfor--mations of a finite dimensional vector space.Which of these is/are correct(a) 1 Only (b) 2 Only(c) 1 & 2 (d) 2 & 3

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5. Consider the linear transformation4 4:T →¡ ¡ given by

4( , , , ) ( , ,0,0) ( , , , )T x y z u x y x y z u= ∀ ∈¡Then which one of the following is correct(a) Rank of T > Nullity of T(b) Rank of T = Nullity of T = 3(c) Nullity of T > Rank of T(d) Rank of T = Nullity of T = 2

6. If V is the real vector space of all mappings

from to ¡ ¡ , { }1 ( ) ( )V f V f x f x= ∈ − =

and { }2 ( ) ( ) ,V f V f x f x= ∈ − = − thenwhich one of the following is correct(a) Neither V1 nor V2 is a subspace of V.(b) V1 is a subspace of V, but V2 is not a

subspace of V.(c) V1 is not a subspace of V, but V2 is a

subspace of V.(d) Both V1, and V2 are subspaces of V.

7. Consider the field ¡ of real numbers and thefollowing spaces:1. Set of real valued functions on [0, 1]

having discontinuity at 12

x = .

2. { }2( , ) 0x y x∈ >¡

Which of the above is/are vector spaces(s)?(a) 1 Only (b) 2 Only(c) Both 1 & 2 (d) Neither 1 nor 2

8. The determinant of the matrix

1 0 0 0 0 20 1 0 0 2 00 0 1 2 0 00 0 2 1 0 00 2 0 0 1 02 0 0 0 0 1

(a) 0 (b) –9(c) –27 (d) 1

9. The eigenvalues of the matrix

1 0 02

0 1 02

0 0 12

0 0 0

a

a

a

a

− − −

are

(a) 1 and a (b) 1 and –a

(c)12 and 2

a− (d)

12 and 2

a

10. Let 1 1 2

1 2 32 3 3

i iM i i

i i

+ − = − + + −

. If

1 1 1

2 2 2

3 3 3

,x y z

B x y zx y z

=

where 1 1

2 2

3 3

,x yx yx y

and 1

2

3

zzz

are linearly

independent eigenvectors of M, then the maindiagonal of the matrix B–1MB has(a) exactly one real entry(b) exactly two real entries(c) exactly three real entries(d) no real entry

11. Let A be a 3×3 matrix with eigenvalues 1, –1and 3. Then(a) A2 + A is non-singular(b) A2 – A is non-singular(c) A2 + 3A is non-singular(d) A2 – 3A is non-singular

12. A 2×2 real matrix A is diagonalizable if andonly if:(a) (tr A)2 < 4Det A(b) (tr A)2 > 4Det A.(c) (tr A)2 = 4Det A.(d) Tr A = Det A.

13. The minimal polynomial of the 3×3 real matrix

0 00 00 0

aa

b

is

(a) (X – a) (X – b) (b) (X – a)2 (X –b)(c) (X – a)2 (X – b)2 (d) (X – a) (X – b)2

14. Let V be the subspace of R3 spanned by(1,1,1)u = and (1,1, 1)v = − . The orthonormal

basis of V obtained by the Gram-Schmidtprocess on the ordered basis (u,v) of V is

(a)1 1 1 2 2 4, , , , ,

3 3 33 3 3 −

(b) {(1, 1, 0), (1, 0, 1)}






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(c)1 1 1 1 1 2, , , , ,3 3 3 6 6 6

−

(d)1 1 1 2 1 1, , , , ,3 3 3 6 6 6

− −

15. Let 2 /5ie πα = and the matrix

2 3 4

2 3 4

2 3 4

3 4

4

100 00 0 00 0 0 0

M

α α α αα α α α

α α αα α

α

=

Then the trace of the matrix 2I M M+ + is(a) –5 (b) 0(c) 3 (d) 5

16. Let

1 1 2 91 3 4 7

2 4 59 7 7

i ii i

Mi i

i i

+ − − = −

+ −

. Then

(a) M has purely imaginary eigen values(b) M is not diagonolizable(c) M has eigen values which are neither real

nor purely imaginary(d) M has only real eigen values

17. Which one of the following group is simple?(a) 3S (b) (2, )GL R

(c) 2 2Z Z× (d) A5

18. Let M denote the set of all 2×2 matrices overthe reals. Addition and multiplication on Mare as follows:

( ) and ( ),ij ijA a B b= = then A+B=(cij),where

,ij ij ijc a b= + and . ( )ijA B d= where

ij ijdij a b= Then which one of the following

is valid for ( ), , ?M + g(a) M is a field(b) M is an integral domain which is not a field(c) M is a commutative ring which is not an

integral domain(d) M is a non-commutative ring

19. In ( , ), n+¢ ¢ denotes the subgroup of allintegral multiples of n. If ,P j k= ∩¢ ¢ ¢ then

what is P equal to ?(a) j k (b) j + k(c) LCM of j & k (d) GCD of j & k

20. Which one of the following group is cyclic ?(a) 12 9×¢ ¢ (b) 10 85×¢ ¢

(c) 4 25 6× ×¢ ¢ ¢ (d) 22 21 65× ×¢ ¢ ¢

21. Let { }:1 55,gcd( ,56) 1G n Z n n= ∈ ≤ ≤ = bea multiplicative group modulo 56.Consider the sets

{ }1 1,9,17,25,33,41S = and

{ }2 1,15,29,43S =

which one of the following is TRUE ?(a) 1S is a subgroup of G but 2S is NOT a

subgroup of G.(b) S1 is NOT a subgroup of G but S2 is a

subgroup of G.(c) Both S1 & S2 subgroups of G(d) Neither S1 nor S2 is a subgroup of G.

22. Consider the alternating group

{ }4 4 : is an even permutationA Sσ σ= ∈

which of the following is FALSE ?(a) A4 has 12 elements(b) A4 has exactly one subgroup of order 4.(c) A4 has a subgroup of order 6.(d) Number of 3 cycles in A4 is 8.

23. G is a group of order 51. Then which one ofthe following statement is false ?(a) All proper subgroups of G are cyclic(b) If G has only one subgroup of order 3

and only one subgroup of order 17, thenG is cyclic

(c) G must have an element of order 17.(d) If G is abelian then there exists no proper

subgroup H of G such that product of allelements of H is Identity.

24. Let M3 (R) be the ring of all 3×3 real matrices.If 3, ( )I J M R⊆ are defined as

0 0 0 , , ,0 0 0

a b cI a b c R

= ∈

0 00 0 , , ,0 0

aJ b a b c R

c

= ∈






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then,(a) I is a right ideal and J a left ideal(b) I and J are both left ideals(c) I and J are both right ideals(d) I is a left ideal and J a right ideal

25. Let F4, F8 and F16 be finite fields of 4, 8 and16 elements respectively. Then,(a) F4 is isomorphic to a subfield of F8(b) F8 is isomorphic to a subfield of F16(c) F4 is isomorphic to a subfield of F16(d) none of the above

26. Let G be the group with the generators a andb given by

4 2 1, : 1,G a b a b ba a b−= = = =

If Z(G) denotes the centre of G, then / ( )G Z Gis isomorphic to(a) the trivial group(b) C2, the cyclic group of order 2(c) C2×C2 (d) C4

27. Let I denote the ideal generated by4 3 2 1x x x x+ + + + in 2[ ]Z x and 2 [ ] .F Z x I=

Then,(a) F is an infinite field(b) F is a finite field of 4 elements(c) F is a finite field of 8 elements(d) F is a finite field of 16 elements

28. Let { }0 1 2 3 0 1 2 3 3: , , ,R i j k Zα α α α α α α α= + + + ∈

be the ring of quaternions over Z3, where2 2 2 1i j k ijk= = = = − ; ; ;ij ji k jk kj i= − = = − =

.ki ik j= − = Then,(a) R is a field(b) R is a division ring(c) R has zero divisors(d) none of the above

29. Let , ,a b c ∈¢ be integers. Consider the

polynomial 5 3( ) 12 34 43 .p x x ax bx c= + + +

(a) ( )p x is irreducible over ¡ if and onlyif ( )p x is reducible over .£

(b) ( )p x is irreducible over ¡ if and onlyif ( )p x is irreducible over .¤

(c) ( )p x is irreducible over ¢ if and onlyif ( )p x is irreducible over .¤

(d) ( )p x is irreducible over ¤ if and onlyif ( )p x is irreducible over .£

30. The number of ideals in the ring 2| |

( 1)x

x −¡ is

(a) 1 (b) 2(c) 3 (d) 4

31. A cyclic group of order 60 has(a) 12 generators (b) 15 generators(c) 16 generators (d) 20 generators

32. Which of the following is false?(a) Any abelian group of order 27 is cyclic.(b) Any abelian group of order 14 is cyclic.(c) Any abelian group of order 21 is cyclic.(d) Any abelian group of order 30 is cyclic.

33. Consider the following statements:S1) Every group of prime order must be

cyclic.S2) Every group of prime order must be

abelian.S3) Every group of prime order has only one

subgroup other than itself.Which of the following is always true?(a) S1 and S2 are true but S3 is false.(b) S2 and S3 are true but S1 is false.(c) S1and S3 are true but S2 is false.(d) S1, S2 and S3 are true.(e) None of the above.

34. The number of solutions of 5 1(mod163)X ≡ in

163¢ is(a) 1 (b) 2(c) 3 (d) 4

35. Pick out the correct statements from thefollowing list:(a) A homomorphic image of a UFD

(unique factorization domain) is againa UFD.

(b) The element 2 [ 5]∈ −¢ is irreducible

in [ 5]−¢ .

(c) Units of the ring [ 5]−¢ are the units of .¢(d) the element 2 is a prime element in

[ 5].−¢36. Pick out the true statement(s):

(a) The set of all 2×2 matrices with rationalentries (with the usual operations ofmatrix addition and matrixmultiplication) is a ring which has nonontrivial ideals.






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(b) Let R=C[0,1] be considered as a ringwith the usual operations of pointwiseaddition and pointwise multipliction. Let

{ }:[0,1] | (1 2) 0 .I f f= → =¡Then I is a maximal ideal.

(c) Let R be a commutative ring and let P bea prime ieal of R. Then R/P is an integraldomain.

37. Pick out the cases where the given subgroupH is a normal subgroup of the group G.(a) G is the group of all 2×2 invertible upper

triangular matrices with real entries,under matrix multiplication, and H is thesubgroup of all such matrices (aij) suchthat a11=1.

(b) G is the group of all 2×2 invertible uppertriangular matrices with real entries,under matrix multiplication, and H is thesubgroup of all such matrices (aij) suchthat a11= a22.

(c) G is the group of all n×n invertiblematrices with real entries, under matrixmultiplication, and H is the subgroup ofsuch matrices with positive determinant.

38. Pick out the cases where the given ideal is amaximal ideal.(a) The ideal 15¢ in ¢ .

(b) The ideal { }: (0) 0I f f= = in the ring

[0,1]C of all continuous real valuedfunctions on the interval [0, 1].

(c) The ideal generated by 3 1x x+ + in the

ring of polynomials 3[ ],xF where 3F isthe field of three elements.

39. Which of the following integral domains anEuclidean domains?

(a) { }3 3 : ,a b a b − = + − ∈ ¢ ¢

(b) [ ]x¢

(c) 2 31

0

[ ]: 0n

ii

i

x x f a x x a=

= = ∈ = ∑¡ ¡

(d)[ ] [ ]

(2, )x yx

¢ where x, y are independent

variables and (2, x) is the ideal generatedby 2 and x

40. Let Q[ ] / I where IR x= is the ideal generatedby 1 + x2. Let y to the coset of x in R. Then(a) 2 1y + is irreducible over R.

(b) 2 1y y+ + is irreducible over R.

(c) 2 1y y− + is irreducible over R.

(d) 3 2 1y y y+ + + is irreducible over R.41. For the linear programming problem

Maximize 1 2 3 42 3 4z x x x x= + + −

Subject to 1 2 3 42 3 15x x x x+ − − =

1 2 3 46 3 21x x x x+ + − =

1 2 3 48 2 3 4 30x x x x+ + − =

1 2 3 4, , , , 0x x x x ≥

1 2 3 44, 3, 0, 2x x x x= = = = is(a) an optimal solution(b) a degenerate basic feasible solution(c) a non-degenerate basic feasible solution(d) a non-basic feasible solution

42. Which one of the following is TRUE ?(a) Every linear programming problem has

a feasible solution.(b) If a linear programming problem has an

optimal solution then it is unique.(c) The union of two convex sets is

necessarily convex.(d) Extreme points of the disk 2 2 1x y+ ≤ are

the points on the circle 2 2 1.x y+ =

43. The bilinear transformation 2 ( 2)w Z Z= −

maps { }: 1 1Z Z − < onto

(a) { }: Re 0w w < (b) { }: Im 0w w >

(c) { }: Re 0w w > (d) { }: 2 1w w + <

44. Let :f →£ £ be a complex valued function ofthe form ( , ) ( , ) ( , ),f x y u x y iv x y= +

Suppose that 2( , ) 3 .u x y x y=Then(a) f cannot be holomorphic on £ for any

choice of v.(b) f is holomorphic on £ for a suitable

choice of v.(c) f is holomorphic on £ for all choices of v..(d) u is not differentiable






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45. Let ,( 1)( 2)

r

rC

dzIz z z

=− −∫

where { }: , 0rC z z r r= ∈ = >£ .Then(a) 2rI iπ− if (2,3)r ∈

(b)12rI = if (0,1)r ∈

(c) 2rI iπ= − if (1,2)r ∈

(d) 0rI = if 3r >

46. Let :[0,1]n nf g R′ → be defined by2 2 1( ) (1 )n

nf x x x −= − and

2 21( )

(1 )ng xn x

=+

for .n N∈

Then, on [0, 1],

(a)1

( )nn

f x∞

=∑ converges uniformly but not

1

( )nn

g x∞

=∑

(b)1

( )nn

g x∞

=∑ converges uniformly but not

1

( )nn

f x∞

=∑

(c) both 1

( )nn

f x∞

=∑ and

1

( )nn

g x∞

=∑ convergege

uniformly

(d) neither 1 1

( ) nor ( )n nn n

f x g x∞ ∞

= =∑ ∑

converges uniformly

47. Let { } 10 : 1,2,.... .4 7

S nn

= = + U Then the

number of analytic functions which vanishonly on S is(a) inifinite (b) 0(c) 1 (d) 2

48. Consider the function

( ) ( ) ( )23( ) cos 5 sin 3 10 4f x x x x x= − + − + + − −

At which of the following points is f notdifferentiable?(a) 5x − (b) 3x =(c) 10x = − (d) 0x =

49. Which of the following is/are true?

(a)log loglog for all , 0.

2 2x y x y x y+ +

≤ >

(b) 2 for all , 0.2

x y x ye ee x y+ +

≤ >

(c)sin sinsin for all , 0.

2 2x y x y x y+ +

≤ >

(d) { }( ) max ,2

kk k

kx y x y+

≤ for all , 0x y >

and all 1.k ≥50. Which of the following is/are correct?

(a)111 as

n

e nn

− + → → ∞

(b)11 as .

1

n

e nn

− → → ∞ +

(c)2

11 as .n

e nn

+ → → ∞

(d) 211 as .

n

e nn

− → → ∞

óvvvñ






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CLASS SCHEDULEThe classroom coaching will be conducted at Mukherjee Nagar (New Delhi) Centre. Theduration of the entire course is 5 months. This particular course is punctuated for cateringthe need of students who are going to appear for CSIR-NET. For those aspirants who willappear for GATE examination, their course will also be finished within the same timeframe as they will get ample time for revision. The classes will be held in live environmentkeeping an eye on each and every individual/ aspirant so that they can get the best out ofit. The interacting classroom session will have its repercussions manifold. The timings forthe class and other requisite things will facilited to the aspirants at the time of admission.

Salient Features of Classroom Coachingv Daily 4hrs. classv Proper strategy and guidance in answer writing skill with the short-cuts.v To the point and very effective study material.v Proper coverage, conceptual and in depth analysis of the syllabus.v Individual interaction and feedback.v Every week one module-wise test equipped with doubt session, feedback and

one-to-one interaction.v Final Mock test series will be conducted for the revision.v Time-to-Time interaction with successful candidates.

ADMISSION PROCEDURE

The aspirants who want to be the part of our institute, they can take admission by depositingRs. 15000/- as registration fee to book your seat either in cash or through demand draft(DD) in favour of IMS (institute of mathematical sciences) payable at New Delhi. Thebalance fee must be submitted atleast on or before 3 days the commencement of classes.If it is possible, you can also deposite full fee through DD/Cash. Seats are limited, thereforethe candidates are requested to take admission as soon as possible and reserve theirseat in advance within the stipulated time so that they should not have to face any problemin admission. Once the seats will be filled, no admission will be entertained.


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Contact Nos.011-45629987, 09999329111, 09999197625

Email:[email protected] || Website: ims4maths.com



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