Exam Details
| Subject | mathematics | |
| Paper | ||
| Exam / Course | m.sc | |
| Department | ||
| Organization | central university | |
| Position | ||
| Exam Date | 2017 | |
| City, State | telangana, hyderabad |
Question Paper
1. Let S ... Define a binary operation on S by a remainder when ab is divided by 13 (remainder when x is divided by 13 is the unique number y in S such that x is divisible by 13). Then the smallest n such that 2^n 1 in S is
3.
4.
6.
12.
2. Let z be elements in a group such that z y3 x2y^-3. The order of x is 12. Then the order of z is
2.
3.
6.
12.
3. Let Q be the set of rationals and define a binary operation on Q by a b a b -ab. Then
is a group.
(S. is a group for some subset S c Q.
is not associative.
has no identity.
4. An element a in a group has order 100. Then the order of a^65 is
5.
10.
20.
25.
5. Consider the following two statements.
S1 There exists a 3 x 3 matrix with integer entries with eigenvalues 1. w^2 (cube roots of 1).
S2: A 2 x 2 matrix with eigenvalues w and w^2 cannot have all real entries. Then
both S1 and S2 are true.
S1 is true but ,S2 is false.
S1 is false but S2 is true.
both S1 and S2 are false.
6. Let
3 6 12 6 9
0 1 1 0 and 0
1 0 2 2 1
2 0 4 2
be five vectors in R^4. The maximum number of linearly independent vectors among them is
2.
3.
4.
5.
7. The determinant of A where
0 0 0 a b
0 0 0 c d
A 0 0 0 e f
p q r s t
v w x y z
aderw.
ad(pw
ad -be.
0.
8. Let X1, X2, X3 be three non-zero vectors in R^3 such that Xi.Xj when i j. Then
such vectors cannot exist.
such vectors form an orthonormal basis of R63
the vectors must have integer coordinates.
the vectors must be linearly independent.
9. Let f R be continuous. Consider the following statements
S1: If f 0 and Integral(f(X))dx 0 then limits a to b
S2 If Integral(f(x)dx) V c then f 0. Then
both S1 and S2 are true.
S1 is true but S2 is false.
S1 is false but S2 is true.
both S1 and S2 are false.
10. Consider the following statements
S1 A bounded sequence of real numbers must have a convergent subsequence.
S2 A Cauchy sequence of real numbers must be bounded.
Then
both S1 and S2 are true.
S1 is true but S2 is false.
S1 is false but S2 is true.
both S1 and S2 are false.
11. Let f R R be a continuous and non-negative function. Consider the following statements:
S1: If J E such that 100 then Integral(f(x)) dx limits 0 to 1
S2: If Integral(f(x)dx)> limits 0 to 1 then 1/2 for some c E
S3: If Integral(f(x)dx)> limits 0 to 1 then 3c E such that 1/2. Then
only S1 is false.
only S2 is false.
only S3 is false.
S1, S2 and S3 are false.
12. Let X n Q and Y n Q. Then
every function f X Y fails to be injective.
every function f Y X fails to be surjective.
every function f X Y satisfies for all x E X.
none of the above.
13. Which of the following sets is countable?
The set of all rational numbers Q.
The set of all subsets of N.
The set of all real numbers.
The set of irrational numbers.
14. If f R R. is a cubic polynomial then which of the following is an empty set?
E R. Sqrt(2)}.
E R. all m,n E N}.
E R. for some y E R where is the derivative of f·
E R. O}.
15. Consider the statement: If f R R is a polynomial of even degree then it is not surjective. Which of these would follow from this?
If f R R is a surjective polynomial function, then its degree should be odd.
If I R R is a non-surjective polynomial function, then its degree can be any integer.
All non-surjective functions have to be polynomials of even degree.
If f C is a polynomial of even degree, then it is not surjective.
16. If the line intersects the planes 6x 4y -5z 4 and x -5y az 12 in the same point then a is
-2.
0.
2.
4.
17. The equation of the plane through the points and parallel to the x-I y-1 z-2 line (x-1)/1 is
4x -2z 0.
x 0.
none of the above.
18. Let R be a differentiable function such that for all x E Then which of the following statements is False?
f is monotone.
f is one-one.
f is uniformly continuous.
None of these.
19. Let x E Q
x Q
The function f at the point x 0
is differentiable.
has the left derivative but not the right derivative.
has the right derivative but not the left derivative.
has neither the left derivative nor the right derivative.
20. The value of integral f f x .ndS where
Y 3xyj (2xz ans S is the surface of the hemisphere x^2 y^2 z^2 16 above the xy-plane is
0.
-16T.
16T.
32T.
21. Consider f R. defined by
x E n
x n Q.
Then at x f is
continuous and Riemann integrable in every interval containing 0.
continuous and Riemann integrable in exactly one interval containing 0.
continuous but not Riemann integrable in any interval containing x.
discontinuous.
22. A ring A is called local if it has a unique maximal ideal. Which of the following is a local ring?
Z.
Z/6Z
23. Let be the set all 2 x 2 matrices with real coefficients. Then which of the following maps is a ring homomorphism?
M2 given by a.
c d
R given by 0).
0 a
given by Tr(A).
R given by det(A).
24. Which of the following limits exists in 1
lim 1/x
lim|sin
lim x sin
lim 1/x sin
25. Let a and b in R. Consider the following statements.
S1: If an b and bn a for infinitely many n E N then a b.
S2 :If an bn for infinitely many n E N then a b.
Then
both S1 and S2 are true.
S1 is true but S2 is false.
S1 is false but S2 is true.
both S1 and S2 are false.
26. Let A be the closure in R of the set 1/m k E N}. For S c let denote the set of limit points of S in R. Then
N c
N U 1/m m E c
N U 1/m m E N}U 1/m 1/k k E c mm Ill}
N U 1/m 1/k 1/l E c .
27. The equation of the sphere through the circle x^2 y^2 Z^2 2x 4y 5z 6 and touching the plane z is
x^2 y^2 z^2 -2x -4y -5z 5 0.
y^2 -2x -4y -5z 1 0.
x^2 y^2 z^2 2x 4y 5z 0.
y^2 2x 4y 5z 0.
28. The volume of revolution of the region between y x2 and y 5x when revolved about the x-axis is
5^4 T/3
5^4 T/3
29. The equation of the plane which is perpendicular to the plane 5x 3y 6z 8 0 and contains a line of intersection of the planes x +2y +3z 2x y 0 is
6x 18y -14z 1.
75x -123y 5.
51x 15y -50z 173 O.
23x -5y -50z 1.
30. Let f R be a twice differentiable function. Which of the following statements are False?
If f is bounded then is bounded.
f and are bounded then f" is bounded.
If 0 and for all x E then for all x E
If f" is a polynomial then f is a polynomial.
31. Let A be a non-empty subset of R. Which of the following statements are True?
If A is closed and dense in JR then A R.
If A is closed then A is infinite.
If A is open then A is infinite.
If A is infinite then A is either open or closed or dense in
32. Let A c R. Let A0 denote the set of interior points of A and denote the set of all limit points of A in R. Then which of the following statements are True?
If A is finite then A0 is empty.
If A is finite then is empty,
If A is countably infinite then A0 is countably infinite
If A is countably infinite then is countably infinite
33. Let f R be continuous on and twice differentiable on Let and 1. Then which of the following are always True?
there exists c E such that 1.
there exists c E such that 0.
for all c E
0.
34, Let Xn 00) be such that E xn^2 to infinity Infinity. Which of the following series are convergent?
E n=1 to infinity
E n=1 to infinity
E n=1 to infinity
E n=1 to infinity
35. Let an bn then which of the following statements are True?
If E bn converges then E an always converges.
If E bn diverges then E an always diverges.
If an 0 and E bn converges then E an always converges.
If an 0 and E bn diverges then E an always diverges.
36. Which of the following functions f R R are differentiable at x
max{|x
eX.
37. Which of the following permutations in S5 are even?
(12345).
(1234).
38. The center of GL2(R) is
a c a c E R}.
0 a
a c a
0 a
GL2(R).
E GL2(R) det(a) 1}.
39. Which of the following statements are True?
Every group whose order is the power of a prime has a non-trivial centre.
Every group of order p^2 is cyclic, where p is a prime number.
Every non-abelian group of order 6 is isomorphic to S3·
Every finite group of order n is isomorphic to a subgroup of Sn.
40. The automorphism group of Z/10Z is isomorphic to
Z/4Z.
x
Z/10Z.
S4
41. Which of the following are integral domains?
Z.
The ring of n x n matrices with entries in R
The ring of continuous functions on the unit interval E R
42. The integer 11213 is a prime number and 11213 82^2 67^2 . Which of the following statements are True in the Gaussian ring
The ideal generated by (82 is a maximal ideal.
±82 ± 67i, ±67 ± 82i are irreducible elements of
The Gaussian ring is not a PID.
67i) is isomorphic to Z/11213Z.
43. Let A be a n x n matrix with real coefficients. Then A is non-singular if and only if
for all n x p matrices rank(AB) rank(B).
A has a left inverse.
0.
the row rank of A is n.
44. If a matrix A in Mnxn where n 2 is nilpotent, then which of the following statements are True?
I A is non-singular.
If A and B commute then AB is nilpotent.
If P be any invertible matrix then PAis nilpotent.
All eigenvalues of A are zero.
45. The solution of 2xy dx dy y^2 dy 0 is
y2 Y =constant.
y^3 3ysin(x^2) =constant.
x^3 3x =constant.
x^2 x =constant.
46. The differential equation dx dy 0 is exact and has the of the form
A and 2x^2 cxy^2 where c is a constant.
A 2 and 2x^2 cxy where c is a constant.
A and 2x^2 cxy where c is a constant.
A 2 and 2x^2 cxy^2 where c is a constant.
47. Let be a sequence in R Consider the following statements:
S1: If {xn is a Cauchy sequence, a and then a b.
S2: If every subsequence {xnk has a convergent subsequence {Xnki then is convergent. Then
both S1 and S2 are true.
S1 is true but S2 is false.
S1 is false and S2 is true
both S1 and S2 are false.
48. If V1, V2, V3 are linearly dependent vectors in a vector space V over a field F. If V3 is not a linear combination of V1 and V2 then
V1 is a linear combination of V2 and V3.
dim(V) 2.
V3 must be a zero vector.
VI and V2 are linearly dependent.
49. The number of words consisting of 4 letters from the letters of word CHEESE so that no two are together is
24.
42.
88.
94.
50. Suppose a bag contains 4 white balls and 3 black balls. If two draws of 2 balls are successively made then the probability of getting 2 white balls at first draw and 2 black balls at second draw when the balls drawn at first draw are replaced is
3/7.
1/7.
19/49.
2/49.
3.
4.
6.
12.
2. Let z be elements in a group such that z y3 x2y^-3. The order of x is 12. Then the order of z is
2.
3.
6.
12.
3. Let Q be the set of rationals and define a binary operation on Q by a b a b -ab. Then
is a group.
(S. is a group for some subset S c Q.
is not associative.
has no identity.
4. An element a in a group has order 100. Then the order of a^65 is
5.
10.
20.
25.
5. Consider the following two statements.
S1 There exists a 3 x 3 matrix with integer entries with eigenvalues 1. w^2 (cube roots of 1).
S2: A 2 x 2 matrix with eigenvalues w and w^2 cannot have all real entries. Then
both S1 and S2 are true.
S1 is true but ,S2 is false.
S1 is false but S2 is true.
both S1 and S2 are false.
6. Let
3 6 12 6 9
0 1 1 0 and 0
1 0 2 2 1
2 0 4 2
be five vectors in R^4. The maximum number of linearly independent vectors among them is
2.
3.
4.
5.
7. The determinant of A where
0 0 0 a b
0 0 0 c d
A 0 0 0 e f
p q r s t
v w x y z
aderw.
ad(pw
ad -be.
0.
8. Let X1, X2, X3 be three non-zero vectors in R^3 such that Xi.Xj when i j. Then
such vectors cannot exist.
such vectors form an orthonormal basis of R63
the vectors must have integer coordinates.
the vectors must be linearly independent.
9. Let f R be continuous. Consider the following statements
S1: If f 0 and Integral(f(X))dx 0 then limits a to b
S2 If Integral(f(x)dx) V c then f 0. Then
both S1 and S2 are true.
S1 is true but S2 is false.
S1 is false but S2 is true.
both S1 and S2 are false.
10. Consider the following statements
S1 A bounded sequence of real numbers must have a convergent subsequence.
S2 A Cauchy sequence of real numbers must be bounded.
Then
both S1 and S2 are true.
S1 is true but S2 is false.
S1 is false but S2 is true.
both S1 and S2 are false.
11. Let f R R be a continuous and non-negative function. Consider the following statements:
S1: If J E such that 100 then Integral(f(x)) dx limits 0 to 1
S2: If Integral(f(x)dx)> limits 0 to 1 then 1/2 for some c E
S3: If Integral(f(x)dx)> limits 0 to 1 then 3c E such that 1/2. Then
only S1 is false.
only S2 is false.
only S3 is false.
S1, S2 and S3 are false.
12. Let X n Q and Y n Q. Then
every function f X Y fails to be injective.
every function f Y X fails to be surjective.
every function f X Y satisfies for all x E X.
none of the above.
13. Which of the following sets is countable?
The set of all rational numbers Q.
The set of all subsets of N.
The set of all real numbers.
The set of irrational numbers.
14. If f R R. is a cubic polynomial then which of the following is an empty set?
E R. Sqrt(2)}.
E R. all m,n E N}.
E R. for some y E R where is the derivative of f·
E R. O}.
15. Consider the statement: If f R R is a polynomial of even degree then it is not surjective. Which of these would follow from this?
If f R R is a surjective polynomial function, then its degree should be odd.
If I R R is a non-surjective polynomial function, then its degree can be any integer.
All non-surjective functions have to be polynomials of even degree.
If f C is a polynomial of even degree, then it is not surjective.
16. If the line intersects the planes 6x 4y -5z 4 and x -5y az 12 in the same point then a is
-2.
0.
2.
4.
17. The equation of the plane through the points and parallel to the x-I y-1 z-2 line (x-1)/1 is
4x -2z 0.
x 0.
none of the above.
18. Let R be a differentiable function such that for all x E Then which of the following statements is False?
f is monotone.
f is one-one.
f is uniformly continuous.
None of these.
19. Let x E Q
x Q
The function f at the point x 0
is differentiable.
has the left derivative but not the right derivative.
has the right derivative but not the left derivative.
has neither the left derivative nor the right derivative.
20. The value of integral f f x .ndS where
Y 3xyj (2xz ans S is the surface of the hemisphere x^2 y^2 z^2 16 above the xy-plane is
0.
-16T.
16T.
32T.
21. Consider f R. defined by
x E n
x n Q.
Then at x f is
continuous and Riemann integrable in every interval containing 0.
continuous and Riemann integrable in exactly one interval containing 0.
continuous but not Riemann integrable in any interval containing x.
discontinuous.
22. A ring A is called local if it has a unique maximal ideal. Which of the following is a local ring?
Z.
Z/6Z
23. Let be the set all 2 x 2 matrices with real coefficients. Then which of the following maps is a ring homomorphism?
M2 given by a.
c d
R given by 0).
0 a
given by Tr(A).
R given by det(A).
24. Which of the following limits exists in 1
lim 1/x
lim|sin
lim x sin
lim 1/x sin
25. Let a and b in R. Consider the following statements.
S1: If an b and bn a for infinitely many n E N then a b.
S2 :If an bn for infinitely many n E N then a b.
Then
both S1 and S2 are true.
S1 is true but S2 is false.
S1 is false but S2 is true.
both S1 and S2 are false.
26. Let A be the closure in R of the set 1/m k E N}. For S c let denote the set of limit points of S in R. Then
N c
N U 1/m m E c
N U 1/m m E N}U 1/m 1/k k E c mm Ill}
N U 1/m 1/k 1/l E c .
27. The equation of the sphere through the circle x^2 y^2 Z^2 2x 4y 5z 6 and touching the plane z is
x^2 y^2 z^2 -2x -4y -5z 5 0.
y^2 -2x -4y -5z 1 0.
x^2 y^2 z^2 2x 4y 5z 0.
y^2 2x 4y 5z 0.
28. The volume of revolution of the region between y x2 and y 5x when revolved about the x-axis is
5^4 T/3
5^4 T/3
29. The equation of the plane which is perpendicular to the plane 5x 3y 6z 8 0 and contains a line of intersection of the planes x +2y +3z 2x y 0 is
6x 18y -14z 1.
75x -123y 5.
51x 15y -50z 173 O.
23x -5y -50z 1.
30. Let f R be a twice differentiable function. Which of the following statements are False?
If f is bounded then is bounded.
f and are bounded then f" is bounded.
If 0 and for all x E then for all x E
If f" is a polynomial then f is a polynomial.
31. Let A be a non-empty subset of R. Which of the following statements are True?
If A is closed and dense in JR then A R.
If A is closed then A is infinite.
If A is open then A is infinite.
If A is infinite then A is either open or closed or dense in
32. Let A c R. Let A0 denote the set of interior points of A and denote the set of all limit points of A in R. Then which of the following statements are True?
If A is finite then A0 is empty.
If A is finite then is empty,
If A is countably infinite then A0 is countably infinite
If A is countably infinite then is countably infinite
33. Let f R be continuous on and twice differentiable on Let and 1. Then which of the following are always True?
there exists c E such that 1.
there exists c E such that 0.
for all c E
0.
34, Let Xn 00) be such that E xn^2 to infinity Infinity. Which of the following series are convergent?
E n=1 to infinity
E n=1 to infinity
E n=1 to infinity
E n=1 to infinity
35. Let an bn then which of the following statements are True?
If E bn converges then E an always converges.
If E bn diverges then E an always diverges.
If an 0 and E bn converges then E an always converges.
If an 0 and E bn diverges then E an always diverges.
36. Which of the following functions f R R are differentiable at x
max{|x
eX.
37. Which of the following permutations in S5 are even?
(12345).
(1234).
38. The center of GL2(R) is
a c a c E R}.
0 a
a c a
0 a
GL2(R).
E GL2(R) det(a) 1}.
39. Which of the following statements are True?
Every group whose order is the power of a prime has a non-trivial centre.
Every group of order p^2 is cyclic, where p is a prime number.
Every non-abelian group of order 6 is isomorphic to S3·
Every finite group of order n is isomorphic to a subgroup of Sn.
40. The automorphism group of Z/10Z is isomorphic to
Z/4Z.
x
Z/10Z.
S4
41. Which of the following are integral domains?
Z.
The ring of n x n matrices with entries in R
The ring of continuous functions on the unit interval E R
42. The integer 11213 is a prime number and 11213 82^2 67^2 . Which of the following statements are True in the Gaussian ring
The ideal generated by (82 is a maximal ideal.
±82 ± 67i, ±67 ± 82i are irreducible elements of
The Gaussian ring is not a PID.
67i) is isomorphic to Z/11213Z.
43. Let A be a n x n matrix with real coefficients. Then A is non-singular if and only if
for all n x p matrices rank(AB) rank(B).
A has a left inverse.
0.
the row rank of A is n.
44. If a matrix A in Mnxn where n 2 is nilpotent, then which of the following statements are True?
I A is non-singular.
If A and B commute then AB is nilpotent.
If P be any invertible matrix then PAis nilpotent.
All eigenvalues of A are zero.
45. The solution of 2xy dx dy y^2 dy 0 is
y2 Y =constant.
y^3 3ysin(x^2) =constant.
x^3 3x =constant.
x^2 x =constant.
46. The differential equation dx dy 0 is exact and has the of the form
A and 2x^2 cxy^2 where c is a constant.
A 2 and 2x^2 cxy where c is a constant.
A and 2x^2 cxy where c is a constant.
A 2 and 2x^2 cxy^2 where c is a constant.
47. Let be a sequence in R Consider the following statements:
S1: If {xn is a Cauchy sequence, a and then a b.
S2: If every subsequence {xnk has a convergent subsequence {Xnki then is convergent. Then
both S1 and S2 are true.
S1 is true but S2 is false.
S1 is false and S2 is true
both S1 and S2 are false.
48. If V1, V2, V3 are linearly dependent vectors in a vector space V over a field F. If V3 is not a linear combination of V1 and V2 then
V1 is a linear combination of V2 and V3.
dim(V) 2.
V3 must be a zero vector.
VI and V2 are linearly dependent.
49. The number of words consisting of 4 letters from the letters of word CHEESE so that no two are together is
24.
42.
88.
94.
50. Suppose a bag contains 4 white balls and 3 black balls. If two draws of 2 balls are successively made then the probability of getting 2 white balls at first draw and 2 black balls at second draw when the balls drawn at first draw are replaced is
3/7.
1/7.
19/49.
2/49.