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Topic:- MATHS MA S2
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Let {xn} and {yn} be sequences of real numbers such that xn ≤ yn for all n ≥ N, where N is some positive integer. Consider the following statements:
(a) lim inf xn ≤ lim inf yn
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(b) lim sup xn ≤ lim sup yn
Which of the above statements is(are) correct?
- Neither (a) nor (b)
- Only (a)
- Only (b)
- Both (a) and (b)
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Correct Answer :-
• Both (a) and (b)
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Which of the sequences {an} and {bn} of real numbers with nth terms
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an = (n2+20n +35) sin n3 / (n2 + n + 1)
bn = 2 cosn - sin n
has(have) convergent subsequences?
- Neither {an} nor {bn}
- Only {an}
- Only {bn}
- Both {an} and {bn}
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Correct Answer :-
• Both {an} and {bn}
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Consider the following series:
(a) ∑ xn / n!, x ∈ R, n=1 to ∞
(b) ∑ 1 / (n + sinn), n=1 to ∞
(c) ∑ 1 / (n2√n), n=1 to ∞
(d) ∑ sin n, n=1 to ∞
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Which of the above series is (are) convergent?
- Only (a), (c) and (d)
- Only (a), (c) and (d)
- Only (a) and (c)
- Only (c)
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Correct Answer :-
• Only (a) and (c)
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The union of infinitely many closed subsets of the real line is
- uncountable
- finite
- always closed
- need not be closed
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Correct Answer :-
• need not be closed
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Consider the series ∑ an, where an = (2 + sin(nπ/2))nrn, r > 0. What are the values of lim inf (an+1/an) and lim sup (an+1/an)?
- r/2 and 2r
- r/3 and r
- 2r/3 and 3r/2
- 0 and 1
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Correct Answer :-
• r/2 and 2r
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Consider the following series:
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(a) ∑ 3-n sin 3nx on R, n=1 to ∞
(b) ∑ n-2xn on (-2,2), n=1 to ∞
(c) ∑ (1/n) cosnx on R, n=1 to ∞
Which of the above series converge uniformly on the indicated domain?
- Only (a) and (b)
- Only (b) and (c)
- Only (a) and (c)
- All of (a), (b) and (c)
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Correct Answer :-
• Only (a) and (c)
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Let {fn} be a sequence of continuous functions on [a, b] converging uniformly to the function f. Consider the following statements:
(a) f is bounded on [a, b]
If fn' exists and the sequence {fn'} converges uniformly to f' on [a, b], f' is the derivative of f.
Which of the following statements is(are) correct?
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- Only (a) and (b)
- Only (a) and (c)
- Only (c)
- Only (b)
Correct Answer :-
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• Only (a) and (b)
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Let G(x) be a real-valued function defined by G(x) = ∫04x2 cos √t dt. If G' is the derivative of G, then
- G'(π/2) = −4π
- G'(π/2) = −4π – 1
- G'(π/2) = -π
- G'(π/2) = 0
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Correct Answer :-
• G'(π/2) = −4π
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Let f(x) = {(4-x2)/2, |x| < 2; 0, |x| ≥ 2
Consider the following statements:
a. f is not continuous on R
b. f is continuous on R but not differentiable at x = 2, -2
c. f is differentiable on R but f' is not continuous on R
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d. f is differentiable on R and f' is continuous on R
Which of the above statements is(are) correct?
- Only (a) and (d)
- Only (b) and (c)
- Only (c)
- Only (d)
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Correct Answer :-
• Only (d)
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Let f(x) be a real-valued function defined on R lie on the interval
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- (-1, 1)
- [3, 4]
- [-2, -1]
- [-5, -3]
Correct Answer :-
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• [-2, -1]
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The Wronskian of cosx, sin x and ex at x = 0 is
- 1
- 2
- -1
- -2
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Correct Answer :-
• 2
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The solution of the initial value problem y' = 1 + y2, y(0) = 1, is:-
- y = cosec(x + π/4)
- y = tan(x + π/4)
- y = sec(x + π/4)
- y = cot(x + π/4)
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Correct Answer :-
• y = tan(x + π/4)
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How many solution(s) does the initial value problem y' - 2y = 0, y(0) = 0 have?
- No solution
- Unique solution
- Two solutions
- Infinitely many solutions
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Correct Answer :-
• Infinitely many solutions
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The general solution of the equation y'' + y = x cosx is (c1 and c2 are arbitrary constants)
- c1 cosx + c2 sin x - x cosx + sin x ln(sin x)
- c1 cosx + c2 sin x + x cosx + sin x ln(sin x)
- c1 cosx + c2 sin x - x sinx + cos x ln(sin x)
- c1 cosx + c2 sin x + x sin x + cosx ln(sin x)
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Correct Answer :-
• c1 cosx + c2 sin x + x cosx + sin x ln(sin x)
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The particular integral of the differential equation is y'' + y = x3 is
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- x2 + 6x
- x2-6x
- x3 + 6x
- x3-6x
Correct Answer :-
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• x3-6x
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The complete integral of the partial differential equation p2z2 + q2 = 1, where p = ∂z/∂x, q = ∂z/∂y is (a, b are arbitrary constants)
- z + √(z2 + a2) + a2 ln (z + √(z2 + a2) / a) = 0
- a2z + by + x2 = 0
- z + √(z2 + a2) + a2 ln (z + √(z2 + a2) / a) = 2x + 2ay + 2b
- z2 + y2 = x2 + 2x + 2ay + 2b
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Correct Answer :-
• z + √(z2 + a2) + a2 ln (z + √(z2 + a2) / a) = 2x + 2ay + 2b
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The complete integral of the partial differential equation z = px+qy - sin(pq) where p = ∂z/∂x, q = ∂z/∂y is
- z = ax + by - sm(ab)
- z = ax + y + sin b
- z = x + by - sin a
Correct Answer :-
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• z = ax + by - sin(ab)
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The partial differential equation y uxx + 2xy uxy + x uyy = ux + uy, is
- Hyperbolic in {(x,y) | 0 < xy < 1}
- Hyperbolic in {(x,y) | xy > 1}
- Elliptic in {(x, y) | xy > 1}
- Elliptic in {(x, y) | xy < 0}
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Correct Answer :-
• Hyperbolic in {(x,y) | xy > 1}
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The general solution of the equation ∂2z/∂y2 = x - y is
- (1/4) x(x - y)2 + Ø1 (x2 + y) + Ø2 (x - y)
- (1/4) x(x - y)2 + Ø1 (x + y) + Ø2 (x - y)
- Ø1 (x + y) + Ø2 (x2 - y)
- Ø1 (x2 + y) + Ø2 (x2 - y) - (1/4) x(x + y)
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Correct Answer :-
• (1/4) x(x - y)2 + Ø1 (x + y) + Ø2 (x - y)
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The general solution of ∂2u/∂t2 = c2 ∂2u/∂x2 with u(0,t) = u(2,t) = 0, u(x, 0) = sin3(πx/2) and ut(x, 0) = 0 is
- (3/4) sin(πx/2) sin(πct/2) - (1/4) sin(3πx/2) sin(3πct/2)
- (3/4) sin(πx/2) cos(πct/2) - (1/4) sin(3πx/2) cos(3πct/2)
- (3/4) cos(πx/2) sin(πct/2) - (1/4) cos(3πx/2) sin(3πct/2)
- (3/4) sin(πx/2) cos(πct/2) - (1/4) cos(3πx/2) sin(3πct/2)
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Let f: R2 → R be given by f(x) = {(x2 + y2) ln(x2 + y2), if (x, y) ≠ (0,0); 0, if (x,y) = (0,0)}
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Then,
- fxy and fyx are continuous at (0, 0), and fxy(0,0) = fyx (0,0)
- fxy and fyx are discontinuous at (0, 0), but fxy (0,0) = fyx (0,0)
- fxy and fyx are continuous at (0, 0), but fxy (0,0) ≠ fyx (0,0)
- fxy and fyx are discontinuous at (0, 0) and fxy(0,0) ≠ fyx (0,0)
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Correct Answer :-
• fxy and fyx are discontinuous at (0, 0), but fxy (0,0) = fyx (0,0)
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The directional derivative of f(x,y,z) = xy2 + yz2 + zx2 defined on R3 along the tangent to the curve x = t, y = t2, z = t3 at the point (1, 1, 1) is
- 18/√14
- 13/√14
- 13/√14
- 18/√14
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Correct Answer :-
• 18/√14
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The unique polynomial of degree 2 passing through (1, 1), (3, 27) and (4, 64) obtained by Lagrange interpolation is
- 8x2-17x + 12
- 8x2-19x- 12
- 8x2+14x- 12
- 8x2-19x + 12
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Correct Answer :-
• 8x2-19x + 12
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The approximate value of ∫01 dx/(1+x)2 by Simpson's 1/3-rd rule, using the least number of equal subintervals, is
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- 0.8512
- 0.8125
- 0.7625
- 0.6702
Correct Answer :-
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• 0.8512
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Consider the differential equation, dy/dx = y - x, y(0) = 2. The absolute value of the difference in the solutions obtained by Euler method and Runge-Kutta second order method at y(0.1) using step size 0.1 is
- 2.205
- 2.252
- 0.005
- 0.055
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Correct Answer :-
• 0.005
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The approximate value of (17)1/3 obtained after two iterations of Newton-Raphson method starting with initial approximation x0 = 2 is
- 2.7566
- 2.5826
- 2.6713
- 2.4566
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Correct Answer :-
• 2.5826
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For an infinite discrete metric space (X, d), which of the following statements is correct?
- x is compact
- For every A ⊆ X, A° ∪ Ā = X, where A and A° denote respectively the closure and interior of A in X
- x is connected
- x is not totally bounded
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Correct Answer :-
• y is not totally bounded
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Consider the metric space (l2,d) of square summable sequences with the Euclidean metric. Y = {e1, e2, ...} ⊆ l2 where ei is the sequence of 1 at the i – th place and 0 elsewhere. Then,
- y is not compact and has no limit point
- y is compact and each ei is a limit point of y
- y is not compact and has a limit point
- y is compact and has no limit point
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Correct Answer :-
• y is not compact and has no limit point
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Let C[0, 1] be the set of real valued continuous functions on [0, 1] with sup-metric. Let A = {f ∈ C[0, 1] | f(0) = 0} and B = {f ∈ C[0, 1] | f(0) > 0} be the subspaces of C[0, 1]. Then,
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- Both A and B are complete
- A is complete but B is incomplete
- A is incomplete but B is complete
- Neither A nor B is complete
Correct Answer :-
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• A is complete but B is incomplete
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Let (R, d) and (R, u) be the metric spaces with the discrete metric space d and usual metric u respectively. Let f: (R, d) → (R, u) and g: (R, u) → (R, d) be the functions given by f(x) = g(x) = {x + 1, x ≤ 0; x > 0}
Then,
- Both f and g are continuous
- Neither f nor g is continuous
- f is continuous but g is not
- g is continuous but f is not
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Correct Answer :-
• f is continuous but g is not
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Let Y1 = {(x,y) ∈ R2 | y = sin(1/x), 0 < x ≤ π} and Y2 = {(0, y) ∈ R2 | y ∈ [-2,2]} be subspaces of the metric space (R2) being the Euclidean metric. For any A ⊆ R2, Ā denotes the closure of A
- Y1 ∪ Y2 is connected
- Y1 ∪ Y2 is connected
- Y1 ∩ Y2 is disconnected
- Y1 ∩ Y2 is a non-empty bounded subset of R2
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Correct Answer :-
• Y1 ∪ Y2 is connected
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Let R be the set of all real-valued Riemann integrable functions on and let f be the function given by f(x) = {0 if x = 0; 1/(n+1) if 1/(n+1) < x ≤ 1/n for n ∈ N}
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Which of the following statements is correct?
- f is monotonically decreasing on [0, 1] but f ∉ R[0, 1]
- f is monotonically decreasing on [0, 1] and f ∈ R[0, 1]
- f is discontinuous at infinitely many points in [0, 1] but f ∉ R[0, 1]
- f is discontinuous at infinitely many points in [0, 1] and f ∈ R[0, 1]
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Correct Answer :-
• f is discontinuous at infinitely many points in [0, 1] and f ∈ R[0, 1]
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The improper integral ∫-∞∞ dx/(x2+1)
- Converges to π
- Converges to π/2
- Converges to 0
- Diverges
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Correct Answer :-
• Converges to π
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Consider the functions f(x) = (x2-1)/(x-1) and g(x) = |x-1|/(x-1), x ≠ 1. Then
- Both f and g have removable discontinuity at x = 1
- f has a removable discontinuity at x = 1, while g has a jump discontinuity at x = 1
- f has a jump discontinuity at x = 1 while g has a removable discontinuity at x = 1
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Correct Answer :-
• f has a removable discontinuity at x = 1, while g has a jump discontinuity at x = 1
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What is the length of the interval on which the function f(x) = x3 - 6x2 + 15x + 8 is decreasing?
- 8
- 6
- 4
- 2
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Correct Answer :-
• 6
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Let f: [a, b] → R be a monotonic function. Consider the following statements:
a. The function f obeys the maximum principle
b. The function f is Riemann integrable on [a, b]
Which of the above statement(s) is (are) true?
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- Only (a)
- Only (b)
- Both (a) and (b)
- Neither (a) nor (b)
Correct Answer :-
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• Both (a) and (b)
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Consider the following:
a. ((a,b), (c, d)) = ac - bd, (a, b), (c, d) ∈ R2
b. (f(x), g(x)) = ∫01 f'(x)g(x) dx, where f(x), g(x) are polynomials over R
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Which of the above is(are) an inner product?
- Neither (a) nor (b)
- Both (a) and (b)
- Only (a)
- Only (b)
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Let T = ((0,1), (1,0), (0,0), (0,2)). Then T3 + 4T2 + 5T - 2I is equal to
- 10T +4I
- 10T-4I
- -10T +4I
- -10T-4I
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Correct Answer :-
• 10T-4I
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Let V be an infinite dimensional vector space over a field F.
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Consider the following statements:
a. Any one-one linear transformation from V to itself is onto
b. Any onto linear transformation from V to itself must be one-one
Which of the above statements is (are) correct?
- Both (a) and (b)
- Only (a)
- Only (b)
- Neither (a) nor (b)
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Correct Answer :-
• Neither (a) nor (b)
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Let Pn(R) be the set of all polynomials over R of degree at most n. Let T: Pn(R) → Pn+1(R) be given by T(f(x)) = xf(x). Then
- T is one-one and onto linear transformation
- T is an onto function but neither a linear transformation nor one-one
- T is not onto but a one-one linear transformation
- T is one-one but neither a linear transformation nor onto
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Correct Answer :-
• T is not onto but a one-one linear transformation
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Let Z is the set of integers. The inverse of a is
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- a-6
- a-4
- 4-a
- 6-a
Correct Answer :-
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• 4-a
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Let G be a group of even order. Suppose that exactly half of G consists of elements of order 2 and the rest forms a subgroup H of G. Which of the following statements is incorrect?
- H is a normal subgroup of G
- Order of H is even
- H is abelian
- G: H = 2
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Correct Answer :-
• Order of H is even
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Let G and K be finite groups such that |G| = 21 and |K| = 49. Suppose G does not have a normal subgroup of order 3. Let be the set of all group homomorphism from G to K. Then the number of elements in is
- 1
- 3
- 5
- 7
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Correct Answer :-
• 1
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Let G be a finite group of a ∈ G has exactly two conjugates. Suppose that C(a) = {x-1ax | x ∈ G} and N(a) = {x ∈ G | ax = xa}.
Which of the following statements is incorrect?
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- The number of elements in C(a) is a prime number
- G is a simple group
- N(a) is a subgroup of G
Correct Answer :-
• G is a simple group
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Let G be a finite group of order 385. Let H, K and L be p-Sylow subgroups of G for p = 5,7 and 11, respectively. Which of the following statements is incorrect?
- K is a normal subgroup of G
- L is normal subgroup of G
- HK is a non-abelian subgroup of G
- G = HKL
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Correct Answer :-
• HK is a non-abelian subgroup of G
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The remainder when 20202020 is divided by 12 is
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- 0
- 2
- 4
- 8
Correct Answer :-
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• 4
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The smallest integer a > 2 such that 2|a, 3|(a+1), 4|(a + 2), 5|(a + 3) and 6|(a + 4) is
- 14
- 56
- 122
- 62
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Correct Answer :-
• 62
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Let R = {(a, b) | a, b ∈ Z} be a ring and f: R → Z be given by f((a, b)) = a - b. Which of the following statements is incorrect?
- f is a ring homomorphism
- ker f is a prime ideal but not maximal
- ker f is maximal ideal
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Consider the following statements
a. A polynomial is irreducible over a field F if it has no zeros in F
b. Let f(x) ∈ Z[x]. If f(x) is reducible over Q, then it is reducible over Z
c. For any prime p, the polynomial xp-1 + xp-2 + ... + x +
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This download link is referred from the post: DUET Last 10 Years 2011-2021 Question Papers With Answer Key || Delhi University Entrance Test conducted by the NTA
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