DU MA MSc Mathematics
Topic:- DU_J18_MA_MATHS_Topic01
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The complete integral of the partial differential equation xpq + yq² - 1 = 0 where p = dz/dx and q = dz/dy is
- (z + b)² = 4(ax + y).
- z + b = 2(ax + y).
- z + b = 4(ax + y)².
- z + b = 2(ax + y)².
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Correct Answer :-
(z + b)² = 4(ax + y).
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Let P be the set of all the polynomials with rational coefficients and S be the set of all sequences of natural numbers. Then which one of the following statements is true?
- S is countable but P is not.
- Both the sets P and S are uncountable.
- Both the sets P and S are countable.
- P is countable but S is not.
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Correct Answer :-
P is countable but S is not.
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For the differential equation
x dy/dx + 6y = 3xy4/3
consider the following statements:
(i) The given differential equation is a linear equation.
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(ii) The differential equation can be reduced to linear equation by the transformation V = y-1/3.
(iii) The differential equation can be reduced to linear equation by the transformation V = x-1/3.
Which of the above statements are true?
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Which one of the following statements is not true for Simpson's 1/3 rule to find approximate value of the definite integral I = ? f(x)dx?
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If y0 = f(0), y1 = f(0.5), y2 = f(1), the approximate value of I is 1/3 [y0 + 3y1 + y2].
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The approximating function has odd number of points common with the function f(x).
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Simpson's 1/3 rule improves trapezoidal rule.
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The function f(x) is approximated by a parabola.
Correct Answer :-
If y0 = f(0), y1 = f(0.5), y2 = f(1), the approximate value of I is 1/3 [y0 + 3y1 + y2].
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The equation of the tangent plane to the surface z = 2x² - y² at the point (1, 1, 1) is
- x - y - 2z = 2.
- 4x - y - 3z = 1.
- 2x - y - 2z = 1.
- 4x - 2y - z = 1.
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Correct Answer :-
4x - 2y - z = 1.
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If {x,y} is an orthonormal set in an inner product space then the value of ||x - y|| + ||x + y || is
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- 2v2.
- 2 + v2.
- v2.
- 2.
Correct Answer :-
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2v2.
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Which one of the following spaces, with the usual metric, is not separable?
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The space C[a, b] of the set of all real valued continuous functions defined on [a, b].
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The space l8 of all bounded real sequences with supremum metric.
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The Euclidean space Rn.
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The space l¹ of all absolutely convergent real sequences.
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Correct Answer :-
The space l8 of all bounded real sequences with supremum metric.
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Let G be an abelian group of order 2018 and f: G? G be defined as f(x) = x5. Then
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- f is not injective.
- f is not surjective.
- there exists e ? x ? G such that f(x) = x-1.
- f is an automorphism of G.
Correct Answer :-
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f is an automorphism of G.
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If f: R ? R is a continuous function such that
f(x + y) = f(x) + f(y), for all x, y ? R,
then
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- f is increasing if f (1) = 0 and decreasing if f (1) = 0.
- f is increasing if f (1) = 0 and decreasing if f (1) = 0.
- f is a not an increasing function.
- f is neither an increasing nor a decreasing function.
Correct Answer :-
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f is increasing if f (1) = 0 and decreasing if f (1) = 0.
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The central difference operator d and backward difference operator ? are related as
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- d = v(1 - ?)½.
- d = (1 + ?)½.
- d = v(1 - v)-½.
- d = v(1 + ?)-½.
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Correct Answer :-
d = v(1 - v)-½.
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How many continuous real functions f can be defined on R such that (f(x))² = x² for every x?R?
- Infinitely many.
- None.
- 4.
- 2.
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Correct Answer :-
4.
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The greatest common divisor of 11 + 7i and 18-i in the ring of Gaussian integers Z[i] is
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- 3i.
- 1.
- 1+i.
- 2+i.
Correct Answer :-
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1.
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The complete integral of the partial differential equation
?²z/?x² - 2 ?²z/?x?y + ?²z/?y² = ex+2y
is
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- f1(y - x) + xf2(y + x) + ex+2y.
- f1(y + x) + xf2(y + x) + xex+2y.
- f1(y - x) + f2(y + x) + ex+2y.
- f1(y + x) + xf2(y + x) + ex+2y.
Correct Answer :-
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f1(y + x) + xf2(y + x) + ex+2y.
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If S = {(1, 0, i), (1, 2, 1)} ? C³ then S? is
- span {(i, -½(i + 1), -1)}.
- span {(-i, -½(i + 1), 1)}.
- span {(i, -½(i + 1), 1)}.
- span {(i, ½(i + 1), -1)}.
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Correct Answer :-
span {(i, -½(i + 1), 1)}.
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The improper integral ?-88 2-x²dx is
- convergent and converges to 2.
- divergent.
- convergent and converges to 1/ln2.
- convergent and converges to -ln2.
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Correct Answer :-
convergent and converges to 1/ln2.
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Let f: R ? R be a continuous function which takes irrational values at rational points and rational values at irrational points. Then which one of the following statements is true?
- f is uniformly continuous on Q.
- f is uniformly continuous on R.
- f is uniformly continuous on Qc.
- No such function exists.
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Correct Answer :-
No such function exists.
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If f: [0,10] ? R is defined as
f(x) = { 0, 0 = x < 2, 1, 2 = x = 5 0, 5 < x = 10,
and F(x) = ?0x f(t)dt then
- F(x) = 3 for x = 5.
- F'(x) = f(x) for every x.
- F is not differentiable at x = 2 and x = 5.
- F is differentiable everywhere on [0, 10].
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Correct Answer :-
F is not differentiable at x = 2 and x = 5.
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The Maclaurin series expansion
ln(1 + x) = x - x²/2 + x³/3 ...
is valid
- only if x ? [-1,1].
- if x > -1.
- only if x ? (-1,1].
- for every x ? R.
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Correct Answer :-
only if x ? (-1,1].
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If 4x = 2(mod 6) and 3x = 5(mod 8) then one of the value of x is
- 32
- 34
- 26
- 23
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Correct Answer :-
23
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If f(x) = limn?8 Sn(x), where
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Sn(x) = x/(x+1)(2x+1) + x/(2x+1)(3x+1) + ... + x/(nx+1)((n+1)x+1)
then the function f is
- continuous nowhere.
- continuous everywhere.
- continuous everywhere except at countably many points.
- continuous everywhere except at one point.
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Correct Answer :-
continuous everywhere except at one point.
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The rate of change of f(x, y) = 4y - x² at the point (1, 5) in the direction from (1, 5) to the point (4, 3) is
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- -6/v5
- -14/v13
- -12/v5
- -19/v13
Correct Answer :-
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-14/v13
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Let G = {a1, a2,...., a25} be a group of order 25. For b, c ? G let
bG = {ba1, ba2,..., ba25}, Gc = {a1c, a2c,...., a25c}.
Then
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- bG = Gc only if b = c.
- bG ? Gc ? b, c ? G.
- bG = Gc only if b-1 = c.
- bG ? Gc, if b ? c.
Correct Answer :-
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If (xn) is a sequence such that xn = 0, for every n ? N and if limn?8((-1)nxn) exists then which one of the following statements is true?
- The sequence (xn) is a Cauchy sequence.
- The sequence (xn) is not a Cauchy sequence.
- The sequence (xn) is unbounded.
- The sequence (xn) is divergent.
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Correct Answer :-
The sequence (xn) is a Cauchy sequence.
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If n > 2, then n5 - 5n³ + 4n is divisible by
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- 80
- 120
- 100
- 125
Correct Answer :-
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120
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Let
S = nn=18 [2 - 1/n, 3 + 1/n].
Then S equals
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- (2, 3].
- [2, 3].
- [2, 3).
- (2, 3).
Correct Answer :-
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[2, 3].
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If an = nsin(np/2) then
- lim sup an = +8, lim inf an = -1.
- lim sup an = +8, lim inf an = 0.
- lim sup an = +8, lim inf an = -8.
- lim sup an = 1, lim inf an = -1.
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Correct Answer :-
lim sup an = +8, lim inf an = 0.
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Let f: R²? R be defined as f (x, y) = |x| + |y|. Then which one of the following statements is true?
- f is continuous at (0, 0) and fx(0,0) ? fy(0,0).
- f is continuous at (0, 0) and fx(0,0) = fy(0,0).
- f is discontinuous at (0, 0) and fx(0,0) = fy(0,0).
- f is continuous at (0, 0) but fx and fy does not exist at (0, 0).
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Correct Answer :-
f is continuous at (0, 0) but fx and fy does not exist at (0, 0).
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Let A and B be two subsets of a metric space X. If intA denotes the interior A of then which one of the following statements is not true?
- A ? B ? intA ? intB.
- int(A ? B) = intA ? intB.
- int(A n B) = intA n intB.
- int(A ? B) ? intA ? intB.
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Correct Answer :-
int(A ? B) = intA ? intB.
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Which one of the following statements is false?
- A subring of a field is a subfield.
- A subring of the ring of integers Z, is an ideal of Z.
- A commutative ring with unity is a field if it has no proper ideals.
- A field has no proper ideals.
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Correct Answer :-
A subring of a field is a subfield.
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Let s = (37125)(43216) ? S7, the symmetric group of degree 7. The order of s is
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- 7
- 4
- 5
- 2
Correct Answer :-
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4
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Let
S = nn=18 [0, 1/n].
Then which one of the following statements is true?
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- inf S > 0.
- sup S=1 and inf S = 0.
- sup S > 0.
- sup S = inf S = 0.
Correct Answer :-
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sup S = inf S = 0.
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The characteristics of the partial differential equation
36 ?²z/?x² - 14 x12 ?²z/?x?y - 8 ?²z/?y² = 0
when it is of hyperbolic type are given by
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- x + 36/y6 = C1, x - 36/y6 = C2.
- x + 1/y6 = C1, x - 1/y6 = C2.
- x + 36/y7 = C1, x - 36/y7 = C2.
- x + 36/y7 = C1, x + 36/y7 = C2.
Correct Answer :-
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x + 1/y6 = C1, x - 1/y6 = C2.
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A bound for the error for the trapezoidal rule for the definite integral ?01 1/(1+x²) dx is
- 1/6
- 2/25
- 1/15
- 1/20
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Correct Answer :-
1/6
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Exact value of the definite integral ?01 f(x)dx using Simpson's rule
- cannot be given for any polynomial.
- is given when f (x) is a polynomial of degree 4.
- is given when f (x) is a polynomial of degree 5.
- is given when f (x) is a polynomial of degree 3.
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Correct Answer :-
is given when f (x) is a polynomial of degree 3.
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Let p be a prime and let G be a non-abelian p-group. The least value of m such that pm|(G/Z(G)) is
- 0
- 1
- 3
- 2
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Correct Answer :-
0
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If f is Euler's Phi function then the value of f(720) is
- 248
- 144
- 192
- 72
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Correct Answer :-
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The total number of arithmetic operations required to find the solution of a system of n linear equations in n unknowns by Gauss elimination method is
- 2/3 n³ + 1/2 n² - 5/6 n.
- 1/3 n³ + 1/2 n² - 1/6 n.
- 2/3 n³ + 3/2 n² - 7/6 n.
- 1/3 n³ + 3/2 n² - 5/6 n.
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Correct Answer :-
2/3 n³ + 3/2 n² - 7/6 n.
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If (xn) is a sequence defined as
xn = [(5+n)/2n] for every n ? N
where [.] denotes the greatest integer function then limn?8 xn
- 1.
- 1/2
- does not exist.
- 0.
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Correct Answer :-
0.
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Let R be a ring with characteristic n where n = 2. If M is the ring of 2 × 2 matrices over R then the characteristic of M is
- 1.
- 0.
- n - 1.
- n.
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Correct Answer :-
n.
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If A = [2 1; a b] is a matrix with eigen values v6 and -v6, then the values of a and b are respectively,
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- 2 and -1.
- 2 and -2.
- 2 and 1.
- -2 and 1.
Correct Answer :-
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2 and -2.
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The dimension of the vector space of all 6 × 6 real skew-symmetric matrices is
- 36
- 21
- 30
- 15
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Correct Answer :-
15
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Let (x0, f(x0)) = (0,1), (x1, f(x1)) = (1, a) and (x2, f (x2)) = (2,b). If the first order divided differences f[x0, x1] = 5 and f[x1, x2] = c and the second order divided difference f[x0, x1, x2] = -3/2, then the values of a, b and c are
- 4, 2, 4.
- 2, 4, 6.
- 4, 6, 2.
- 6, 2, 4.
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Correct Answer :-
4, 6, 2.
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Let the polynomial f(x) = 3x5 + 15x4 - 20x3 + 10x + 20 ? Z[x], and f0(x) be the polynomial in Z3[x] obtained by reducing the coefficients of f(x) modulo 3. Which one of the following statements is true?
- f(x) is reducible over Q, f0(x) is reducible over Z3.
- f(x) is irreducible over Q, f0(x) is reducible over Z3.
- f(x) is reducible over Q, f0(x) is irreducible over Z3.
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