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Let f(x) = x^{2 }– 25 for all x ∈ R. The total number of points of R at which f attains a local extremum (minimum or maximum) is
The coefficient of (x – 1)^{2} in the Taylor series expansion of f(x) = xe^{x} (x ∈ R) about the point x = 1 is
For a, b, c ∈ R, if the differential equation (ax^{2} + bxy + y^{2})dx + (2x^{2} + cxy + y^{2})dy = 0 is exact, then
If f(x, y, z) = x^{2}y + y^{2}z + z^{2}x for all (x, y, z) ∈ R^{3} and then the value of
at (1, 1, 1) is
The value of the radius of convergence of f(n) = 2n, n<0 is ____________
The series is convergent if  z / 2  < 1, that is  z  < 2
Hence, ROC is  z  < 2.
Let G be a group of order 17. The total number of non isomorphic subgroups of G is
As 17 is a prime, and the order of a subgroup must divide the order of the group, G has only two subgroups, the unity subgroup and G itself. So the answer is 2.
Which one of the following is a subspace of the vector space R^{3} ?
Let T : R^{3} → R^{3} be the linear transformation defined by T(x, y, z) = (x + y, z + z, z + x) for all (x, y, z) ∈ R^{3}. Then
Let f: R → R be a continuous function satisfying for all x ∈ R . Then the set
is the interval
Directions: Q.11 – Q.35 carry two marks each.
The system of linear equations
x – y + 2z = b_{1}
x + 2y – z = b_{2}
2y – 2z = b_{3}
is inconsistent when (b_{1}, b_{2}, b_{3}) equals
be a matrix with real entries. If the sum and the product of all the eigenvalues
of A are 10 30 respectively, then a^{2} + b^{2} equals
Consider the subspace W = {(x_{1}, x_{2}, ..., x_{10}) ∈ R^{10 }: x_{ n} = x_{n–1 }+ x_{n–2} for 3 ≤ n ≤ 10} of the vector space R^{10}. The dimension of W is
Let y_{1}(x) and y_{2}(x) be two linearly independent solutions of the differential equation
x_{2} y”(x) – 2xy’(x) – 4y(x) = 0 for x ∈ [1, 10].
Consider the Wronskian W(x) = y_{1}(x)y_{2}’(x) – y_{2}(x)y_{1}’(x). If W(1) = 1, then W(3) – W(2) equals
The equation of the curve passing through the point ,(π/2, 1) and having slope at each
point (x, y) with x ≠ 0 is
The value of α ∈ R for which the curves x^{2} + αy^{2} = 1 and y = x^{2} intersect orthogonally is
Let {x_{n}} be a sequence of real numbers such that where c is a positive real number. Then then sequence {x_{n }/n} is
Let S = [0, 1] ∪ [2, 3) and let f : S → R be defined by
If T = {f(x) : x ∈ S}, then the inverse function f^{–1} : T → S
Let f(x) = x^{3} + x and g(x) = x^{3} – x for all x ∈ R. If f^{–1} denotes the inverse function of f, then the derivative of the composite function g o f^{–1} at the point 2 is
Let f : R → R be a function with continuous derivative such that f ( √2 ) = 2 and f(x) =
for all x ∈ R. Then f(3) equals
If C is a smooth curve in R^{3} from (–1, 0, 1) to (1, 1, –1), then the value of
is
Let C be the boundary of the region R = {(x, y) ∈ R^{2} : –1 ≤ y ≤ 1, 0 ≤ x ≤ 1 – y^{2}} oriented in the counterclockwise direction. Then the value of
Let G be a cyclic group of order 24. The total number of group isomorphism’s of G onto itself is
Let S_{n} be the group of all permutations on the set {1, 2, ..., n} under the composition of mappings.
For n > 2, if H is the smallest subgroup of S_{n} containing the transposition (1, 2) and the cycle (1, 2, ..., n), then
Let S be the oriented surface x^{2} + y^{2} + z^{2} = 1 with the unit normal n pointing outward. For the vector field F(x, y, z) = xi + yj + zk, the value of
Let f : (0, ∞ ) → R be a differentiable function such that f'(x^{2}) = 1  x^{3} for all x > 0 and f(1) = 0.
Then f(4) equals
Which one of the following conditions on a group G implies that G is abelian?
Let S = {x ∈ R : x^{6} – x^{5} ≤ 100} and T = {x^{2 }– 2x : x ∈ (0, ∞)}. Then set S ∩ T is
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