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Let where b_{1} = 1, b_{2} = 1 and b_{n+2} = b_{n} + b_{n+1}, Then is:
If {v_{1}, v_{2}, v_{3}} is a linearly independent set of vectors in a vector space over then which one of the following sets is also linearly independent?
Let a be a positive real number. If f is a continuous and even function defined on the interval [–a, a], then is equal to :
The tangent plane to the surface at (1, 1, 2) is given by
In , the cosine of the acute angle between the surfaces x^{2} + y^{2} + z^{2}  9 = 0 and z  x^{2}  y^{2} + 3 = 0 at the point (2, 1, 2) is :
Let f : be a scalar field, be a vector filed and let be a constant vector. If represents the position vector then which one of the following is FALSE?
In , the family of trajectories orthogonal to the family of asteroids x^{2/3} + y^{2/3} = a^{2/3} is given by
Consider the vector space V over of polynomial functions of degree less then or equal to 3 defined on . Let T : V → V be defined by (Tf)(x) = f(x) – xf’(x). Then the rank of T is
Let Then which one of the following is True for the sequence ?
Let a, b, c . Which of the following values of a, b, c do NOT result in the convergence of the series
Let and let where Then which one of the following is true?
Suppose that f, g are differentiable functions such that f is strictly increasing and g is strictly decreasing. Define p(x) = f(g(x)) and q(x) = g(f(x)), Then, for t > 0, the sign of is :
Let Then which one of the following is true for f at the point (0, 0)?
Let a, b be a thrice differentiable function. If z = e^{u} f(v), where u = ax + by and v = ax – by, then which one of the following is true?
Consider the region D in the yz plane bounded by the line y = 1/2 and the curve y^{2} + z^{2} = 1, where y≥0. If the region D is revolved about the z axis in , then the volume of the resulting solid is :
If where C is the boundary of the triangular region bounded by the lines x = 0, y = 0 and x + y = 1 oriented in the anti clockwise direction, is :
Let U, V and W be finite dimensional real vector spaces, T : U → V, S : V → W and P : W → U be linear transformations. If range (ST) = nullspace (P), nullspace (ST) = range (P) and rank (T) = rank (S), then which one of the following is true?
Let y(x) be the solution of the differential equation dy/dx + = , for x ≥ 0, y(0) = 0, where Then y(x) =
An integrating factor of the differential equation is:
A particular integral of the differential equation is:
Let G be a group satisfying the property that f : is a homomorphism implies f(g) = 0, Then a possible group G is :
Let H be the quotient group Consider the following statements.
I. Every cyclic subgroup of H is finite.
II. Every finite cyclic group is isomorphic to a subgroup of H.
Which one of the following holds?
Let I denote the 4 × 4 identity matrix. If the roots of the characteristic polynomial of a 4 × 4 matrix M are , then M^{8} =
Consider the group under component wise addition. Then which of the following is a subgroup of ?
Let f : be a function and let J be a bounded open interval in Define
Which one of the following is false?
Let f : be defined by f ( x) = On which of the following interval(s) is f one one?
The solution(s) of the differential equation satisfying y(0) = 0 is (are)
Suppose f, g, h are permutations of the set where
f interchanges ∝ and β but fixes γ and δ
g interchanges β and γ but fixes ∝ and δ,
h interchanges γ and δ but fixes ∝ and β.
Which of the following permutations interchange(s) ∝ and δ but fix(es) β and γ?
Let P and Q be two non empty disjoint subsets of Which of the following is (are) false?
Let denote the group of non zero complex numbers under multiplication. Suppose Which of the following is (are) subgroup(s) of ?
Suppose Consider the following system of linear equations. x + y + z = α, x + βy + z = γ, x + y + αz = β. If this system has at least one solution, then which of the following statements is (are) true?
Let m, Then which of the following is (are) Not possible?
One among the following is the correct explanation of pedal equation of an polar curve, r = f (θ), p = r sin(∅) (where p is the length of the perpendicular from the pole to the tangent & ∅ is the angle made by tangent to the curve with vector drawn to curve from pole)is _______.
It is expressed in terms of p & r only
where p = r
& r = f (θ) or after solving we get direct relationship between p & r as
Which of the following subsets of is (are) connected?
Let S be a subset of ¡ such that 2018 is an interior point of S. Which of the following is (are) true?
The order of the element (1 2 3)(2 4 5)(4 5 6) in the group S_{6} is _______ .
Let Then the absolute value of the directional derivative of φ in the direction of the line at the point (1, – 2, 1) is _______ .
Let be given by
Then at the point (0, 0) is _______ .
Let for (x, y) Then f_{x}(1, 1) + f_{y}(1, 1) = _______ .
Let be continuous on and differentiable on then f(6) = _______ .
Let Then the radius of convergence of the power series about x = 0 is _______ .
Let A_{6} be the group of even permutations of 6 distinct symbols. Then the number of elements of order 6 in A_{6} is _______ .
Let W_{1} be the real vector space of all 5 × 2 matrices such that the some of the entries in each row is zero. Let W_{2} be the real vector space of all 5 × 2 matrices such that the sum of the entries in each column is zero. Then the dimension of the space is _______ .
The coefficient of x^{4} in the power series expansion of e^{sin x} about x = 0 is _______ (correct up to three decimal places).
Let where k, Then is _______ . (correct up to one decimal places).
Let be such that f" is continuous on and f(0) = 1, f'(0) = 0 and f"(0) = – 1. Then is _______ (correct up to three decimal places).
Suppose x, y, z are positive real number such that x + 2y + 3z = 1. If M is the maximum value of xyz^{2}, then the value of 1/M is _______ .
If the volume of the solid in bounded by the surfaces x = – 1, x = 1, y = – 1, y = 1, z = 2, y^{2} + z^{2} = 2 is α– π, then a = _______ .
The value of the integral is _______ . (correct up to three decimal places).
Suppose is a matrix of rank 2. Let T : be the linear transformation defined by T(P) = QP. Then the rank of T is _______ .
The area of the parametrized surface is _______ (correct up to two decimal places).
If x(t) is the solution to the differential equation satisfying x(0) = 1, then the value of x (√2 ) is _______ (correct up to two decimal places).
If y(x) = v(x) sec x is the solution of y" – (2 tan x)y' + 5y = 0, satisfying y(0) = 0 and y '(0) = √6, then v is _______ . (correct up to two decimal places).
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