Let a1 = b1 = 0, and for each n ≥ 2, let an and bn be real numbers given by
Then which one of the following is TRUE about the sequences {an} and {bn}?
Let Let V be the subspace of defined by
Then the dimension of V is
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Let be a twice differentiable function. Define
f(x,y,z) = g(x2 + y2 - 2z2).
be sequences of positive real numbers such that nan < bn < n2an for
all n > 2. If the radius of convergence of the power series then the power series
Let S be the set of all limit points of the set be the set of all positive
rational numbers. Then
If xhyk is an integrating factor of the differential equation y(1 + xy) dx + x(1 — xy) dy = 0, then the ordered pair (h, k) is equal to
If y(x) = λe2x + eβx, β ≠ 2, is a solution of the differential equation
satisfying dy/dx (0) = 5, then y(0) is equal to
The equation of the tangent plane to the surface at the point (2, 0, 1) is
The area of the surface generated by rotating the curve x = y3, 0 ≤ y ≤ 1, about the y-axis, is
Let H and K be subgroups of If the order of H is 24 and the order of K is 36, then the order of the subgroup H ∩ K is
Let P be a 4 × 4 matrix with entries from the set of rational numbers. If with is a root of the characteristic polynomial of P and I is the 4 × 4 identity matrix, then
Let be a differentiable function such that f'(x) > f(x) for all and f(0) = 1. Then f( 1) lies in the interval
For which one of the following values of k, the equation 2x3 + 3x2 − 12x − k = 0 has three distinct real roots?
Let S be the family of orthogonal trajectories of the family of curves 2x2 + y2 = k, for and k > 0. If passes through the point (1, 2), then passes through
Let x, x + ex and 1 + x + ex be solutions of a linear second order ordinary differential equation with constant coefficients. If y(x) is the solution of the same equation satisfying y(0) = 3 and y'(0) = 4, then y(1) is equal to
The function f(x,y) = x3 + 2xy + y3 has a saddle point at
The area of the part of the surface of the paraboloid x2 + y2 + z = 8 lying inside the cylinder x2 + y2 = 4 is
be the circle (x − 1)2 + y2 = 1, oriented counterclockwise. Then the value of the line integral
is
be the curve of intersection of the plane x + y + z = 1 and the cylinder x2 + y2 = 1. Then the value of
is
The tangent line to the curve of intersection of the surface x2 + y2 − z = 0 and the plane x + y = 3 at the point (1, 1, 2) passes through
The set of eigenvalues of which one of the following matrices is NOT equal to the set of eigenvalues of
Let {an} be a sequence of positive real numbers such that a1 = 1, for all n ≥ 1.
Then the sum of the series lies in the interval
Let {an} be a sequence of positive real numbers. The series converges if the series