The orthogonal trajectories of the family of curves y = c1x3, where c1 is arbitary costant, is
Suppose f ; ℝ→ℝ is an odd and differentiable fraction. Then for every x0 ∈ ℝ. f'(-x0) is equal to;
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Let denote the eigenvalues of the matrix
If , then the set of possible values of t, -π ≤ t < π, is
f T = | eigen values of B are in Z}, then which of the following statement(s) is true?
Number of elements of order p in Zp2q where p and q are distinct prime is;
An object moves in the force field How much work is performed on the object moves from (2, 0) counter clockwise along the elliptical path x2 +4y2 = 4 to (0. 1), then back to (2,0) along the line segment joining the two points.
The area bounded by x2 + y2 = 25, 4y = |4 - 2| and x = 0 in the first quadrant is
Then k is
where and 'c' be the quarter circular path x2 + y2 = a2 from (a, 0) to (0, a)
Let σ be the 12- cycles (1 2 3 4 5 6 7 8 9 10 11 12) for which positive integer i is σi also a 12 cycle?
If R→R is given by f(x) = x3 + x2f'(1) + xf''(2) + f'''(3) for all x in R. then f(2) - f(1) is
Let . If f(x) is continuous in the interval [-1, 1], then p equals
If x3y2 is an integrating factor of (6y2 + axy)dx + (6xy + bx2)dy =0, a, b ∈ ℝ then
The radius of convergence of the series , where a0 = 1. an = 3-n an-1 for n ∈ ℕ, is
Let be a smooth vector function of a real variable. Consder two statements
S1 div curl = 0
S2; grad div = 0
Then
Let G be the group with the generators a and b given byG = {(a,b)|σ(a) = 4,σ(b) = 2, ba = a-1b}. Let Z(G) denotes the centre of G. Then G/Z(G) is isomorphic to.
Suppose that L(y) = y'' + a1y' + a2y = b(x), where a1, a2 are constants and b(x) is a continuous function on Then consider the statements
I. If b(x) is bounded on [0, ∞), then every solution of L(y) = b(x) is bounded on [0, ∞).
II. If b(x) → 0 as x → ∞, then every solution of L(y) = b(x) tends too as x → ∞.
Let y(x) be a non-trivial solution of the second order linear differential equation
where c < 0. k > 0 and C2 > k , then
Let P1, P2 and P3 denote. respectively, the planes defined by
a1x+b1y+c1z= = α1
a2x + b2y+c2z= α2
a3x-b3y + c3z = α3
It is given that P1, P2, P3 intersect exactly at one point when α1 = α2 = α3 = 1, If now
α1 = 2, α2 = 3 and α3 = 4 then the planes
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