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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
Let be a twice differentiable function. Define
f(x,y,z) = g(x2 + y2 - 2z2).
Correct Answer :- A
Explanation : f(x,y,z) = g(x2 + y2 - 2z2).
df'/dx = g'(x2 + y2 - 2z2) (2x)
df"/dx” = g"(x2 + y2 - 2z2) (4x2) + g'(x2 + y2 - 2z2)*2.........(1)
df/dy = g'(x2 + y2 - 2z2) (2y)
df"/dy” = g"(x2 + y2 - 2z2) (4y2) + g'(x2 + y2 - 2z2)*2.........(2)
df'/dz = g'(x2 + y2 - 2z2) (2y)
df"/dz” = g"(x2 + y2 - 2z2) (4z2) + g'(x2 + y2 - 2z2)*2.........(3)
Adding (1), (2) and (3)
g"(x2 + y2 - 2z2)(4x2 + 4y2 + 16z2) + g'(x2+ y2 - 2z2) (2 + 2 - 4)
= 4(x2 + y2 + 4z2) 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
If the radius of convergenceis 4, Then
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 xh yk is an I.F. of differential equation, Then given equation become exact differential equation.
xh uk+1 (1 + xy)dx + xh +1 yk (1 – xy) dy = 0
So
=> (k + 1)ykxh + (k + 2)xh + 1yk + 1
= (h + 1)xhyk - (h + 1)xk + 1yk + 1
Comparing coefficients of both the sides, we have
h – k = 0
h + k – 4
⇒ h = –2, k = – 2
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
equation of the tangent plane at point (2, 0, 1) is (x – 2) fx(2, 0, 1) + (y – 0) fy(2, 0, 1) + (z – 1) fz(0, 0, 1) = 0
Here fx(2,0,1) = 3, fy(2,0,1) =0, fz(2,0,1) = 4
so, we have
(x – 2).3 + (y – 0).0 + (z – 1)4 = 0
⇒ 3x + 4z = 10
which is required tangent plane.
By the change of order of integration
Let t = (1 – x)2
dt = –2(1 – x) dx
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
Given P is a 4 × 4 matrix with rational entries.
Let characteristic polynomial of P is
hp(x) = x4 + ax3 + bx2 + cx + d, ...(i)
where a, b, c and d are rational.
Since √2 + i is a root of (1), Then √2 − i is also a root of (1)
⇒ (x2 − 22x + 3) is a factor of (1)
Since x2 + 3 − 2√2x is factor of (1), Then
x2 + 3 + 2√2x is also a factor of (1)
⇒ (x2 + 3 − 2√2x) (x2 + 3 + 2√2x) is a factor of (1)
x4 + 6x2 + 9 – 8x2.
= x4 – 2x2 + 9 ...(ii)
From (1) & (2), we have
P4 – 2P2 + 9I = 0
⇒ P4 = 2P2 – 9I
Let be a differentiable function such that f'(x) > f(x) for all
and f(0) = 1. Then f( 1) lies in the interval
Let be defined as
f(x) = eax, a > 1 x ← R
f′(x) = aeax.
f(o) = aeao = 1 & f′(x) = aeax > eax = f(x) ∀ x ∈ R.
hence f(1) = ea
.
e < ea < ∞
⇒ f(1). lies in the interval (e, ∞)
For which one of the following values of k, the equation 2x3 + 3x2 − 12x − k = 0 has three distinct real roots?
Let g(x) = 2x3 + 3x2 – 12x.
For the roots of g(x) = 0 x(2x2 + 3x – 12) = 0
for the graph of g(x):
g′(x) = 6x2 + 6x – 12
g′(x) = 0 ⇒ x = 1, –2
function has maximum at x = –2 & minimum at x = 1 in between the roots of g(x) = 0. The graph of function shown as.
g(–2) = 2(–2)3 + 3(–2)2 – 12(–2) = 20
If y = K intersect at three distinct points with g(x), then K should be less than 20, So K = 16
For option (b);
Let vn = 1 / n3
Then,
Asis convergent, so
is convergent.
For option (c);
Let vn = 1 / n3
Then
Thenis also convergent.
For option (d)
Sois also convergent.
Hence option (a) is divergent
(By Cauchy-condensation test)
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
For the orthogonal trajectories of the family of curves,
integrating, we have
loge y = loge√x + log c1 ⇒ y = c1√x
The curve passes through the point (1, 2), then c1 = 2
Now, orthogonal trajectories in y = 2√x
At point (2, 2√2) satisfy the given condition.
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
Here x, x + ex are two linear independent
solution. so general solution can be written as
y = c1x + c2 (x + ex)
= (c1+ c2) x + c2ex
y′= (c1+ c2) + c2ex
.
y(0) = c2 = 3
y′ (0) = 4 = c1+ 2c2
⇒ c1 = 4 – 6 = –2
so, y(x) (–2 + 3) x + 3ex = 3ex+ x
y(1) = 3e + 1
The function f(x,y) = x3 + 2xy + y3 has a saddle point at
Here
fx = 3x2 + 2y ⇒ fxx = 6x, fxy = 2
fy = 2x + 3y2 ⇒ fyy = 6y
then fxx fyy – (fxy)2 = 36xy – 4
so at the point (0, 0) fxx fyy – (fxy)2 < 0
⇒ (0, 0) is a saddle point.
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
By Green's Theorem,
Now by using polar x = rcosθ, y = r sinθ. r = 0 to 2 cosθ, θ = 0 to 2θ.
we obtain the solution.
be the curve of intersection of the plane x + y + z = 1 and the cylinder x2 + y2 = 1. Then the value of
is
By stoke theorem,
Here
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
Here, {an} is a sequence of a positive real number.
The series converges if the series
converges.
Let G be a noncyclic group of order 4. Consider the statements I and II:
I. There is NO injective (one-one) homomorphism from
II. There is NO surjective (onto) homomorphism from
Then
Let G be a nonabelian group, y ∈ G, and let the maps f, g, h from G to itself be defined by f(x) = yxy-1, g(x) = x-1 and h = g ° g.
Then
Let S and T be linear transformations from a finite dimensional vector space V to itself such that S(T(v)) = 0 for all v ∈ V. Then
Let be differentiable vector fields and let g be a differentiable scalar function. Then
Consider the intervals S = (0, 2] and T = [1, 3). Let S° and T° be the sets of interior points of S and T, respectively. Then the set of interior points of S\T is equal to
Let {an} be the sequence given by
Then which of the following statements is/are TRUE about the subsequences {a6n−1} and {a6n+4}?
Let f(x) = cos(|π - x|) + (π - n) sin |x| and g(x) = x2 If h(x) = f(g(x)), then
be given by f(x) = (sin x)π − π sin x + π.
Then which of the following statements is/are TRUE?
Let {an} be the sequence of real numbers such that a1 = 1 and an+1 = an + a2n for all n > 1.
Then
Let x be the 100-cycle (1 2 3 ⋯ 100) and let y be the transposition (49 50) in the permutation group S100. Then the order of xy is ______
= (1 2 ------ 48, 50, 51 ------ 100) So the order of xy is 99
Let W1 and W2 be subspaces of the real vector space defined by
W1 = {(x1,x2, ...,x100) : xi = 0 if i is divisible by 4},
W2 = { (x1;x2, ....x100) : xi = 0 if i is divisible by 5}.
Then the dimension of W1 ∩ W2 is ____
Consider the following system of three linear equations in four unknowns x1, x2, x3 and x4
x1 + x2 + x3 + x4 = 4,
x1 + 2x2 + 3x3 + 4x4 = 5,
x1 + 3x2 + 5x3 + kx4 = 5.
If the system has no solutions, then k = _____
Let and
be the ellipse
oriented counter clockwise. Then the value of (round off to 2 decimal places) is_______________
The coefficient of in the Taylor series expansion of the function
about x = π/2, is ____________
Let be given by
Then
max {f(x): x ∈ [0,1]} - min [f(x):x ∈ [0,1]}
is ___________
Let
Then the directional derivative of f at (0, 0) in the direction of
The value of the integral (round off to 2 decimal places) is ___________
The volume of the solid bounded by the surfaces x = 1 - y2 and x = y2 - 1, and the planes z = 0 and z = 2 (round off to 2 decimal places) is
The volume of the solid of revolution of the loop of the curve y2 = x4 (x + 2) about the x-axis (round off to 2 decimal places) is ___________
The greatest lower bound of the set (round off to 2 decimal places) is ______________
Let G = : n < 55, gcd(n, 55) = 1} be the group under multiplication modulo 55. Let x ∈ G be such that x2 = 26 and x > 30. Then x is equal to _______
The number of critical points of the function is ___________
The number of elements in the set {x ∈ S3: x4 = e), where e is the identity element of the permutation group S3, is
If is an eigenvector corresponding to a real eigenvalue of the matrix
then z − y is equal to__________
Let M and N be any two 4 × 4 matrices with integer entries satisfying
Then the maximum value of det(M) + det(N) is ___________
Let M be a 3 × 3 matrix with real entries such that M2 = M + 2I, where I denotes the 3 × 3 identity matrix. If α, β and γ are eigenvalues of M such that αβγ = -4, then α+β+γ is equal to ______
Let y(x) = xv(x) be a solution of the differential equation If v(0) = 0 and v(1) = 1, then v(-2) is equal to ______
If y(x) is the solution of the initial value problem then y(ln 2) is (round off to 2 decimal places) equal to ____________
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