Let a_{1} = b_{1} = 0, and for each n ≥ 2, let a_{n} and b_{n} be real numbers given by
Then which one of the following is TRUE about the sequences {a_{n}} and {b_{n}}?
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(x^{2} + y^{2}  2z^{2}).
Correct Answer : A
Explanation : f(x,y,z) = g(x^{2} + y^{2}  2z^{2}).
df'/dx = g'(x^{2} + y^{2}  2z^{2}) (2x)
df"/dx” = g"(x^{2} + y^{2}  2z^{2}) (4x^{2}) + g'(x^{2} + y^{2}  2z^{2})*2.........(1)
df/dy = g'(x^{2} + y^{2}  2z^{2}) (2y)
df"/dy” = g"(x^{2} + y^{2}  2z^{2}) (4y^{2}) + g'(x^{2} + y^{2}  2z^{2})*2.........(2)
df'/dz = g'(x^{2} + y^{2}  2z^{2}) (2y)
df"/dz” = g"(x^{2} + y^{2}  2z^{2}) (4z^{2}) + g'(x^{2} + y^{2}  2z^{2})*2.........(3)
Adding (1), (2) and (3)
g"(x^{2} + y^{2}  2z^{2})(4x^{2} + 4y^{2} + 16z^{2}) + g'(x^{2}+ y^{2}  2z^{2}) (2 + 2  4)
= 4(x^{2} + y^{2} + 4z^{2}) g"(x^{2} + y^{2}  2z^{2})
be sequences of positive real numbers such that nan < bn < n^{2}a_{n} 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 x^{h}y^{k} 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 x^{h} y^{k} is an I.F. of differential equation, Then given equation become exact differential equation.
x^{h} u^{k+1} (1 + xy)dx + x^{h +1} yk (1 – xy) dy = 0
So
=> (k + 1)y^{k}x^{h} + (k + 2)x^{h + 1}y^{k + 1}
= (h + 1)x^{h}y^{k}  (h + 1)x^{k + 1}y^{k + 1}
Comparing coefficients of both the sides, we have
h – k = 0
h + k – 4
⇒ h = –2, k = – 2
If y(x) = λe^{2x} + e^{βx}, 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.
The value of the integral is
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 = y^{3}, 0 ≤ y ≤ 1, about the yaxis, 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) = x^{4} + ax^{3} + bx^{2} + 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)
⇒ (x^{2} − 22x + 3) is a factor of (1)
Since x^{2} + 3 − 2√2x is factor of (1), Then
x^{2} + 3 + 2√2x is also a factor of (1)
⇒ (x^{2} + 3 − 2√2x) (x^{2} + 3 + 2√2x) is a factor of (1)
x^{4} + 6x^{2} + 9 – 8x^{2}.
= x^{4} – 2x^{2} + 9 ...(ii)
From (1) & (2), we have
P^{4} – 2P^{2} + 9I = 0
⇒ P^{4} = 2P^{2} – 9I
The set as a subset of
The set as a subset of
For −1 < x < 1, the sum of the power series
Let f(x) = (ln x)^{2} , x > 0. 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
Let be defined as
f(x) = e^{ax}, a > 1 x ← R
f′(x) = ae^{ax}.
f(o) = ae^{ao} = 1 & f′(x) = ae^{ax }> e^{ax} = f(x) ∀ x ∈ R.
hence f(1) = e^{a}
.
e < ea < ∞
⇒ f(1). lies in the interval (e, ∞)
For which one of the following values of k, the equation 2x^{3} + 3x^{2} − 12x − k = 0 has three distinct real roots?
Let g(x) = 2x^{3} + 3x^{2} – 12x.
For the roots of g(x) = 0 x(2x^{2} + 3x – 12) = 0
for the graph of g(x):
g′(x) = 6x^{2} + 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
Which one of the following series is divergent?
For option (b);
Let vn = 1 / n3
Then,
Asis convergent, sois convergent.
For option (c);
Let vn = 1 / n3
Then
Thenis also convergent.
For option (d)
Sois also convergent.
Hence option (a) is divergent
(By Cauchycondensation test)
Let S be the family of orthogonal trajectories of the family of curves 2x^{2} + y^{2} = 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
log_{e} y = log_{e}√x + log c_{1} ⇒ y = c_{1}√x
The curve passes through the point (1, 2), then c_{1} = 2
Now, orthogonal trajectories in y = 2√x
At point (2, 2√2) satisfy the given condition.
Let x, x + e^{x} and 1 + x + e^{x} 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 + e^{x} are two linear independent
solution. so general solution can be written as
y = c_{1}x + c_{2} (x + e^{x})
= (c_{1}+ c_{2}) x + c_{2}e^{x}
y′= (c_{1}+ c_{2}) + c_{2}e^{x}
.
y(0) = c_{2} = 3
y′ (0) = 4 = c_{1}+ 2c_{2}
⇒ c_{1} = 4 – 6 = –2
so, y(x) (–2 + 3) x + 3e^{x }= 3e^{x}+ x
y(1) = 3e + 1
The function f(x,y) = x^{3} + 2xy + y^{3} has a saddle point at
Here
f_{x} = 3x^{2} + 2y ⇒ f_{xx} = 6x, f_{xy} = 2
f_{y} = 2x + 3y^{2} ⇒ f_{yy} = 6y
then f_{xx} f_{yy} – (f_{xy})^{2} = 36xy – 4
so at the point (0, 0) f_{xx} f_{yy} – (f_{xy})^{2} < 0
⇒ (0, 0) is a saddle point.
The area of the part of the surface of the paraboloid x^{2 }+ y^{2} + z = 8 lying inside the cylinder x^{2} + y^{2} = 4 is
be the circle (x − 1)^{2} + y^{2} = 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 x^{2} + y^{2} = 1. Then the value of
is
By stoke theorem,
Here
The tangent line to the curve of intersection of the surface x^{2} + y^{2} − 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
For , define
Then, at (0, 0), the function f is
Let {a_{n}} be a sequence of positive real numbers such that a_{1} = 1, for all n ≥ 1.
Then the sum of the series lies in the interval
Let {a_{n}} 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 (oneone) 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 {a_{n}} be the sequence given by
Then which of the following statements is/are TRUE about the subsequences {a_{6n−1}} and {a_{6n+4}}?
Let f(x) = cos(π  x) + (π  n) sin x and g(x) = x^{2} 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
Then at (0, 0),
Let {a_{n}} be the sequence of real numbers such that a_{1} = 1 and a_{n+1 }= a_{n} + a^{2}_{n} for all n > 1.
Then
Let x be the 100cycle (1 2 3 ⋯ 100) and let y be the transposition (49 50) in the permutation group S_{100}. Then the order of xy is ______
= (1 2  48, 50, 51  100) So the order of xy is 99
Let W_{1} and W_{2} be subspaces of the real vector space defined by
W_{1} = {(x_{1,}x_{2}, ...,x_{100}) : x_{i} = 0 if i is divisible by 4},
W_{2} = { (x_{1};x_{2}, ....x_{100}) : x_{i} = 0 if i is divisible by 5}.
Then the dimension of W_{1} ∩ W_{2} is ____
Consider the following system of three linear equations in four unknowns x_{1}, x_{2}, x_{3} and x_{4}
x_{1} + x_{2} + x_{3} + x_{4} = 4,
x_{1} + 2x_{2} + 3x_{3} + 4x_{4} = 5,
x_{1} + 3x_{2} + 5x_{3} + kx_{4} = 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 ___________
If then g′(1) = _______
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  y^{2} and x = y^{2}  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 y^{2} = x^{4} (x + 2) about the xaxis (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 x^{2} = 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 ∈ S_{3}: x^{4} = e), where e is the identity element of the permutation group S_{3}, 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 M^{2} = 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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