Math - 2019 Past Year Paper

be sequences of positive real numbers such that nan < bn < n²a_n 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 x^hy^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 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

The value of the integral 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

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

For which one of the following values of k, the equation 2x³ + 3x² − 12x − k = 0 has three distinct real roots?

Which one of the following series is divergent?

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 + 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

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 

For , define 

Then, at (0, 0), the function f is

Let {a_n} 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 {a_n} be a sequence of positive real numbers. The series  converges if the series

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 {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) = 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 

Then at (0, 0),

Let {a_n} be the sequence of real numbers such that a₁ = 1 and a_n+1 = a_n + a²_n 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 ______

Let W₁ and W₂ be subspaces of the real vector space defined by
W₁ = {(x_1,x₂, ...,x₁₀₀) : x_i = 0 if i is divisible by 4},
W₂ = { (x₁;x₂, ....x₁₀₀) : x_i = 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 ___________

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 - 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 ____________