Expert verified Math Q&As, notes, & guides

Recent questions

• For a > b > 0, consider D = {(x,y,z) ∈ R^3 : x^2 + y^2 + z^2 ≤ a^2 and x^2 + y^2 ≥ b^2}. Then, the surface area of the boundary of the solid 𝐷 is
• Let 𝑓: ℝ → ℝ be defined by f(x) = (x^2 + 1)^2/(x^4 + x^2 + 1) for x ∈ ℝ. Then, which one of the following is TRUE?
• Consider the following two statements. P: There exist functions 𝑓: ℝ → ℝ, 𝑔: ℝ → ℝ such that 𝑓 is continuous at x = 1 and 𝑔 is discontinuous at 𝑥 = 1 but 𝑔 ∘ 𝑓 is continuous at 𝑥 = 1. Q: There exist functions 𝑓: ℝ → ℝ, 𝑔: ℝ → ℝ such that both 𝑓 and 𝑔 are discontinuous at 𝑥 = 1 but 𝑔 ∘ 𝑓 is continuous at 𝑥 = 1. Which one of the following holds?
• For p,q,r ∈ R, r ≠ 0 and n ∈ N, let a_n = p^n·n^q(n/(n+2))^(n^2) and b_n = ((n^n)/(n! r^n))(sqrt((n+2)/n)). Then, which one of the following statements is TRUE?
• Let 𝑃_7(𝑥) be the real vector space of polynomials, in the variable 𝑥 with real coefficients and having degree at most 7, together with the zero polynomial. Let 𝑇: 𝑃_7(𝑥) → 𝑃_7(𝑥) be the linear transformation defined by T(f(x)) = f(x) + df(x)/dx. Then, which one of the following is TRUE?
• For a matrix 𝑀, let Rowspace(𝑀) denote the linear span of the rows of 𝑀 and Colspace(𝑀) denote the linear span of the columns of 𝑀. Which of the following hold(s) for all 𝐴, 𝐵, 𝐶 ∈ 𝑀_{10}(ℝ) satisfying 𝐴 = 𝐵𝐶 ?
• Let 𝑓: ℝ → ℝ be a solution of the differential equation d^2y/dx^2 - 2dy/dx + y = 2e^x for x ∈ ℝ. Consider the following statements. P: If 𝑓(𝑥) > 0 for all 𝑥 ∈ ℝ, then 𝑓′(𝑥) > 0 for all 𝑥 ∈ ℝ . Q: If 𝑓′(𝑥) > 0 for all 𝑥 ∈ ℝ, then 𝑓(𝑥) > 0 for all 𝑥 ∈ ℝ . Then, which one of the following holds?
• Consider G = {m + n·sqrt2 : m,n ∈ Z} as a subgroup of the additive group ℝ. Which of the following statements is/are TRUE?
• Let S = {f : R → R : f is polynomial and f(f(x)) = (f(x))^2024 for x ∈ R}. Then, the number of elements in S is _____________
• Let M = ([0, 0, 0, 0, -1], [2, 0, 0, 0, -4], [0, 2, 0, 0, 0], [0, 0, 2, 0, 3], [0, 0, 0, 2, 2]). If 𝑝(𝑥) is the characteristic polynomial of 𝑀, then 𝑝(2) − 1 equals _____________
• Consider the 4 × 4 matrix M = ([0, 1, 2, 3], [1, 0, 1, 2], [2, 1, 0, 1], [3, 2, 1, 0]). If a_{i,j} denotes the (i,j)^{th} entry of M^{-1}, then a_{4,1} equals _____________ (rounded off to two decimal places).
• Let 𝑃_{12}(𝑥) be the real vector space of polynomials in the variable 𝑥 with real coefficients and having degree at most 12, together with the zero polynomial. Define V = {f ∈ P_{12}(x): f(-x) = f(x) for all x ∈ R and f(2024) = 0}. Then, the dimension of V is _____________
• For 𝛼 ∈ (−2𝜋,0), consider the differential equation x^2·d^2y/dx^2 + 𝛼x·dy/dx + y = 0 for x > 0. Let 𝐷 be the set of all 𝛼 ∈ (−2𝜋,0) for which all corresponding real solutions to the above differential equation approach zero as 𝑥 → 0^+. Then, the number of elements in 𝐷 ∩ ℤ equals _____________
• Let 𝑦: ℝ → ℝ be the solution to the differential equation d^2y/dx^2 + 2dy/dx + 5y = 1 satisfying 𝑦(0) = 0 and 𝑦′(0) = 1. Then, lim_{x to infty}y(x) equals _____________ (rounded off to two decimal places).
• The area of the region R = {(x, y) ∈ R^2 : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 and 1/4 ≤ xy ≤ 1/2} is _____________ (rounded off to two decimal places).
• For 𝛼 > 0, let 𝑦_𝛼(𝑥) be the solution to the differential equation 2d^2y/dx^2 - dy/dx - y = 0 satisfying the conditions 𝑦(0) = 1, 𝑦′(0) = 𝛼. Then, the smallest value of 𝛼 for which 𝑦_𝛼(𝑥) has no critical points in ℝ equals _____________ (rounded off to the nearest integer).
• Let S = {(x,y,z) ∈ R^3 : x^2 + y^2 + z^2 = 4, (x-1)^2 + y^2 ≤ 1, z ≥ 0}. Then, the surface area of 𝑆 equals _____________ (rounded off to two decimal places).
• For 𝛼 ∈ ℝ, let 𝑦_𝛼(𝑥) be the solution of the differential equation dy/dx + 2y = 1/(1 + x^2) for x ∈ ℝ satisfying 𝑦(0) = 𝛼. Then, which one of the following is TRUE?
• Let c > 0 be such that ∫^c_0{e^{s^2}}ds = 3 Then, the value of ∫^c_0(∫^c_x e^{x^2 + y^2}dy)dx equals _____________ (rounded off to two decimal places).
• For x ∈ R, let ⌊x⌋ denote the greatest integer less than or equal to x. For x, y ∈ R, define min{x,y} = {(x, if x le y),(y, otherwise.):} Let f: [-2pi, 2pi] → R be defined by f(x) = sin(min{x, x - ⌊x⌋}) for x ∈ [-2pi, 2pi]. Consider the set S = {x ∈ [-2pi, 2pi]: f is discontinuous at x}. Which one of the following statements is TRUE?
• Let F_{11}(x) be the real vector space of polynomials, in the variable x with real coefficients and having degree at most 11, together with the zero polynomial. Let E = {s_{0}(x), s_{1}(x), . . . , s_{11}(x)}, F = {r_{0}(x), r_{1}(x), . . . , r_{11}(x)} be subsets of P_{11}(x) having 12 elements each and satisfying, s_{0}(3) = s_{1}(3), = . . . = s_{11}(3) = 0, r_{0}(4) = r_{1}(4) = . . . = r_{11}(4) = 1. Then, which one of the following is TRUE?
• Let a = [(1/sqrt{3}), (-1/sqrt{2}), (1/sqrt{6}), (0)]. Consider the following two statements. P: The matrix I_4 - aa^T is invertible. Q: The matrix I_4 - 2aa^T is invertible. Then, which one of the following holds?
• For the differential equation y(8x - 9y)dx + 2x(x - 3y)dy = 0, which one of the following statements is TRUE?
• Let V be a nonzero subspace of the complex vector space M_7(C) such that every nonzero matrix in V is invertible. Then, the dimension of V over C is
• For n ∈ N, let a_n = 1/(3n+2)(3n+4) and b_n = (n^3 + cos(3^n))/(3^n + n^3). Then, which one of the following is TRUE?
• Let G be a group of order 39 such that it has exactly one subgroup of order 3 and exactly one subgroup of order 13. Then, which one of the following statements is TRUE?
• For a twice continuously differentiable function g: R → R, define u_(g)(x, y) = 1/y int_{-y}^{y}g(x + t)dt for (x, y) in R^2, y > 0. Which one of the following holds for all such g?
• Define the sequences {a_n}_{n=3}^{infty} and {b_n}_{n=3}^{infty} as a_{n} = (log(n) + \log(log(n)))^{log(n)} and b_{n} = n^{(1 + 1/{log(n)})}. Which one of the following is TRUE?
• Let A be a 6 × 5 matrix with entries in R and B be a 5 × 4 matrix with entries in R. Consider the following two statements. P: For all such nonzero matrices A and B, there is a nonzero matrix Z such that AZB is the 6 × 4 zero matrix. Q: For all such nonzero matrices A and B, there is a nonzero matrix Y such that BYA is the 5 × 5 zero matrix. Which one of the following holds?
• Which one of the following groups has elements of order 1,2,3,4,5 but does not have an element of order greater than or equal to 6 ?
• Consider the group G = {A ∈ M_2(R): AA^{T} = I_2} with respect to matrix multiplication. Let Z(G) = {A ∈ G : AB = BA, for all B ∈ G}. Then, the cardinality of Z(G) is
• Which one of the following is TRUE for the symmetric group S_{13}?
• For a positive integer n, let U(n) = {bar r ∈ Z_n : gcd(r, n) = 1} be the group under multiplication modulo n. Then, which one of the following statements is TRUE?
• Let G be a finite group containing a non-identity element which is conjugate to its inverse. Then, which one of the following is TRUE?
• Consider the following statements. P : If a system of linear equations Ax = b has a unique solution, where A is an m×n matrix and b is an m × 1 matrix, then m = n. Q : For a subspace W of a nonzero vector space V, whenever u ∈ V \ W and v ∈ V \ W, then u + v ∈ V\W. Which one of the following holds?
• Let g: R → R be a continuous function. Which one of the following is the solution of the differential equation d^2y/dx^2 + y = g(x) for x ∈ R, satisfying the conditions y(0) = 0, y'(0) = 1 ?
• Let F be the family of curves given by x^2 + 2hxy + y^2 = 1, -1 < h < 1. Then, the differential equation for the family of orthogonal trajectories to F is
• Let y_c : R rightarrow (0, infty) be the solution of the Bernoulli’s equation dy/dx - y + y^3 = 0, y(0) = c > 0. Then, for every c > 0, which one of the following is true?
• Let y(x) be the solution of the differential equation dy/dx = 1 + y.sec{x}, for x in (-pi/2, pi/2) that satisfies y(0) = 0. Then, the value of y(pi/6) equals
• A line is passing through (α, β, γ) and its direction cosines are l, m, n then the equations of the line are -
• {d(sin^{-1}x)}/dx = __________
• If A and B are two independent events then P(A cap B) =
• If S be the sample space and E be the event then P (E) = __________
• If A, B and C are three events independent of each other then P(A cap B cap C) =
• If vec{OA} = 2vec{i}+5vec{j}-2vec{k} and vec{OB} = 3vec{i}+6vec{j}+5vec{k} then vec{AB} =
• If vec{a} = vec{i}+vec{j}+3vec{k} ; vec{b} = 2vec{i}+3vec{j}-5vec{k} then vec{a}.vec{b} =
• If vec{a} and vec{b} are mutually perpendicular then vec{a}.vec{b} =
• Let a, b, c be the direction ratios of a line then direction cosines are-
• Let l_1, m_1, n_1 and l_2, m_2, n_2 be the direction cosines of two st-lines. Both the lines are perpendicular to each other, if-
• If P(A) = 3/8 ; P(B) = 1/2 and P(A cap B) = 1/4 then P(A cup B) = __________
• |2vec{i} - 3vec{j} + vec{k}| =
• The solution of dy/dx = x/y is-
• The direction cosines of z-axis are-
• The direction ratio of the normal to the plane 7x + 4y - 2z + 5 = 0 are-
• The degree of the equation ((d^2y)/(dx^2))^3-4(dy)/(dx)=2 is
• The order of the differential equation dy/dx + 4y = 2x is-
• The position vector of the point (4, 5, 6) is
• The solution of the differential equation dy/dx = e^{x-y} is
• {d(sec(x))}/dx =
• (d)/(dx)(sin ^(-1)x+cos ^(-1)x) = ___________
• If y = sin(log(x)), then dy/dx = ___________
• If y = x^5 then dy/dx = ________
• int 0 dx = _________
• int x^5 dx = _________
• int_a^b x^3 dx = _________
• int dx/x = _________
• vec{j} × vec{k} =
• {d(sin(x))}/dx =
• vec{k} \cdot vec{k} =
• If A = [[9, 10, 11], [12, 13,14]] and B = [[11,10,9], [8,7,6]] then A + B =
• If |(10, 2), (35, 7)| = 0 then x =
• If |(x, 5), (5, x)| = 0 then x =
• If f : R → R such that f(x) = 3x - 4 then which of the following is f^{-1}(x)?
• If n(A) = 3 and n(B) = 2 then n(A × B) = . . . . . . .
• |(1,1,2),(2,2,4),(3,5,6)| =
• f : A = {1, 2, 3} , then how many equivalence relation can be defined on A containing (1, 2)
• tan^{-1}(1) =
• tan^{-1}(1/2) + tan^{-1}(1/4) =
• {d(tan(ax))}/dx =