conversations listlengths 1 2 |
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[
{
"from": "human",
"value": "What is an equivalence relation, and how can you verify that a relation is an equivalence relation?"
},
{
"from": "gpt",
"value": "An equivalence relation on a set A is a relation R that satisfies three properties: reflexive, symmetric, and transitive. To verify that... |
[
{
"from": "human",
"value": "Let R be the relation defined on the set of all triangles in a plane, where R is given by T₁ is congruent to T₂. Show that R is an equivalence relation."
},
{
"from": "gpt",
"value": "To show that R is an equivalence relation, we need to verify that it satisfies the ... |
[
{
"from": "human",
"value": "Prove that the relation R on the set of all lines in a plane, defined by L₁ is perpendicular to L₂, is symmetric but neither reflexive nor transitive."
},
{
"from": "gpt",
"value": "We verify the properties of the relation R defined as L₁ is perpendicular to L₂: 1. R... |
[
{
"from": "human",
"value": "Show that the relation R defined on the set {1, 2, 3}, given by R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)}, is reflexive but neither symmetric nor transitive."
},
{
"from": "gpt",
"value": "We check the properties of the relation R = {(1, 1), (2, 2), (3, 3), (1, 2)... |
[
{
"from": "human",
"value": "Prove that the relation R on the set of integers Z, defined by (a, b) ∈ R if and only if 2 divides (a - b), is an equivalence relation."
},
{
"from": "gpt",
"value": "We verify the properties of the relation R defined by (a, b) ∈ R if and only if 2 divides (a - b): 1... |
[
{
"from": "human",
"value": "Show that the relation R in the set A = {1, 2, 3, 4, 5}, defined by R = {(a, b) : |a – b| is even}, is an equivalence relation. Also, explain how the elements of {1, 3, 5} and {2, 4} are related."
},
{
"from": "gpt",
"value": "We verify the properties of R = {(a, b) ... |
[
{
"from": "human",
"value": "Show that the relation R in the set A = {1, 2, 3, 4, 5}, defined by R = {(a, b) : |a – b| is even}, is an equivalence relation. Also, explain how the elements of {1, 3, 5} and {2, 4} are related."
},
{
"from": "gpt",
"value": "We verify the properties of R = {(a, b) ... |
[
{
"from": "human",
"value": "Show that the relation R defined in the set A of all triangles, given by R = {(T1, T2) : T1 is similar to T2}, is an equivalence relation. Also, determine which triangles among T1, T2, and T3 (with sides 3, 4, 5; 5, 12, 13; and 6, 8, 10 respectively) are related."
},
{
"... |
[
{
"from": "human",
"value": "Show that the relation R defined in the set A of all polygons, given by R = {(P1, P2) : P1 and P2 have the same number of sides}, is an equivalence relation. Also, determine the set of all elements in A related to the right-angle triangle T with sides 3, 4, and 5."
},
{
... |
[
{
"from": "human",
"value": "Let L be the set of all lines in the XY-plane, and R be the relation in L defined as R = {(L1, L2) : L1 is parallel to L2}. Show that R is an equivalence relation. Also, find the set of all lines related to the line y = 2x + 4."
},
{
"from": "gpt",
"value": "We verif... |
[
{
"from": "human",
"value": "Let R be the relation in the set {1, 2, 3, 4} given by R = {(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)}. Determine which of the following statements is correct:\n(A) R is reflexive and symmetric but not transitive.\n(B) R is reflexive and transitive but not symmetric... |
[
{
"from": "human",
"value": "Show that each of the relations R in the set A = {x ∈ Z : 0 ≤ x ≤ 12}, given by (i) R = {(a, b) : |a – b| is a multiple of 4} and (ii) R = {(a, b) : a = b}, is an equivalence relation. Also, find the set of all elements related to 1 in each case."
},
{
"from": "gpt",
... |
[
{
"from": "human",
"value": "Show that the relation R in the set A of points in a plane, given by R = {(P, Q) : distance of the point P from the origin is the same as the distance of the point Q from the origin}, is an equivalence relation. Further, show that the set of all points related to a point P ≠ (0,... |
[
{
"from": "human",
"value": "Show that the relation R defined in the set A of all triangles as R = {(T1, T2) : T1 is similar to T2}, is an equivalence relation. Consider three right-angle triangles T1 with sides 3, 4, 5, T2 with sides 5, 12, 13, and T3 with sides 6, 8, 10. Which triangles among T1, T2, and ... |
[
{
"from": "human",
"value": "Show that the relation R defined in the set A of all polygons as R = {(P1, P2) : P1 and P2 have the same number of sides}, is an equivalence relation. What is the set of all elements in A related to the right-angle triangle T with sides 3, 4, and 5?"
},
{
"from": "gpt",
... |
[
{
"from": "human",
"value": "Let L be the set of all lines in the XY-plane and R be the relation in L defined as R = {(L1, L2) : L1 is parallel to L2}. Show that R is an equivalence relation. Find the set of all lines related to the line y = 2x + 4."
},
{
"from": "gpt",
"value": "We verify the p... |
[
{
"from": "human",
"value": "Let R be the relation in the set {1, 2, 3, 4} given by R = {(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)}. Choose the correct answer: (A) R is reflexive and symmetric but not transitive. (B) R is reflexive and transitive but not symmetric. (C) R is symmetric and transi... |
[
{
"from": "human",
"value": "Let R be the relation in the set N given by R = {(a, b) : a = b – 2, b > 6}. Choose the correct answer: (A) (2, 4) ∈ R (B) (3, 8) ∈ R (C) (6, 8) ∈ R (D) (8, 7) ∈ R"
},
{
"from": "gpt",
"value": "We analyze the relation R = {(a, b) : a = b – 2, b > 6}:\n\n1. For (2, 4... |
[
{
"from": "human",
"value": "Show that the function f : N → N, given by f(1) = f(2) = 1 and f(x) = x – 1 for every x > 2, is onto but not one-one."
},
{
"from": "gpt",
"value": "We analyze the given function:\n\n1. **Not One-One**: The function is not one-one because f(1) = 1 and f(2) = 1. Since... |
[
{
"from": "human",
"value": "Show that the function f : R → R, defined as f(x) = x², is neither one-one nor onto."
},
{
"from": "gpt",
"value": "We analyze the given function:\n\n1. **Not One-One**: The function is not one-one because f(-1) = 1 and f(1) = 1. Since two distinct inputs (-1 and 1) ... |
[
{
"from": "human",
"value": "Show that the function f : N → N, given by f(x) = x + 1 if x is odd and f(x) = x – 1 if x is even, is both one-one and onto."
},
{
"from": "gpt",
"value": "We analyze the given function:\n\n1. **One-One**: Suppose f(x₁) = f(x₂). If x₁ is odd and x₂ is even, we have x... |
[
{
"from": "human",
"value": "Show that an onto function f : {1, 2, 3} → {1, 2, 3} is always one-one."
},
{
"from": "gpt",
"value": "We analyze the given function:\n\n1. Suppose f is not one-one. Then there exist two distinct elements, say 1 and 2, in the domain such that f(1) = f(2). The image o... |
[
{
"from": "human",
"value": "Show that a one-one function f : {1, 2, 3} → {1, 2, 3} must be onto."
},
{
"from": "gpt",
"value": "We analyze the given function:\n\n1. Since f is one-one, each element of {1, 2, 3} in the domain must map to a unique element in the co-domain. This means no two eleme... |
[
{
"from": "human",
"value": "Show that the function f : R∗ → R∗ defined by f(x) = x is one-one and onto, where R∗ is the set of all non-zero real numbers. Is the result true if the domain R∗ is replaced by N with co-domain being the same as R∗?"
},
{
"from": "gpt",
"value": "We analyze the funct... |
[
{
"from": "human",
"value": "Check the injectivity and surjectivity of the following functions:\n(i) f : N → N given by f(x) = x²\n(ii) f : Z → Z given by f(x) = x²\n(iii) f : R → R given by f(x) = x²\n(iv) f : N → N given by f(x) = x³\n(v) f : Z → Z given by f(x) = x³"
},
{
"from": "gpt",
"valu... |
[
{
"from": "human",
"value": "Prove that the Greatest Integer Function f : R → R, given by f(x) = [x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x."
},
{
"from": "gpt",
"value": "We analyze the function f(x) = [x]:\n\n1. **Not One-One**: The functio... |
[
{
"from": "human",
"value": "Show that the Modulus Function f : R → R, given by f(x) = |x|, is neither one-one nor onto, where |x| is x if x ≥ 0 and -x if x < 0."
},
{
"from": "gpt",
"value": "We analyze the function f(x) = |x|:\n\n1. **Not One-One**: The function is not one-one because f(-1) = ... |
[
{
"from": "human",
"value": "Show that the Signum Function f : R → R, given by\nf(x) = 1 if x > 0,\nf(x) = 0 if x = 0,\nf(x) = -1 if x < 0, is neither one-one nor onto."
},
{
"from": "gpt",
"value": "We analyze the function f(x):\n\n1. **Not One-One**: The function is not one-one because f(x) = ... |
[
{
"from": "human",
"value": "Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let f = {(1, 4), (2, 5), (3, 6)} be a function from A to B. Show that f is one-one."
},
{
"from": "gpt",
"value": "We analyze the function f:\n\n1. **One-One**: For f to be one-one, distinct elements in the domain must map to d... |
[
{
"from": "human",
"value": "Let f : {2, 3, 4, 5} → {3, 4, 5, 9} and g : {3, 4, 5, 9} → {7, 11, 15} be functions defined as f(2) = 3, f(3) = 4, f(4) = f(5) = 5, and g(3) = g(4) = 7 and g(5) = g(9) = 11. Find gof."
},
{
"from": "gpt",
"value": "We calculate gof as follows:\n\n1. **gof(2)** = g(f(... |
[
{
"from": "human",
"value": "Find gof and fog, if f : R → R and g : R → R are given by f(x) = cos x and g(x) = 3x². Show that gof ≠ fog."
},
{
"from": "gpt",
"value": "We calculate gof and fog:\n\n1. **gof(x)** = g(f(x)) = g(cos x) = 3(cos x)² = 3cos²x.\n2. **fog(x)** = f(g(x)) = f(3x²) = cos(3x... |
[
{
"from": "human",
"value": "Let f : N → Y be a function defined as f(x) = 4x + 3, where Y = {y ∈ N : y = 4x + 3 for some x ∈ N}. Show that f is invertible. Find the inverse."
},
{
"from": "gpt",
"value": "We analyze the function f(x) = 4x + 3:\n\n1. **One-One**: Suppose f(x₁) = f(x₂). Then, 4x₁... |
[
{
"from": "human",
"value": "If R1 and R2 are equivalence relations in a set A, show that R1 ∩ R2 is also an equivalence relation."
},
{
"from": "gpt",
"value": "We analyze R1 ∩ R2:\n\n1. **Reflexivity**: Since R1 and R2 are equivalence relations, (a, a) ∈ R1 and (a, a) ∈ R2 for all a ∈ A. Thus,... |
[
{
"from": "human",
"value": "Let R be a relation on the set A of ordered pairs of positive integers defined by (x, y) R (u, v) if and only if xv = yu. Show that R is an equivalence relation."
},
{
"from": "gpt",
"value": "We analyze the relation R:\n\n1. **Reflexivity**: For any (x, y) ∈ A, xv =... |
[
{
"from": "human",
"value": "Let X = {1, 2, 3, 4, 5, 6, 7, 8, 9}. Let R1 = {(x, y) : x – y is divisible by 3} and R2 = {(x, y): {x, y} ⊂ {1, 4, 7}} or {x, y} ⊂ {2, 5, 8} or {x, y} ⊂ {3, 6, 9}}. Show that R1 = R2."
},
{
"from": "gpt",
"value": "We compare R1 and R2:\n\n1. For R1: (x, y) ∈ R1 if x... |
[
{
"from": "human",
"value": "Find the number of all one-one functions from the set A = {1, 2, 3} to itself."
},
{
"from": "gpt",
"value": "The number of one-one functions from {1, 2, 3} to itself is equivalent to the number of permutations of the set {1, 2, 3}.\n\n1. The total number of permutat... |
[
{
"from": "human",
"value": "Let A = {1, 2, 3}. Then show that the number of relations containing (1, 2) and (2, 3) which are reflexive and transitive but not symmetric is three."
},
{
"from": "gpt",
"value": "We analyze the relations:\n\n1. The smallest reflexive and transitive relation contain... |
[
{
"from": "human",
"value": "Show that the function f : R → {x ∈ R : – 1 < x < 1} defined by f(x) = x / (1 + |x|), x ∈ R, is a one-one and onto function."
},
{
"from": "gpt",
"value": "We analyze the function f(x) = x / (1 + |x|):\n\n1. **One-One**: Suppose f(x₁) = f(x₂). Then, x₁ / (1 + |x₁|) =... |
[
{
"from": "human",
"value": "Show that the function f : R → R given by f(x) = x³ is injective."
},
{
"from": "gpt",
"value": "We analyze the function f(x) = x³:\n\n1. **One-One**: Suppose f(x₁) = f(x₂). Then, x₁³ = x₂³. Taking cube roots on both sides, we get x₁ = x₂.\n\n2. Since distinct inputs... |
[
{
"from": "human",
"value": "Given a non-empty set X, consider P(X), which is the set of all subsets of X. Define the relation R in P(X) as follows: For subsets A, B in P(X), ARB if and only if A ⊂ B. Is R an equivalence relation on P(X)? Justify your answer."
},
{
"from": "gpt",
"value": "We an... |
[
{
"from": "human",
"value": "Find the number of all onto functions from the set {1, 2, 3, ..., n} to itself."
},
{
"from": "gpt",
"value": "The number of onto functions from a set of size n to itself is equivalent to the number of permutations of the set. This is given by n! (n factorial).\n\n**... |
[
{
"from": "human",
"value": "Let A = {–1, 0, 1, 2}, B = {–4, –2, 0, 2} and f, g : A → B be functions defined by f(x) = x² – x and g(x) = (2x – 1) / 2. Are f and g equal? Justify your answer."
},
{
"from": "gpt",
"value": "We compare f and g:\n\n1. For x = -1: f(-1) = (-1)² – (-1) = 1 + 1 = 2, an... |
[
{
"from": "human",
"value": "Let A = {1, 2, 3}. Then the number of relations containing (1, 2) and (1, 3) which are reflexive and symmetric but not transitive is:\n(A) 1\n(B) 2\n(C) 3\n(D) 4"
},
{
"from": "gpt",
"value": "To find reflexive and symmetric relations containing (1, 2) and (1, 3):\n\... |
[
{
"from": "human",
"value": "Let A = {1, 2, 3}. Then the number of equivalence relations containing (1, 2) is:\n(A) 1\n(B) 2\n(C) 3\n(D) 4"
},
{
"from": "gpt",
"value": "To find equivalence relations containing (1, 2):\n\n1. The smallest equivalence relation containing (1, 2) is R1 = {(1, 1), (2... |
[
{
"from": "human",
"value": "Find the principal value of sin⁻¹(1/2)."
},
{
"from": "gpt",
"value": "Let sin⁻¹(1/2) = y. Then, sin(y) = 1/2.\n\nWe know that the range of the principal value branch of sin⁻¹ is [-π/2, π/2]. Also, sin(π/6) = 1/2.\n\n**Conclusion**: The principal value of sin⁻¹(1/2) ... |
[
{
"from": "human",
"value": "Find the principal value of cot⁻¹(-√3)."
},
{
"from": "gpt",
"value": "Let cot⁻¹(-√3) = y. Then, cot(y) = -√3.\n\nWe know that the range of the principal value branch of cot⁻¹ is (0, π). Also, cot(π/3) = √3 and cot(2π/3) = -√3.\n\n**Conclusion**: The principal value ... |
[
{
"from": "human",
"value": "Find the principal value of sin⁻¹(-1/2)."
},
{
"from": "gpt",
"value": "Let sin⁻¹(-1/2) = y. Then, sin(y) = -1/2.\n\nWe know that the range of the principal value branch of sin⁻¹ is [-π/2, π/2]. Also, sin(-π/6) = -1/2.\n\n**Conclusion**: The principal value of sin⁻¹(... |
[
{
"from": "human",
"value": "Find the principal value of cos⁻¹(√3/2)."
},
{
"from": "gpt",
"value": "Let cos⁻¹(√3/2) = y. Then, cos(y) = √3/2.\n\nWe know that the range of the principal value branch of cos⁻¹ is [0, π]. Also, cos(π/6) = √3/2.\n\n**Conclusion**: The principal value of cos⁻¹(√3/2) ... |
[
{
"from": "human",
"value": "Find the principal value of cosec⁻¹(2)."
},
{
"from": "gpt",
"value": "Let cosec⁻¹(2) = y. Then, cosec(y) = 2.\n\nWe know that the range of the principal value branch of cosec⁻¹ is [-π/2, π/2], y ≠ 0. Also, cosec(π/6) = 2.\n\n**Conclusion**: The principal value of co... |
[
{
"from": "human",
"value": "Find the principal value of tan⁻¹(-√3)."
},
{
"from": "gpt",
"value": "Let tan⁻¹(-√3) = y. Then, tan(y) = -√3.\n\nWe know that the range of the principal value branch of tan⁻¹ is (-π/2, π/2). Also, tan(-π/3) = -√3.\n\n**Conclusion**: The principal value of tan⁻¹(-√3)... |
[
{
"from": "human",
"value": "Find the principal value of cos⁻¹(-1/2)."
},
{
"from": "gpt",
"value": "Let cos⁻¹(-1/2) = y. Then, cos(y) = -1/2.\n\nWe know that the range of the principal value branch of cos⁻¹ is [0, π]. Also, cos(2π/3) = -1/2.\n\n**Conclusion**: The principal value of cos⁻¹(-1/2)... |
[
{
"from": "human",
"value": "Find the principal value of tan⁻¹(-1)."
},
{
"from": "gpt",
"value": "Let tan⁻¹(-1) = y. Then, tan(y) = -1.\n\nWe know that the range of the principal value branch of tan⁻¹ is (-π/2, π/2). Also, tan(-π/4) = -1.\n\n**Conclusion**: The principal value of tan⁻¹(-1) is -... |
[
{
"from": "human",
"value": "Find the principal value of sec⁻¹(2√3/3)."
},
{
"from": "gpt",
"value": "Let sec⁻¹(2√3/3) = y. Then, sec(y) = 2√3/3.\n\nWe know that the range of the principal value branch of sec⁻¹ is [0, π], y ≠ π/2. Also, sec(π/6) = 2√3/3.\n\n**Conclusion**: The principal value of... |
[
{
"from": "human",
"value": "Find the principal value of cot⁻¹(√3)."
},
{
"from": "gpt",
"value": "Let cot⁻¹(√3) = y. Then, cot(y) = √3.\n\nWe know that the range of the principal value branch of cot⁻¹ is (0, π). Also, cot(π/6) = √3.\n\n**Conclusion**: The principal value of cot⁻¹(√3) is π/6."
... |
[
{
"from": "human",
"value": "Find the principal value of cos⁻¹(-1/2)."
},
{
"from": "gpt",
"value": "Let cos⁻¹(-1/2) = y. Then, cos(y) = -1/2.\n\nWe know that the range of the principal value branch of cos⁻¹ is [0, π]. Also, cos(2π/3) = -1/2.\n\n**Conclusion**: The principal value of cos⁻¹(-1/2)... |
[
{
"from": "human",
"value": "Find the principal value of cosec⁻¹(-2)."
},
{
"from": "gpt",
"value": "Let cosec⁻¹(-2) = y. Then, cosec(y) = -2.\n\nWe know that the range of the principal value branch of cosec⁻¹ is [-π/2, π/2], y ≠ 0. Also, cosec(-π/6) = -2.\n\n**Conclusion**: The principal value ... |
[
{
"from": "human",
"value": "Find the value of tan⁻¹(1) + cos⁻¹(-1/2) + sin⁻¹(-1/2)."
},
{
"from": "gpt",
"value": "We calculate each term:\n\n1. **tan⁻¹(1)**: Let tan⁻¹(1) = x. Then, tan(x) = 1. From the principal value branch of tan⁻¹, x = π/4.\n2. **cos⁻¹(-1/2)**: Let cos⁻¹(-1/2) = y. Then, c... |
[
{
"from": "human",
"value": "Find the value of cos⁻¹(1/2) + 2sin⁻¹(1/2)."
},
{
"from": "gpt",
"value": "We calculate each term:\n\n1. **cos⁻¹(1/2)**: Let cos⁻¹(1/2) = x. Then, cos(x) = 1/2. From the principal value branch of cos⁻¹, x = π/3.\n2. **sin⁻¹(1/2)**: Let sin⁻¹(1/2) = y. Then, sin(y) = ... |
[
{
"from": "human",
"value": "If sin⁻¹(x) = y, then which of the following is true?\n(A) 0 ≤ y ≤ π\n(B) -π/2 ≤ y ≤ π/2\n(C) 0 < y < π\n(D) -π < y < π"
},
{
"from": "gpt",
"value": "The range of the principal value branch of sin⁻¹ is [-π/2, π/2].\n\n**Conclusion**: The correct answer is (B) -π/2 ≤... |
[
{
"from": "human",
"value": "Find the value of tan⁻¹(3) - sec⁻¹(-2)."
},
{
"from": "gpt",
"value": "We calculate each term:\n\n1. **tan⁻¹(3)**: Let tan⁻¹(3) = x. Then, tan(x) = 3. From the principal value branch of tan⁻¹, x = π/3.\n2. **sec⁻¹(-2)**: Let sec⁻¹(-2) = y. Then, sec(y) = -2. From the... |
[
{
"from": "human",
"value": "Show that sin⁻¹(2√x(1−x)) = 2sin⁻¹(x)."
},
{
"from": "gpt",
"value": "Let x = sin(θ). Then, sin⁻¹(x) = θ. Substituting:\n\nsin⁻¹(2√x(1−x)) = sin⁻¹(2sin(θ)√(1−sin²(θ))).\n\nUsing the identity 1−sin²(θ) = cos²(θ), this becomes:\n\nsin⁻¹(2sin(θ)cos(θ)) = sin⁻¹(sin(2θ)) ... |
[
{
"from": "human",
"value": "Show that sin⁻¹(2√x(1−x)) = 2cos⁻¹(x)."
},
{
"from": "gpt",
"value": "Let x = cos(θ). Then, cos⁻¹(x) = θ. Substituting:\n\nsin⁻¹(2√x(1−x)) = sin⁻¹(2cos(θ)√(1−cos²(θ))).\n\nUsing the identity 1−cos²(θ) = sin²(θ), this becomes:\n\nsin⁻¹(2cos(θ)sin(θ)) = sin⁻¹(sin(2θ)) ... |
[
{
"from": "human",
"value": "Express tan⁻¹((1−sin(x))/(cos²(x))) in the simplest form."
},
{
"from": "gpt",
"value": "We simplify tan⁻¹((1−sin(x))/cos²(x)) step by step:\n\n1. Rewrite the expression: \n tan⁻¹((1−sin(x))/(cos²(x))) = tan⁻¹((cos²(x) − sin²(x)) / cos²(x)).\n\n2. Use the trigonome... |
[
{
"from": "human",
"value": "Write cot⁻¹((x²−1)/x), x > 1, in the simplest form."
},
{
"from": "gpt",
"value": "Let x = sec(θ), where θ ∈ (0, π/2). Then:\n\nx² − 1 = sec²(θ) − 1 = tan²(θ).\n\nThe given expression becomes:\n\ncot⁻¹((x²−1)/x) = cot⁻¹(tan²(θ)/sec(θ)).\n\nUsing the identities tan²(θ... |
[
{
"from": "human",
"value": "Show that 3sin⁻¹(x) = sin⁻¹(3x - 4x³) for x ∈ [-1/2, 1/2]."
},
{
"from": "gpt",
"value": "To prove: 3sin⁻¹(x) = sin⁻¹(3x - 4x³), we proceed as follows:\n\nLet y = sin⁻¹(x). Then, sin(y) = x. We use the triple angle formula for sine:\n\nsin(3y) = 3sin(y) - 4sin³(y).\n... |
[
{
"from": "human",
"value": "Show that 3cos⁻¹(x) = cos⁻¹(4x³ - 3x) for x ∈ [1/2, 1]."
},
{
"from": "gpt",
"value": "To prove: 3cos⁻¹(x) = cos⁻¹(4x³ - 3x), we proceed as follows:\n\nLet y = cos⁻¹(x). Then, cos(y) = x. We use the triple angle formula for cosine:\n\ncos(3y) = 4cos³(y) - 3cos(y).\n\... |
[
{
"from": "human",
"value": "Simplify tan⁻¹((1 + x² - 1)/x), x ≠ 0."
},
{
"from": "gpt",
"value": "We simplify the expression step by step:\n\n1. **Simplify the numerator**: The given numerator is 1 + x² - 1. Here, the terms +1 and -1 cancel out, leaving only x².\n\n2. **Rewrite the fraction**: ... |
[
{
"from": "human",
"value": "Simplify tan⁻¹((1 + cos(x))/(1 - cos(x))), 0 < x < π."
},
{
"from": "gpt",
"value": "We simplify the expression step by step:\n\n1. Rewrite using the identity for 1 + cos(x) and 1 - cos(x):\n tan⁻¹((1 + cos(x))/(1 - cos(x))) = tan⁻¹((2cos²(x/2))/(2sin²(x/2))).\n2. ... |
[
{
"from": "human",
"value": "Simplify tan⁻¹((cos(x) - sin(x))/(cos(x) + sin(x))) - π/4, 3π/4 < x < 4π/4."
},
{
"from": "gpt",
"value": "We simplify the expression step by step:\n\n1. **Apply the tangent subtraction formula**:\n The given expression is tan⁻¹((cos(x) - sin(x))/(cos(x) + sin(x)))... |
[
{
"from": "human",
"value": "Simplify tan⁻¹((x)/(√(a² - x²))), |x| < a."
},
{
"from": "gpt",
"value": "We simplify the expression step by step:\n\n1. **Understand the structure of the expression**:\n The given expression is tan⁻¹(x / √(a² - x²)). It involves an inverse tangent function where t... |
[
{
"from": "human",
"value": "Simplify tan⁻¹((3a³x² - 3ax³)/(a³ - x³)), a > 0, -a³ < x < a³."
},
{
"from": "gpt",
"value": "We simplify the expression step by step:\n\n1. **Analyze the given expression**:\n The expression is tan⁻¹((3a³x² - 3ax³)/(a³ - x³)). Both the numerator and denominator in... |
[
{
"from": "human",
"value": "Find the value of tan⁻¹[2cos(2x)sin(x)] + tan⁻¹[sin⁻¹(x²) + cos⁻¹(y²)], |x| < 1, y > 0, xy < 1."
},
{
"from": "gpt",
"value": "We calculate each term step by step:\n\n1. **Simplify the first term (tan⁻¹[2cos(2x)sin(x)])**:\n - The term inside the inverse tangent is... |
[
{
"from": "human",
"value": "Find the value of tan⁻¹(3) - cot⁻¹(-3)."
},
{
"from": "gpt",
"value": "We calculate the value step by step:\n\n1. **Simplify tan⁻¹(3):**\n - Let tan⁻¹(3) = θ₁. Then, tan(θ₁) = 3.\n - The angle θ₁ lies in the principal branch of tan⁻¹, which is (-π/2, π/2).\n - ... |
[
{
"from": "human",
"value": "Find the value of sin⁻¹(sin(3π/5))."
},
{
"from": "gpt",
"value": "We calculate the value step by step:\n\n1. **Understand the property of sin⁻¹(sin(x)):**\n - The expression sin⁻¹(sin(x)) = x holds only if x lies within the principal branch of sin⁻¹, which is [-π/... |
[
{
"from": "human",
"value": "Find the value of cos⁻¹(cos(13π/6))."
},
{
"from": "gpt",
"value": "We calculate the value step by step:\n\n1. **Understand the property of cos⁻¹(cos(x)):**\n - The expression cos⁻¹(cos(x)) = x holds only if x lies within the principal branch of cos⁻¹, which is [0,... |
[
{
"from": "human",
"value": "Prove that tan⁻¹(tan(7π/6)) = 7π/6."
},
{
"from": "gpt",
"value": "We prove the value step by step:\n\n1. **Understand the property of tan⁻¹(tan(x)):**\n - The expression tan⁻¹(tan(x)) = x holds only if x lies within the principal branch of tan⁻¹, which is (-π/2, π... |
[
{
"from": "human",
"value": "Prove that tan⁻¹(tan(7π/6)) = tan⁻¹(tan(π/6))."
},
{
"from": "gpt",
"value": "We prove the value step by step:\n\n1. **Understand the property of tan⁻¹(tan(x)):**\n - The expression tan⁻¹(tan(x)) = x holds only if x lies within the principal branch of tan⁻¹, which ... |
[
{
"from": "human",
"value": "Prove that 2sin⁻¹(3/5) = tan⁻¹(24/7)."
},
{
"from": "gpt",
"value": "We prove this step by step:\n\n1. **Let sin⁻¹(3/5) = θ:**\n - Then, sin(θ) = 3/5. Using the Pythagorean identity, cos(θ) = √(1 - sin²(θ)) = √(1 - (3/5)²) = 4/5.\n\n2. **Double-angle formula for si... |
[
{
"from": "human",
"value": "Prove that sin⁻¹(8/17) + sin⁻¹(3/5) = tan⁻¹(77/36)."
},
{
"from": "gpt",
"value": "We prove this step by step:\n\n1. **Let sin⁻¹(8/17) = α and sin⁻¹(3/5) = β:**\n - For α, sin(α) = 8/17. Using the Pythagorean identity, cos(α) = √(1 - sin²(α)) = √(1 - (8/17)²) = 15/... |
[
{
"from": "human",
"value": "Prove that cos⁻¹(4/5) + cos⁻¹(12/13) = cos⁻¹(33/65)."
},
{
"from": "gpt",
"value": "We prove this step by step:\n\n1. **Use the property of cos⁻¹ addition**:\n - The property for cos⁻¹(a) + cos⁻¹(b) is:\n cos⁻¹(a) + cos⁻¹(b) = cos⁻¹(ab - √(1-a²)√(1-b²)), where ... |
[
{
"from": "human",
"value": "Prove that tan⁻¹(63/16) = sin⁻¹(5/13) + cos⁻¹(3/5)."
},
{
"from": "gpt",
"value": "We prove this step by step:\n\n1. **Simplify sin⁻¹(5/13):**\n - Let sin⁻¹(5/13) = α. Then, sin(α) = 5/13.\n - Using the Pythagorean identity, cos(α) = √(1 - sin²(α)) = √(1 - (5/13)... |
[
{
"from": "human",
"value": "Prove that cos⁻¹(12/13) + sin⁻¹(3/5) = sin⁻¹(56/65)."
},
{
"from": "gpt",
"value": "We prove this step by step:\n\n1. **Simplify cos⁻¹(12/13):**\n - Let cos⁻¹(12/13) = α. Then, cos(α) = 12/13.\n - Using the Pythagorean identity, sin(α) = √(1 - cos²(α)) = √(1 - (1... |
[
{
"from": "human",
"value": "Prove that tan⁻¹(x) = cos⁻¹((1 - x) / (1 + x)), x ∈ [0, 1]."
},
{
"from": "gpt",
"value": "We prove this step by step:\n\n1. **Express tan⁻¹(x) using trigonometric identities:**\n - Let θ = tan⁻¹(x). This implies tan(θ) = x.\n - Using the Pythagorean identity for... |
[
{
"from": "human",
"value": "Prove that cot⁻¹((1 + sin(x)) / (1 - sin(x))) = 2x, x ∈ (0, π/4)."
},
{
"from": "gpt",
"value": "We prove this step by step:\n\n1. **Start with the given expression:**\n - Let y = cot⁻¹((1 + sin(x)) / (1 - sin(x))).\n - This implies cot(y) = (1 + sin(x)) / (1 - s... |
[
{
"from": "human",
"value": "Solve 2tan⁻¹(cos(x)) = tan⁻¹(2cosec(x))."
},
{
"from": "gpt",
"value": "We solve step by step:\n\n1. **Start with the given equation:**\n - 2tan⁻¹(cos(x)) = tan⁻¹(2cosec(x)).\n\n2. **Simplify 2tan⁻¹(cos(x)):**\n - Using the double-angle formula for tangent:\n ... |
[
{
"from": "human",
"value": "Find sin(tan⁻¹(x)), |x| < 1."
},
{
"from": "gpt",
"value": "We solve step by step:\n\n1. **Understand the given expression:**\n - Let θ = tan⁻¹(x). This implies tan(θ) = x.\n\n2. **Use the Pythagorean identity for sine:**\n - Recall that sin²(θ) + cos²(θ) = 1.\n ... |
[
{
"from": "human",
"value": "Prove that tan⁻¹((1 - x) / (1 + x)) = tan⁻¹(x), x > 0."
},
{
"from": "gpt",
"value": "We prove this step by step:\n\n1. **Start with the given equation:**\n - Let y = tan⁻¹((1 - x) / (1 + x)).\n - This implies tan(y) = (1 - x) / (1 + x).\n\n2. **Rewrite tan(y):**... |
[
{
"from": "human",
"value": "Solve sin⁻¹(1 - x) - 2sin⁻¹(x) = 0."
},
{
"from": "gpt",
"value": "We solve this step by step:\n\n1. **Start with the given equation:**\n - sin⁻¹(1 - x) - 2sin⁻¹(x) = 0.\n - Rearrange:\n sin⁻¹(1 - x) = 2sin⁻¹(x).\n\n2. **Let sin⁻¹(x) = θ:**\n - Then x = sin... |
[
{
"from": "human",
"value": "Consider the following information regarding the number of men and women workers in three factories I, II, and III:\nMen workers Women workers\nI 30 25\nII 25 31\nIII 27 26\nRepresent the above information i... |
[
{
"from": "human",
"value": "If a matrix has 8 elements, what are the possible orders it can have?"
},
{
"from": "gpt",
"value": "We know that if a matrix is of order m × n, it has mn elements.\n2. To find all possible orders of a matrix with 8 elements, we will find all ordered pairs of natural... |
[
{
"from": "human",
"value": "Construct a 3 × 2 matrix whose elements are given by aᵢⱼ = (1/2) | i − 3j |."
},
{
"from": "gpt",
"value": "In general, a 3 × 2 matrix is given by:\n\nA =\n [a₁₁ a₁₂]\n [a₂₁ a₂₂]\n [a₃₁ a₃₂]\n\nNow, the elements of the matrix are calculated using the formula:\n... |
[
{
"from": "human",
"value": "If\n [ -6 a -1 0 ] = -[ 6 -3 2c + 2 ]\n [ b -3 -21 0 ] [ 2b + 4 -21 0 ]\nFind the values of a, b, c, x, y, and z."
},
{
"from": "gpt",
"value": "As the given matrices are equal, their corresponding elements must also be equal. By ... |
[
{
"from": "human",
"value": "Find the values of a, b, c, and d from the following equation:\n [ 2a + b a - 2b ] = [ 4 -3 ]\n [ 5c - d 4c + 3d ] [ 11 24 ]"
},
{
"from": "gpt",
"value": "By the equality of two matrices, we equate the corresponding elements:\n\n1. 2a + b = 4\n2. a - 2b = ... |
[
{
"from": "human",
"value": "In the matrix A = [[35, -2, 5, 12], [2, 3, 1, -5], [17, 0, 0, 1]]:\n(i) The order of the matrix,\n(ii) The number of elements,\n(iii) Write the elements a13, a21, a33, a24, a23."
},
{
"from": "gpt",
"value": "Matrix A is: \nA = [ [35, -2, 5, 12], [2, 3, 1, -5], [17, ... |
[
{
"from": "human",
"value": "If a matrix has 24 elements, what are the possible orders it can have? What, if it has 13 elements?"
},
{
"from": "gpt",
"value": "To find the possible orders of a matrix with a given number of elements, we look for pairs of natural numbers whose product equals the t... |
[
{
"from": "human",
"value": "If a matrix has 18 elements, what are the possible orders it can have? What, if it has 5 elements?"
},
{
"from": "gpt",
"value": "For 18 elements, the possible orders are:\n- (1, 18)\n- (2, 9)\n- (3, 6)\n- (6, 3)\n- (9, 2)\n- (18, 1)\nThus, the possible orders are 1 ... |
[
{
"from": "human",
"value": "Construct a 2 × 2 matrix, A = [aij], whose elements are given by:\n(i) aij = (i + j)^2\n(ii) aij = i\n(iii) aij = (i + 2)j^2"
},
{
"from": "gpt",
"value": "(i) Constructing the matrix with aij = (i + j)^2:\nFor a 2 × 2 matrix:\n- a11 = (1 + 1)^2 = 4\n- a12 = (1 + 2)^... |
[
{
"from": "human",
"value": "Construct a 3 × 4 matrix, whose elements are given by:\n(i) aij = 1 | −3i + j |\n(ii) aij = 2i − j"
},
{
"from": "gpt",
"value": "For a 3 × 4 matrix, let's compute the elements based on the given formulas:\n\n(i) For aij = 1 | −3i + j |:\n- For i = 1, j = 1: a11 = 1 ... |
[
{
"from": "human",
"value": "Find the values of x, y and z from the following equations:\n(i) [x + y + z] = [9]\n [4x 5y 5z] = [1]\n(ii) [x + y + z] = [5]\n [x + 2y + 2z] = [6]\n [y + z] = [7]"
},
{
"from": "gpt",
"value": "For the given equations, we equate the corresponding element... |
[
{
"from": "human",
"value": " Find the value of a, b, c and d from the equation:\n[a − b 2a + c] = [−1 5]\n[2a − b 3c + d] = [0 13]"
},
{
"from": "gpt",
"value": "We equate the corresponding elements of the matrices:\n\nFrom the equation:\n[a − b, 2a + c] = [−1, 5]\n[2a − b, 3c + d] = [0, 13... |
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