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question:a right triangle ABC with angle BAC as the right angle, let M be a point on side AC such that AM = x. If BC = 2d and AC = h, and we know the sum of distances BM and MC equals the length of the hypotenuse AB, find the value of x. A) frac{h + sqrt{h^2 8d^2}}{2} B) frac{h sqrt{h^2 8d^2}}{2} C) frac{h + 2d}{2} D) frac{h 2d}{2} E) frac{h}{2}
answer:1. **Given Information and Equation Setup:** Given the conditions, we have: [ BM + MC = AB ] Since AM = x, then MC = h - x. Also, since AB is the hypotenuse of triangle ABC, AB = sqrt{h^2 + (2d)^2} = sqrt{h^2 + 4d^2}. 2. **Using Geometry:** Using the Pythagorean theorem in triangle BMC, we find: [ BM = sqrt{x^2 + (2d)^2} = sqrt{x^2 + 4d^2} ] Substituting this into the equation, we get: [ sqrt{x^2 + 4d^2} + (h - x) = sqrt{h^2 + 4d^2} ] 3. **Manipulate and Solve for x:** Isolate the square root term: [ sqrt{x^2 + 4d^2} = sqrt{h^2 + 4d^2} - (h - x) ] Square both sides: [ x^2 + 4d^2 = h^2 + 4d^2 - 2(h - x)sqrt{h^2 + 4d^2} + (h - x)^2 ] Simplify and solve for x: [ x(h - x) = sqrt{h^2 + 4d^2} - 2(h - x) ] Solving this quadratic equation, we find: [ x = frac{h pm sqrt{h^2 - 8d^2}}{2} ] Since x must be real and positive, and less than h: [ x = frac{h - sqrt{h^2 - 8d^2}}{2} ] 4. **Conclusion:** The value of x that satisfies the conditions is frac{h - sqrt{h^2 - 8d^2}{2}}. The final answer is boxed{textbf{(B)} frac{h - sqrt{h^2 - 8d^2}}{2}}
question:Define the operation ⊗ on mathbb{R} as: x⊗y=(1x)(1+y). If the inequality (xa)⊗(x+a) < 1 holds for any real number x, then A: 1 < a < 1 B: 2 < a < 0 C: 0 < a < 2 D: dfrac {3}{2} < a < dfrac {1}{2}
answer:Given x⊗y=(1-x)(1+y), Then (x-a)⊗(x+a)=(1-(x-a))(1+(x+a))=(1+a)^{2}-x^{2}, Therefore, (1+a)^{2}-x^{2} < 1 holds for any real number x, Which means (1+a)^{2}-1 < x^{2} holds for any real number x, Thus, (1+a)^{2}-1 < 0, Solving this, we get -2 < a < 0, Therefore, the correct answer is boxed{text{B}}. This problem tests the students' learning ability and the application of transformational thinking methods in solving problems that always hold true.
question:For a sequence v_1, v_2, dots, define Delta^1(v_n) = v_{n+1} v_n and for all integers k > 1, Delta^k(v_n) = Delta^1(Delta^{k1}(v_n)). If v_n = n^4 + 2n^2, then Delta^k(v_n) = 0 for all n textbf{(A) }text{if }k=1qquad textbf{(B) }text{if }k=2,text{ but not if }k=1qquad textbf{(C) }text{if }k=3,text{ but not if }k=2qquad textbf{(D) }text{if }k=4,text{ but not if }k=3qquad textbf{(E) }text{for no value of }k textbf{(A)} if k=1 textbf{(B)} if k=2, but not if k=1 textbf{(C)} if k=3, but not if k=2 textbf{(D)} if k=4, but not if k=3 textbf{(E)} for no value of k
answer:1. **First Difference Calculation**: [ Delta^1(v_n) = v_{n+1} - v_n = [(n+1)^4 + 2(n+1)^2] - [n^4 + 2n^2] ] Expanding (n+1)^4 and (n+1)^2: [ (n+1)^4 = n^4 + 4n^3 + 6n^2 + 4n + 1 ] [ (n+1)^2 = n^2 + 2n + 1 ] Therefore, [ Delta^1(v_n) = [n^4 + 4n^3 + 6n^2 + 4n + 1 + 2(n^2 + 2n + 1)] - [n^4 + 2n^2] = 4n^3 + 8n^2 + 8n + 3 ] 2. **Second Difference Calculation**: [ Delta^2(v_n) = Delta^1(v_{n+1}) - Delta^1(v_n) = [4(n+1)^3 + 8(n+1)^2 + 8(n+1) + 3] - [4n^3 + 8n^2 + 8n + 3] ] Expanding (n+1)^3 and (n+1)^2: [ (n+1)^3 = n^3 + 3n^2 + 3n + 1 ] [ (n+1)^2 = n^2 + 2n + 1 ] Therefore, [ Delta^2(v_n) = [4(n^3 + 3n^2 + 3n + 1) + 8(n^2 + 2n + 1) + 8n + 8 + 3] - [4n^3 + 8n^2 + 8n + 3] = 12n^2 + 24n + 12 ] 3. **Third Difference Calculation**: [ Delta^3(v_n) = Delta^2(v_{n+1}) - Delta^2(v_n) = [12(n+1)^2 + 24(n+1) + 12] - [12n^2 + 24n + 12] ] Simplifying, [ Delta^3(v_n) = [12(n^2 + 2n + 1) + 24n + 24 + 12] - [12n^2 + 24n + 12] = 36 ] 4. **Fourth Difference Calculation**: [ Delta^4(v_n) = Delta^3(v_{n+1}) - Delta^3(v_n) = 36 - 36 = 0 ] 5. **Conclusion**: The sequence Delta^k(v_n) becomes zero for all n starting from k=4. This means that Delta^4(v_n) = 0 for all n, and Delta^k(v_n) = 0 for all k geq 4. Therefore, the correct answer is textbf{(D)}. Delta^k(v_n) = 0 for all n if k=4, but not if k=3. The final answer is (D) if boxed{k=4}, but not if boxed{k=3}
question:Consider an infinite line ell displaying a repeating pattern that consists of a square, a right triangle, and a section of line segments. Each square side is 1 unit, each triangle hypotenuse is 1 unit, and the line segments together are 2 units long. The triangle's right angle faces away from the square. Determine which of the following rigid motion transformations, excluding the identity transformation, maps this figure onto itself: 1. Some rotation around a point on line ell. 2. Some translation in the direction parallel to line ell. 3. The reflection across line ell. 4. Some reflection across a line perpendicular to line ell. textbf{(A)} 0 textbf{(B)} 1 textbf{(C)} 2 textbf{(D)} 3 textbf{(E)} 4
answer:1. **Rotation around a point on line ell**: - Consider rotating around a midpoint between the square and triangle along ell. A rotation of 180^circ could potentially align the pattern, but the non-symmetrical nature of the triangle (facing away from the square) disrupts the symmetry. - Therefore, the figure will not map onto itself. 2. **Translation in the direction parallel to line ell**: - Since the pattern repeats after a segment containing a square, a triangle, and line segments (totaling 4 units), translating by multiples of 4 units along ell will align each component correctly with their corresponding patterns. - This transformation will map the figure onto itself. 3. **Reflection across line ell**: - A reflection across ell flips the triangle and square, aligning them incorrectly because their orientations will be opposite. This won't map the figure onto itself. 4. **Reflection across a line perpendicular to ell**: - Reflecting the pattern across a vertical line through the center of the square would not match the orientations of the triangles, as they point in specific directions that won't align upon reflection. - Thus, this transformation also doesn't map the figure onto itself. Only the translation along ell by multiples of the repeating pattern's length maps the figure onto itself. textbf{B 1} The final answer is boxed{textbf{(B)} 1}