answer:1. **Identifying Symmetry of Curve (1): (x^2 + y^2 = r^2)** This is the equation of a circle with radius (r) centered at the origin ((0, 0)). - **Symmetry Relative to the x-axis**: If ((x, y)) is a point on the circle, then ((x, -y)) is also a point on the circle since: [ x^2 + (-y)^2 = x^2 + y^2 = r^2 ] - **Symmetry Relative to the y-axis**: If ((x, y)) is a point on the circle, then ((-x, y)) is also a point on the circle since: [ (-x)^2 + y^2 = x^2 + y^2 = r^2 ] - **Symmetry Relative to the Origin**: If ((x, y)) is a point on the circle, then ((-x, -y)) is also a point on the circle since: [ (-x)^2 + (-y)^2 = x^2 + y^2 = r^2 ] Hence, curve (1) is symmetric with respect to both axes and the origin. 2. **Identifying Symmetry of Curve (2): (b^2 x^2 + a^2 y^2 = b^2 a^2)** This is the equation of an ellipse centered at the origin ((0, 0)). - **Symmetry Relative to the x-axis**: If ((x, y)) is a point on the ellipse, then ((x, -y)) is also a point on the ellipse since: [ b^2 x^2 + a^2 (-y)^2 = b^2 x^2 + a^2 y^2 = b^2 a^2 ] - **Symmetry Relative to the y-axis**: If ((x, y)) is a point on the ellipse, then ((-x, y)) is also a point on the ellipse since: [ b^2 (-x)^2 + a^2 y^2 = b^2 x^2 + a^2 y^2 = b^2 a^2 ] - **Symmetry Relative to the Origin**: If ((x, y)) is a point on the ellipse, then ((-x, -y)) is also a point on the ellipse since: [ b^2 (-x)^2 + a^2 (-y)^2 = b^2 x^2 + a^2 y^2 = b^2 a^2 ] Hence, curve (2) is symmetric with respect to both axes and the origin. 3. **Identifying Symmetry of Curve (3): (b^2 x^2 - a^2 y^2 = b^2 a^2)** This is the equation of a hyperbola centered at the origin ((0, 0)). - **Symmetry Relative to the x-axis**: If ((x, y)) is a point on the hyperbola, then ((x, -y)) is also a point on the hyperbola since: [ b^2 x^2 - a^2 (-y)^2 = b^2 x^2 - a^2 y^2 = b^2 a^2 ] - **Symmetry Relative to the y-axis**: If ((x, y)) is a point on the hyperbola, then ((-x, y)) is also a point on the hyperbola, but by equation (b^2 (-x)^2 - a^2 y^2 = b^2 x^2 - a^2 y^2 = b^2 a^2 ). - **Symmetry Relative to the Origin**: If ((x, y)) is a point on the hyperbola, then ((-x, -y)) is also a point on the hyperbola since: [ b^2 (-x)^2 - a^2 (-y)^2 = b^2 x^2 - a^2 y^2 = b^2 a^2 ] Hence, curve (3) is symmetric with respect to both axes and the origin. 4. **Identifying Symmetry of Curve (4): (y^2 = 2px)** This is the equation of a parabola that opens to the right with the vertex at the origin. - **Symmetry Relative to the x-axis**: If ((x, y)) is a point on the parabola, then ((x, -y)) is also a point on the parabola since: [ (-y)^2 = y^2 = 2px ] Hence, curve (4) is symmetric with respect to the x-axis. 5. **Identifying Symmetry of Curve (5): (y = sin x)** This is the equation of the sine function. - **Symmetry Relative to the Origin**: The sine function is an odd function, meaning if ((x, y)) is a point on the graph, then ((-x, -y)) is also a point on the graph since: [ sin(-x) = -sin(x) ] Hence, curve (5) is symmetric with respect to the origin. 6. **Identifying Symmetry of Curve (6): (y = tan x)** This is the equation of the tangent function. - **Symmetry Relative to the Origin**: The tangent function is an odd function, meaning if ((x, y)) is a point on the graph, then ((-x, -y)) is also a point on the graph since: [ tan(-x) = -tan(x) ] Hence, curve (6) is symmetric with respect to the origin. 7. **Identifying Symmetry of Curve (7): (y = log x)** This is the equation of the logarithmic function. - The logarithmic function does not exhibit symmetry relative to the x-axis, y-axis, or the origin. It is defined only for (x > 0) and has no corresponding negative parts to reflect symmetry. Hence, curve (7) does not possess any symmetry properties. # Conclusion: boxed{text{Curves (1), (2), (3) are symmetric with respect to both axes and the origin; Curve (4) is symmetric with respect to the x-axis; Curves (5) and (6) are symmetric with respect to the origin; Curve (7) has no symmetry properties.}}

answer:To find the original value of the tempo, we first need to determine the insured value of the tempo, which is the amount that the premium is based on. Let's denote the original value of the tempo as V. The tempo is insured to an extent of 5/7 of its original value, so the insured value (I) is: I = (5/7) * V The premium is calculated at a rate of 3% on the insured value, and it amounts to 300. So we can write the equation for the premium (P) as: P = (3/100) * I We know that P = 300, so we can substitute this value into the equation: 300 = (3/100) * I Now we can substitute the expression for I from the first equation into this equation: 300 = (3/100) * (5/7) * V Now we can solve for V: 300 = (3 * 5/7 * V) / 100 300 = (15/700) * V 300 = (3/140) * V Now we multiply both sides of the equation by 140/3 to solve for V: V = (300 * 140) / 3 Now we calculate the value of V: V = (42000) / 3 V = 14000 So the original value of the tempo is boxed{14000} .

answer:Given overline{z}=i, where i is the imaginary unit, we aim to find the location of the complex number z on the complex plane. 1. The conjugate of a complex number z = a + bi is overline{z} = a - bi. Given that overline{z} = i, we can deduce that a - bi = 0 + 1i. 2. Comparing real and imaginary parts, we get a = 0 and -b = 1, which implies b = -1. 3. Therefore, z = 0 - 1i = -i. 4. The complex number -i corresponds to the point (0, -1) on the complex plane. 5. The point (0, -1) lies on the imaginary axis, not on the real axis, nor does it lie on the angle bisectors of the first and third or second and fourth quadrants. Hence, the correct answer is boxed{B}, indicating that the point corresponding to the complex number z on the complex plane must lie on the imaginary axis.

answer:1. Let's define the total visible landscape as (1) unit. 2. According to the problem, the cloud covers half of the visible landscape, which is: [ text{Area covered by the cloud} = frac{1}{2} ] 3. The island, which is partially covered by the cloud, represents (frac{1}{4}) of the visible landscape when uncovered. Therefore, the uncovered island area is: [ text{Uncovered island area} = frac{1}{4} ] 4. Since the total visible island area (both covered and uncovered) must be ( frac{1}{3}) of the entire landscape: [ text{Total island area} = frac{1}{3} ] 5. The proportion of the island covered by the cloud is calculated by subtracting the uncovered part of the island from the total island area: [ text{Island area covered by the cloud} = frac{1}{3} - frac{1}{4} ] 6. To find a common denominator: [ frac{1}{3} = frac{4}{12} quad text{and} quad frac{1}{4} = frac{3}{12} ] [ text{Island area covered by the cloud} = frac{4}{12} - frac{3}{12} = frac{1}{12} ] 7. Knowing that the clouds cover a total of (frac{1}{2}) of the landscape, the sea area covered by clouds is: [ text{Sea area covered by the cloud} = frac{1}{2} - text{Island area covered by the cloud} ] [ = frac{1}{2} - frac{1}{12} ] 8. To find a common denominator for the subtraction: [ frac{1}{2} = frac{6}{12} ] [ text{Sea area covered by the cloud} = frac{6}{12} - frac{1}{12} = frac{5}{12} ] 9. Given that the sea occupies (frac{2}{3}) of the scene visible from the airplane: [ text{Sea fraction visible not covered by clouds} = frac{2}{3} - text{Sea area covered by the cloud} ] 10. Substituting the values and finding a common denominator for the entire calculation again: [ = frac{2}{3} - frac{5}{12} ] [ frac{2}{3} = frac{8}{12} ] [ = frac{8}{12} - frac{5}{12} = frac{3}{12} = frac{1}{4} ] 11. Now, calculate the fraction of the sea hidden by the cloud, we subtract (frac{1}{4}) from (frac{2}{3}) like this: [ frac{2}{3} - frac{1}{4} = frac{8}{12} - frac{3}{12} = frac{5}{12} ] 12. Finally, the portion of the sea hidden by the cloud expressed as a fraction of the total visible sea is: [ frac{frac{5}{12}}{frac{2}{3}} = frac{5}{12} div frac{2}{3} = frac{5}{12} times frac{3}{2} = frac{5 times 3}{12 times 2} = frac{15}{24} = frac{5}{8} ] Conclusion: [ boxed{frac{5}{8}} ]