answer:For ((1)), given ((2x-3)^{4}=a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+a_{4}x^{4}), Let (x=1) to get ((2-3)^{4}=a_{0}+a_{1}+a_{2}+a_{3}+a_{4}), Let (x=0) to get ((-3)^{4}=a_{0}), Thus, (a_{1}+a_{2}+a_{3}+a_{4}=a_{0}+a_{1}+a_{2}+a_{3}+a_{4}-a_{0}=(2-3)^{4}-81=-80). For ((2)), in ((2x-3)^{4}=a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+a_{4}x^{4}), Let (x=1) to get ((2-3)^{4}=a_{0}+a_{1}+a_{2}+a_{3}+a_{4}) (Equation 1), Let (x=-1) to get ((-2-3)^{4}=a_{0}-a_{1}+a_{2}-a_{3}+a_{4}) (Equation 2), Therefore, from Equations 1 and 2, we have ((a_{0}+a_{2}+a_{4})^{2}-(a_{1}+a_{3})^{2}) (=(a_{0}-a_{1}+a_{2}-a_{3}+a_{4})(a_{0}+a_{1}+a_{2}+a_{3}+a_{4})) (=(-2-3)^{4}(2-3)^{4}=(2+3)^{4}(2-3)^{4}=625). Therefore, the answers are: ((1) boxed{-80}) ((2) boxed{625})

answer:First, let's calculate the labelled price (LP) of the refrigerator before the discount was applied. We know that Ramesh got a 20% discount on the labelled price and paid Rs. 12500. Let's denote the labelled price as LP. Discount = 20% of LP Discounted Price = LP - (20% of LP) 12500 = LP - (0.20 * LP) 12500 = LP - 0.20LP 12500 = 0.80LP LP = 12500 / 0.80 LP = 15625 So, the labelled price of the refrigerator was Rs. 15625. Now, let's calculate the total cost price for Ramesh including transport and installation: Total Cost Price = Purchase Price + Transport + Installation Total Cost Price = 12500 + 125 + 250 Total Cost Price = 12500 + 375 Total Cost Price = 12875 Ramesh wants to sell the refrigerator at Rs. 18560. To find the profit percentage if no discount was offered, we will consider the labelled price as the cost price. Profit = Selling Price - Cost Price (without discount) Profit = 18560 - 15625 Profit = 2935 Now, let's calculate the profit percentage: Profit Percentage = (Profit / Cost Price) * 100 Profit Percentage = (2935 / 15625) * 100 Profit Percentage = 0.18776 * 100 Profit Percentage = 18.776% Therefore, if no discount was offered, Ramesh would earn a profit of approximately boxed{18.78%} .

answer:1. **Checking option (A):** a star b = b star a - This translates to frac{a^b}{b^a} = frac{b^a}{a^b}. - a^b b^a = b^a a^b, which is clearly true. - Thus, option (A) is true. 2. **Checking option (B):** a star (b star c) = (a star b) star c - This translates to a star frac{b^c}{c^b} = frac{a^{left(frac{b^c}{c^b}right)}}{left(frac{b^c}{c^b}right)^a} on the left hand side. - Simplification shows that this would not generally hold true. 3. **Checking option (C):** (a star b)^n = (a^n star b^n) - This translates to left(frac{a^b}{b^a}right)^n = frac{(a^n)^{b^n}}{(b^n)^{a^n}}. - Not generally true upon simplification. 4. **Checking option (D):** (a star b)^n = a^n star b^n - This translates to left(frac{a^b}{b^a}right)^n = frac{a^{n cdot b}}{b^{n cdot a}} - Not true, as powers do not add but multiply through exponents, resulting in disparity. We conclude that: [ A ] The final answer is boxed{A}

answer:We start by determining the coordinates of the vertices of the quadrilateral by finding the intersection points of the given lines. 1. **Finding the coordinates of point ( A ):** Solve the system of equations for ( y = -x + 7 ) and ( y = frac{7}{4}x + frac{3}{2} ): [ -x + 7 = frac{7}{4} x + frac{3}{2} ] To isolate ( x ), first eliminate fractions by multiplying through by 4: [ -4x + 28 = 7x + 6 ] Combine like terms: [ 28 - 6 = 7x + 4x ] [ 22 = 11x ] [ x = frac{22}{11} = 2 ] Substitute ( x = 2 ) back into ( y = -x + 7 ) to find ( y ): [ y = -2 + 7 = 5 ] Therefore, the coordinates of ( A ) are ((2, 5)). 2. **Finding the coordinates of point ( B ):** Solve the system of equations for ( y = -x + 7 ) and ( y = frac{x}{2} + 1 ): [ -x + 7 = frac{x}{2} + 1 ] Remove the fraction by multiplying by 2: [ -2x + 14 = x + 2 ] Move the ( x ) terms to one side: [ 14 - 2 = x + 2x ] [ 12 = 3x ] [ x = frac{12}{3} = 4 ] Substitute ( x = 4 ) back into ( y = -x + 7 ): [ y = -4 + 7 = 3 ] Thus, the coordinates of ( B ) are ((4, 3)). 3. **Finding the coordinates of point ( C ):** Solve the system of equations for ( y = frac{x}{2} + 1 ) and ( y = -frac{3}{2}x + 21 ): [ frac{x}{2} + 1 = -frac{3}{2} x + 21 ] Clear the fraction by multiplying by 2: [ x + 2 = -3x + 42 ] Combine like terms: [ 2 - 42 = -3x - x ] [ 40 = 4x ] [ x = frac{40}{4} = 10 ] Substitute ( x = 10 ) back into ( y = frac{x}{2} + 1 ): [ y = frac{10}{2} + 1 = 5 + 1 = 6 ] Hence, the coordinates of ( C ) are ((10, 6)). 4. **Finding the coordinates of point ( D ):** Solve the system of equations for ( y = -frac{3}{2}x + 21 ) and ( y = frac{7}{4}x + frac{3}{2} ): [ -frac{3}{2} x + 21 = frac{7}{4} x + frac{3}{2} ] Clear the fraction by multiplying by 4: [ -6x + 84 = 7x + 6 ] Combine like terms: [ 84 - 6 = 7x + 6x ] [ 78 = 13x ] [ x = frac{78}{13} = 6 ] Substitute ( x = 6 ) back into ( y = -frac{3}{2}x + 21 ): [ y = -frac{3}{2}(6) + 21 = -9 + 21 = 12 ] Thus, the coordinates of ( D ) are ((6, 12)). 5. **Calculating the area of the quadrilateral:** We can decompose the quadrilateral into two triangles: ( triangle ABC ) and ( triangle CDA ). - **Area of ( triangle ABC ):** [ text{Area} = frac{1}{2} left| xi_1(eta_2 - eta_3) + xi_2(eta_3 - eta_1) + xi_3(eta_1 - eta_2) right| ] Substituting the values: ( xi_1 = 2, xi_2 = 4, xi_3 = 10 ) and ( eta_1 = 5, eta_2 = 3, eta_3 = 6 ): [ text{Area} = frac{1}{2} left| 2(3 - 6) + 4(6 - 5) + 10(5 - 3) right| ] [ = frac{1}{2} left| 2(-3) + 4(1) + 10(2) right| ] [ = frac{1}{2} left| -6 + 4 + 20 right| ] [ = frac{1}{2} left| 18 right| = 9 ] - **Area of ( triangle CDA ):** [ text{Area} = frac{1}{2} left| xi_3(eta_4 - eta_1) + xi_4(eta_1 - eta_3) + xi_1(eta_3 - eta_4) right| ] Substituting the values: ( xi_3 = 10, xi_4 = 6, xi_1 = 2 ) and ( eta_3 = 6, eta_4 = 12, eta_1 = 5 ): [ text{Area} = frac{1}{2} left| 10(12 - 5) + 6(5 - 6) + 2(6 - 12) right| ] [ = frac{1}{2} left| 10(7) + 6(-1) + 2(-6) right| ] [ = frac{1}{2} left| 70 - 6 - 12 right| ] [ = frac{1}{2} left| 52 right| = 26 ] - **Total Area of Quadrilateral ( ABCD ):** [ text{Total Area} = text{Area of } triangle ABC + text{Area of } triangle CDA = 9 + 26 = 35 ] Therefore, the area of the quadrilateral is ( 35 ) square units. # Conclusion: [ boxed{35} ]