This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are the solutions to the given problems: 1. (i) Simple Interest Step 1: Identify the given values. The amount A = 13600 টাকা. The rate of simple interest R = 6\%. The time T = 6 years. Let the principal be P. Step 2: Use the formula for amount in simple interest. The formula for the amount A is A = P (1 + (RT)/(100)). Step 3: Substitute the values and solve for P. 13600 = P (1 + (6 × 6)/(100)) 13600 = P (1 + (36)/(100)) 13600 = P ((100 + 36)/(100)) 13600 = P ((136)/(100)) P = (13600 × 100)/(136) P = 100 × 100 P = 10000 The principal is 10000 টাকা. The correct option is (c). 10000 টাকা 1. (ii) Compound Interest Step 1: Identify the given values. The principal P = 100 টাকা. The amount A = 144 টাকা. The time T = 2 years. Let the rate of compound interest be R. Step 2: Use the formula for amount in compound interest. The formula for the amount A is A = P (1 + (R)/(100))^T. Step 3: Substitute the values and solve for R. 144 = 100 (1 + (R)/(100))^2 (144)/(100) = (1 + (R)/(100))^2 ((12)/(10))^2 = (1 + (R)/(100))^2 Taking the square root of both sides: (12)/(10) = 1 + (R)/(100) 1.2 = 1 + (R)/(100) 1.2 - 1 = (R)/(100) 0.2 = (R)/(100) R = 0.2 × 100 R = 20\% The compound interest rate is 20\%. The correct option is (b). 20\% 1. (iii) Quadratic Equations Step 1: Identify the coefficients of the quadratic equation. The given equation is 5x^2 + 2x - 3 = 0. Here, a=5, b=2, c=-3. Step 2: Find the sum and product of the roots ( and ). Sum of roots: + = -(b)/(a) = -(2)/(5) Product of roots: = (c)/(a) = (-3)/(5) Step 3: Calculate ^2 + ^2. We use the identity ^2 + ^2 = ( + )^2 - 2. Substitute the values: ^2 + ^2 = (-(2)/(5))^2 - 2(-(3)/(5)) ^2 + ^2 = (4)/(25) + (6)/(5) To add the fractions, find a common denominator: ^2 + ^2 = (4)/(25) + (6 × 5)/(5 × 5) ^2 + ^2 = (4)/(25) + (30)/(25) ^2 + ^2 = (4 + 30)/(25) ^2 + ^2 = (34)/(25) The correct option is (a). (34)/(25) 1. (iv) Mean Proportional Step 1: Identify the two terms. The two terms are (x-y)^2 and (x+y)^2. Step 2: Use the formula for the mean proportional. The mean proportional of two numbers a and b is sqrt(ab). Step 3: Calculate the mean proportional. Mean proportional = sqrt((x-y)^2 (x+y)^2) = sqrt([(x-y)(x+y)]^2) = (x-y)(x+y) Using the difference of squares formula, (a-b)(a+b) = a^2 - b^2: = x^2 - y^2 The correct option is (d). x^2 - y^2 1. (v) Cyclic Quadrilateral Step 1: Recall the property of a cyclic quadrilateral. In a cyclic quadrilateral, the sum of opposite angles is 180^. Step 2: Apply the property to the given angles. Given that ABCD is a cyclic quadrilateral and A = 80^. Therefore, A + C = 180^. Step 3: Solve for C. 80^ + C = 180^ C = 180^ - 80^ C = 100^ The correct option is (b). 100^ 1. (vi) Angle in a Semicircle Step 1: Recall the geometric property of an angle in a semicircle. An angle subtended by the diameter at any point on the circumference of a circle (i.e., an angle in a semicircle) is always a right angle. Step 2: State the value of a right angle. A right angle is 90^.