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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Welcome back dineo❤️! — missed you this week. Here are the solutions to your questions: QUESTION 1 1.1 To find between which two integers sqrt(11) lies, we find the perfect squares closest to 11. Step 1: Identify perfect squares around 11. 3^2 = 9 4^2 = 16 Step 2: Place sqrt(11) between the square roots of these perfect squares. sqrt(9) < sqrt(11) < sqrt(16) 3 < sqrt(11) < 4 Thus, sqrt(11) lies between the integers 3 and 4. 3 and 4 1.2 We classify the numbers using the given symbols: N (Natural numbers), Q (Rational numbers), Q' (Irrational numbers), Z (Integers), R (Real numbers). We will assume R' refers to non-real numbers in this context. 1.2.1 -1.525252... This is a repeating decimal, which can be expressed as a fraction. Therefore, it is a Rational number (Q) and also a Real number (R). Q, R 1.2.2 sqrt(-25) The square root of a negative number is an imaginary number. Imaginary numbers are not real numbers. Therefore, sqrt(-25) is a non-real number (R'). R' 1.3 To write 0,7 as a common fraction, we assume 0,7 means 0.7 (0.7 repeating). Step 1: Let x be the repeating decimal. x = 0.7 Step 2: Multiply by 10 to shift the decimal. 10x = 7.7 Step 3: Subtract the first equation from the second. 10x - x = 7.7 - 0.7 9x = 7 Step 4: Solve for x. x = (7)/(9) (7)/(9) QUESTION 2 2.1 Simplify the following: 2.1.1 (x - 2)(x^2 + 5x - 1) Step 1: Distribute each term from the first parenthesis to the second. x(x^2 + 5x - 1) - 2(x^2 + 5x - 1) Step 2: Multiply the terms. x^3 + 5x^2 - x - 2x^2 - 10x + 2 Step 3: Combine like terms. x^3 + (5x^2 - 2x^2) + (-x - 10x) + 2 x^3 + 3x^2 - 11x + 2 x^3 + 3x^2 - 11x + 2 2.1.2 3^x - 2 · 3^x-13^x+2 + 2 · 3^x+1 Step 1: Factor out the lowest power of 3 from the numerator and denominator. Numerator: 3^x - 2 · 3^x-1 = 3^x-1(3^1 - 2) Denominator: 3^x+2 + 2 · 3^x+1 = 3^x+1(3^1 + 2) Step 2: Simplify the terms in the parentheses. Numerator: 3^x-1(3 - 2) = 3^x-1(1) = 3^x-1 Denominator: 3^x+1(3 + 2) = 3^x+1(5) = 5 · 3^x+1 Step 3: Rewrite the fraction and simplify using exponent rules (a^m/a^n = a^m-n). 3^x-15 · 3^x+1 = (1)/(5) · 3^(x-1) - (x+1) (1)/(5) · 3^x-1-x-1 = (1)/(5) · 3^-2 Step 4: Evaluate 3^-2. (1)/(5) · (1)/(3^2) = (1)/(5) · (1)/(9) (1)/(45) (1)/(45) 2.2 Factorize the following expressions fully: 2.2.1 2x^2 - x - 6 Step 1: Find two numbers that multiply to 2 × (-6) = -12 and add to -1. These numbers are -4 and 3. Step 2: Rewrite the middle term using these numbers. 2x^2 - 4x + 3x - 6 Step 3: Factor by grouping. 2x(x - 2) + 3(x - 2) Step 4: Factor out the common binomial factor (x - 2). (x - 2)(2x + 3) (x - 2)(2x + 3) 2.2.2 a^2 - 2ab + b^2 - 100c^2 Step 1: Recognize the first three terms as a perfect square trinomial. (a^2 - 2ab + b^2) - 100c^2 (a - b)^2 - 100c^2 Step 2: Recognize the expression as a difference of squares, X^2 - Y^2 = (X - Y)(X + Y), where X = (a - b) and Y = 10c. (a - b)^2 - (10c)^2 Step 3: Apply the difference of squares formula. ((a - b) - 10c)((a - b) + 10c) (a - b - 10c)(a - b + 10c) (a - b - 10c)(a - b + 10c) 2.3 Solve the following equations simultaneously for a and b: Equation 1: a + b = 12 Equation 2: 4a + 2b = 44 Step 1: From Equation 1, express a in terms of b. a = 12 - b (Equation 3) Step 2: Substitute Equation 3 into Equation 2. 4(12 - b) + 2b = 44 Step 3: Distribute and solve for b. 48 - 4b + 2b = 44 48 - 2b = 44 -2b = 44 - 48 -2b = -4 b = (-4)/(-2) b = 2 Step 4: Substitute the value of b back into Equation 3 to find a. a = 12 - 2 a = 10 Step 5: State the solution. a = 10, b = 2 a = 10, b = 2 Drop the next question! 📸