This English question involves literary analysis, grammar, or writing skills. The detailed response below provides a well-structured answer with supporting evidence and clear explanations.
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Here are the detailed solutions for each question. Question 1: A shop sells a TV set at a discount of 15% on the normal price. Kina bought the TV set for 357,000 fcfa. Calculate the normal price of the TV set. Step 1: Determine the percentage of the normal price paid. If the discount is 15%, then Kina paid 100\% - 15\% = 85\% of the normal price. Step 2: Set up the equation. Let P be the normal price. We have 0.85 × P = 357\,000. Step 3: Solve for P. P = (357\,000)/(0.85) P = 420\,000 The normal price of the TV set is 420,000 fcfa. 420,000 fcfa Question 2: Given that (x-1) is one of the factors of f(x) = 2x^3 + x^2 - 2x - 1. a) Show that (x-1) is a factor of f(x). Step 1: Apply the Factor Theorem. According to the Factor Theorem, (x-1) is a factor of f(x) if and only if f(1) = 0. Step 2: Calculate f(1). f(1) = 2(1)^3 + (1)^2 - 2(1) - 1 f(1) = 2(1) + 1 - 2 - 1 f(1) = 2 + 1 - 2 - 1 f(1) = 0 Since f(1) = 0, we have shown that (x-1) is a factor of f(x). b) Find the other two factors of f(x). Step 1: Perform polynomial division. Since (x-1) is a factor, we can divide f(x) by (x-1) to find the quotient. Using synthetic division with the root x=1: ` 1 | 2 1 -2 -1 | 2 3 1 ----------------- 2 3 1 0 ` The quotient is 2x^2 + 3x + 1. Step 2: Factor the quadratic quotient. We need to factor the quadratic polynomial 2x^2 + 3x + 1. We look for two numbers whose product is 2 × 1 = 2 and whose sum is 3. These numbers are 1 and 2. 2x^2 + 3x + 1 = 2x^2 + 2x + x + 1 2x^2 + 3x + 1 = 2x(x+1) + 1(x+1) 2x^2 + 3x + 1 = (2x+1)(x+1) The other two factors of f(x) are (2x+1) and (x+1). (2x+1) and (x+1) Question 3: a) Find the value of x, given that 2^x+1 = (1)/(4^3). Step 1: Rewrite the right side with the same base. We know that 4 = 2^2. 2^x+1 = (1)/((2^2)^3) 2^x+1 = (1)/(2^6) Step 2: Use the property of exponents (1)/(a^n) = a^-n. 2^x+1 = 2^-6 Step 3: Equate the exponents. Since the bases are equal, the exponents must be equal. x+1 = -6 x = -6 - 1 x = -7 The value of x is -7. -7 b) Simplify (5-x)/(5) - (2-x)/(10). Step 1: Find a common denominator. The common denominator for 5 and 10 is 10. Rewrite the first fraction with the denominator 10: (5-x)/(5) = (2(5-x))/(2 × 5) = (10-2x)/(10) Step 2: Combine the fractions. (10-2x)/(10) - (2-x)/(10) ((10-2x) - (2-x))/(10) Step 3: Simplify the numerator. (10-2x-2+x)/(10) (8-x)/(10) The simplified expression is (8-x)/(10). (8-x)/(10) Question 4: The function f is defined on R as f(x) = 2-8x^2. a) Evaluate f(2). Step 1: Substitute x=2 into the function. f(2) = 2 - 8(2)^2 f(2) = 2 - 8(4) f(2) = 2 - 32 f(2) = -30 The value of f(2) is -30. -30 b) Factorize f(x) completely. Step 1: Factor out the common factor. f(x) = 2 - 8x^2 f(x) = 2(1 - 4x^2) Step 2: Use the difference of squares formula (a^2 - b^2 = (a-b)(a+b)). Here, 1 = 1^2 and 4x^2 = (2x)^2. f(x) = 2(1^2 - (2x)^2) f(x) = 2(1-2x)(1+2x) The factored form of f(x) is 2(1-2x)(1+2x). 2(1-2x)(1+2x) c) Solve the equation f(x) = 0. Step 1: Use the factored form of f(x) and set it to zero. 2(1-2x)(1+2x) = 0 Step 2: Solve for x using the zero product property. For the product to be zero, at least one of the factors must be zero. 1-2x = 0 or 1+2x = 0 1 = 2x or -1 = 2x x = (1)/(2) or x = -(1)/(2) The solutions to the equation f(x)=0 are x = (1)/(2) and x = -(1)/(2). x = (1)/(2), x = -(1)/(2) Question 5: Copy and complete the table accompanying this question. The truth table needs to be completed for ~p (negation of p) and q ^ ~p (q AND not p). | p | q | ~p | q ^ ~p | |---|---|----|--------| | T | T | F | F | | T | F | F | F | | F | T | T | T | | F | F | T | F | Here is the completed table: |c|c|c|c| p & q & & q \\ T & T & F & F \\ T & F & F & F \\ F & T & T & T \\ F & F & T & F \\ That's 2 down. 3 left today — send the next one.