MATHEMATICS PAPER 1 [40 Marks]

Here are the solutions for the questions you've sent. Question 1: If 4^x = 2^x+3, find the value of x. Step 1: Express both sides of the equation with the same base. Since 4 = 2^2, we can rewrite 4^x as (2^2)^x. (2^2)^x = 2^x+3 2^2x = 2^x+3 Step 2: Since the bases are the same, equate the exponents. 2x = x+3 Step 3: Solve for x. 2x - x = 3 x = 3 The value of x is B. 3. Question 2: Kwaku and Attah formed a company and agreed that their profit will be shared in the ratio 3:2 respectively. At the end of the year, Attah received GH¢5000.00 more than Kwaku. How much profit did the company make in the year? Assumption: The problem states the ratio for Kwaku:Attah is 3:2, but then says Attah received GH¢5000.00 more than Kwaku. This implies Attah's share is larger. To reconcile this, we assume the ratio of shares is Kwaku:Attah = 2:3, which means Attah gets 3 parts and Kwaku gets 2 parts. Step 1: Let the total profit be P. Let Kwaku's share be 2x and Attah's share be 3x. The difference in their shares is GH¢5000.00. 3x - 2x = 5000 Step 2: Solve for x. x = 5000 Step 3: Calculate the total profit, which is the sum of their shares (2x + 3x = 5x). Total Profit = 5x = 5 × 5000 Total Profit = 25000 The company made a profit of B. GH¢25,000.00. Question 3: Find the inverse of the matrix A = 2 & -3 \\ 1 & 4 . Step 1: Calculate the determinant of matrix A, denoted as (A). For a 2 × 2 matrix a & b \\ c & d , the determinant is ad - bc. (A) = (2)(4) - (-3)(1) (A) = 8 - (-3) (A) = 8 + 3 (A) = 11 Step 2: Find the adjugate matrix. For a 2 × 2 matrix a & b \\ c & d , the adjugate is d & -b \\ -c & a . adj(A) = 4 & -(-3) \\ -1 & 2 adj(A) = 4 & 3 \\ -1 & 2 Step 3: Calculate the inverse matrix A^-1 using the formula A^-1 = (1)/((A)) adj(A). A^-1 = (1)/(11) 4 & 3 \\ -1 & 2 A^-1 = (4)/(11) & (3)/(11) \\ -(1)/(11) & (2)/(11) The inverse of the matrix is B. (4)/(11) & (3)/(11) \\ -(1)/(11) & (2)/(11) . Question 4: Find the equation of the line with gradient 3 which passes through the point (-2, 0). Step 1: Use the point-slope form of a linear equation, y - y_1 = m(x - x_1), where m is the gradient and (x_1, y_1) is the given point. Given m = 3 and (x_1, y_1) = (-2, 0). y - 0 = 3(x - (-2)) y = 3(x + 2) Step 2: Distribute the gradient and rearrange the equation into the general form Ax + By + C = 0 to match the options. y = 3x + 6 0 = 3x - y + 6 Or, multiplying by -1 to match option D: y - 3x - 6 = 0 The equation of the line is D. y - 3x - 6 = 0. Question 5: Solve 6x^2 - 13x = -5. Step 1: Rewrite the quadratic equation in standard form ax^2 + bx + c = 0. 6x^2 - 13x + 5 = 0 Step 2: Factor the quadratic expression. We look for two numbers that multiply to ac = 6 × 5 = 30 and add up to b = -13. These numbers are -3 and -10. Rewrite the middle term using these numbers: 6x^2 - 3x - 10x + 5 = 0 Step 3: Factor by grouping. 3x(2x - 1) - 5(2x - 1) = 0 (3x - 5)(2x - 1) = 0 Step 4: Set each factor equal to zero and solve for x. 3x - 5 = 0 3x = 5 x = (5)/(3) 2x - 1 = 0 2x = 1 x = (1)/(2) The solutions are x = (1)/(2) and x = (5)/(3). Note that (5)/(3) can be written as 1(2)/(3). The correct option is C. x = (1)/(2), x = 1(2)/(3). Question 6: Which of the following is the definition of a prism? A prism is a three-dimensional solid object with two identical ends (bases) that are parallel and straight sides. The sides are parallelograms. Based on this definition and the visible options: A. A solid figure with a rectangular base. (This is too specific and could describe other shapes like pyramids.) B. A solid figure with two identical and parallel bases. (This is the defining characteristic of a prism.) C. A solid figure with a circular base. (This describes a cylinder, which is a type of prism, but not the general definition.) D. A solid figure with a triangular cross-section. (This describes a triangular prism, not the general definition.) The most accurate general definition of a prism among the choices is B. A solid figure with two identical and parallel bases. Drop the next question.