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 questions: Question 1: An isosceles triangle MNP has sides MN = MP = 26 cm and NP = 20 cm. To find the area, we first need to find the height. Step 1: Draw an altitude from M to the base NP. Let the point where the altitude meets NP be Q. In an isosceles triangle, the altitude to the base bisects the base. So, NQ = QP = (NP)/(2) = (20)/(2) = 10 cm. Step 2: Use the Pythagorean theorem in the right-angled triangle MQN to find the height MQ. MN^2 = MQ^2 + NQ^2 26^2 = MQ^2 + 10^2 676 = MQ^2 + 100 MQ^2 = 676 - 100 MQ^2 = 576 MQ = sqrt(576) MQ = 24 cm The height of the triangle is 24 cm. Step 3: Calculate the area of the triangle. Area = (1)/(2) × base × height Area = (1)/(2) × NP × MQ Area = (1)/(2) × 20 cm × 24 cm Area = 10 cm × 24 cm Area = 240 cm^2 The correct option is D. D. 240 cm^2 Question 2: Given that (x-2) is a factor of 8 - 2qx - x^2, we need to find the value of q. Step 1: Apply the Factor Theorem. If (x-2) is a factor of a polynomial P(x), then P(2) = 0. Let P(x) = 8 - 2qx - x^2. Step 2: Substitute x=2 into the polynomial and set the expression to zero. P(2) = 8 - 2q(2) - (2)^2 = 0 8 - 4q - 4 = 0 Step 3: Solve for q. 4 - 4q = 0 4 = 4q q = (4)/(4) q = 1 The correct option is D. D. 1 Question 3: Write 6.02517 to three significant figures. Step 1: Identify the first three significant figures. The first significant figure is 6. The second significant figure is 0. The third significant figure is 2. Step 2: Look at the fourth significant figure to determine rounding. The fourth significant figure is 5. Since it is 5 or greater, we round up the third significant figure. The third significant figure (2) becomes 3. Step 3: Write the number rounded to three significant figures. 6.03 The correct option is D. D. 6.03 Question 4: Factorize completely (h^2 - k^2) - p(h+k). Step 1: Recognize the difference of squares pattern in the first term. h^2 - k^2 = (h-k)(h+k) Step 2: Substitute this into the given expression. (h-k)(h+k) - p(h+k) Step 3: Factor out the common term (h+k). (h+k)[(h-k) - p] (h+k)(h-k-p) The correct option is D. D. (h+k)(h-k-p) Question 5: Solve the compound inequality 2x - 3 4x + 5 < 3x + 7. Step 1: Split the compound inequality into two separate inequalities. Inequality 1: 2x - 3 4x + 5 Inequality 2: 4x + 5 < 3x + 7 Step 2: Solve Inequality 1. 2x - 3 4x + 5 Subtract 2x from both sides: -3 2x + 5 Subtract 5 from both sides: -3 - 5 2x -8 2x Divide by 2: -4 x or x -4 Step 3: Solve Inequality 2. 4x + 5 < 3x + 7 Subtract 3x from both sides: x + 5 < 7 Subtract 5 from both sides: x < 7 - 5 x < 2 Step 4: Combine the solutions from both inequalities. We have x -4 and x < 2. Combining these gives: -4 x < 2 The correct option is D. D. -4 x < 2 3 done, 2 left today. You're making progress.