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 problems: 23. A plan of a house is drawn to a scale of 2:1000. A room in the house is 6m long. The length of the room on the map is: Step 1: Convert the real length of the room from meters to centimeters. 6 m = 6 × 100 cm = 600 cm Step 2: Apply the given scale of 2:1000. This means 2 units on the map represent 1000 units in real life. Length on mapActual length = (2)/(1000) Step 3: Substitute the actual length and solve for the length on the map. Length on map600 cm = (2)/(1000) Length on map = (2)/(1000) × 600 cm Length on map = (1200)/(1000) cm Length on map = 1.2 cm The length of the room on the map is 1.2 cm. The correct option is [B] 1.2cm. 24. The y-intercept of a line is -5 and the gradient is -3. The equation of the line is: Step 1: Recall the general equation of a straight line, which is y = mx + c, where m is the gradient and c is the y-intercept. Step 2: Substitute the given values for the gradient (m = -3) and the y-intercept (c = -5) into the equation. y = (-3)x + (-5) y = -3x - 5 The equation of the line is y = -3x - 5. The correct option is [D] y = -3x - 5. 25. The coordinates of the midpoint of the straight line joining the points M(4, y) and N(-6, 4) is P(-1, 2). The value of y is: Step 1: Recall the midpoint formula for two points (x_1, y_1) and (x_2, y_2), which is P((x_1+x_2)/(2), (y_1+y_2)/(2)). Step 2: We are given the points M(4, y) and N(-6, 4), and the midpoint P(-1, 2). We need to find the value of y. We will use the y-coordinate part of the midpoint formula. y_P = (y_M + y_N)/(2) Step 3: Substitute the given y-coordinates into the formula. 2 = (y + 4)/(2) Step 4: Solve for y. 2 × 2 = y + 4 4 = y + 4 y = 4 - 4 y = 0 The value of y is 0. The correct option is [D] 0. 26. Given that (m-n)/(x) = y, then expressing n in terms of m, y and x gives? Step 1: Start with the given equation. (m-n)/(x) = y Step 2: Multiply both sides by x to eliminate the denominator. m-n = xy Step 3: Subtract m from both sides to isolate the term with n. -n = xy - m Step 4: Multiply both sides by -1 to solve for n. n = -(xy - m) n = -xy + m n = m - xy Expressing n in terms of m, y and x gives n = m - xy. The correct option is [C] m - xy. 27. Given that x + 2y = 5 and x - 2y = 8. Then the value of 4x is: Step 1: Write down the given system of equations. 1) x + 2y = 5 2) x - 2y = 8 Step 2: Add equation (1) and equation (2) to eliminate y. (x + 2y) + (x - 2y) = 5 + 8 x + 2y + x - 2y = 13 2x = 13 Step 3: Solve for x. x = (13)/(2) Step 4: Calculate the value of 4x. 4x = 4 × (13)/(2) 4x = 2 × 13 4x = 26 The value of 4x is 26.