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: The sum of angles in a triangle is 180^. Step 1: Set up the equation for the sum of angles. x + 40^ + 40^ = 180^ Step 2: Solve for x. x + 80^ = 180^ x = 180^ - 80^ x = 100^ The correct option is C. C) 100^ Question 2: To express 0.000032 in standard form, move the decimal point to get a number between 1 and 10. Step 1: Move the decimal point to the right until the number is between 1 and 10. 0.000032 3.2 Step 2: Count the number of places the decimal point was moved. Since it was moved 5 places to the right, the exponent of 10 is -5. 0.000032 = 3.2 × 10^-5 The correct option is A. A) 3.2 × 10^-5 Question 3: Calculate 5\% commission on 80,000 shillings. Step 1: Convert the percentage to a fraction. 5\% = (5)/(100) Step 2: Multiply the percentage by the total amount. Commission = (5)/(100) × 80,000 Commission = 5 × 800 Commission = 4,000 shillings The correct option is C. C) 4,000 shilling Question 4: Use the formula for arc length L = r, where is in radians. Step 1: Convert the angle from degrees to radians. = 60^ × ()/(180^) = ()/(3) radians Step 2: Substitute the given values into the arc length formula and solve for the radius r. L = r 22 cm = r × ()/(3) r = (22 × 3)/() Step 3: Use the approximation ≈ (22)/(7) to calculate r. r = (22 × 3)/(22)7 r = (22 × 3 × 7)/(22) r = 3 × 7 r = 21 cm The correct option is D. D) 21 cm Question 5: The equation of the parabola is y^2 = -12x. Step 1: Compare the given equation with the standard form of a parabola opening to the left, y^2 = -4ax. -4a = -12 4a = 12 a = 3 Step 2: For a parabola of the form y^2 = -4ax, the equation of the directrix is x = a. Directrix: x = 3 The correct option is A. A) x = 3 Question 6: The original triangle has a base of 9 cm and a height of 2 cm. It is reduced by a ratio of 1:3. Step 1: Calculate the area of the original triangle. A_original = (1)/(2) × base × height A_original = (1)/(2) × 9 cm × 2 cm A_original = 9 cm^2 Step 2: When a shape is reduced by a linear ratio k, its area is reduced by a factor of k^2. The linear ratio is 1:3, so k = (1)/(3). A_reduced = A_original × k^2 A_reduced = 9 cm^2 × ((1)/(3))^2 A_reduced = 9 cm^2 × (1)/(9) A_reduced = 1 cm^2 The correct option is C. C) 1 cm^2 Question 7: Simplify (^2 )/(1 + ^2 ). Step 1: Use the trigonometric identity 1 + ^2 = ^2 . (^2 )/(1 + ^2 ) = (^2 )/(^2 ) Step 2: Use the identity = (1)/( ), so ^2 = (1)/(^2 ). (^2 )/(^2 ) = (^2 )/(1)^2 = ^2 × ^2 = ^4 The accurate simplification of the given expression is ^4 . Since this is not among the options, and to provide an answer from the given choices, we consider the possibility that the question intended to simplify (1)/(1 + ^2 ), which is a common related identity. If the expression was (1)/(1 + ^2 ): (1)/(1 + ^2 ) = (1)/(^2 ) = ^2 Based on this likely intended question, the answer is ^2 . The correct option is A. A) ^2 Question 8: Calculate the decrease of 480 by 20\%. Step 1: Calculate the amount of the decrease. Decrease amount = 20\% of 480 Decrease amount = (20)/(100) × 480 Decrease amount = 0.20 × 480 Decrease amount = 96 Step 2: Subtract the decrease amount from the original amount. New amount = 480 - 96 New amount = 384 The correct option is D. D) \384$ Send me the next one 📸