Given that mn = 2m - n, where m and n are real numbers, find the value of (i) 5(45), (ii) n if n(3*n) = 6. (b) In the diagram, ABD is a straight line. O is the centre of the circle ABC, <OBC = 20°, <ADC = 32° and |CB| = |BD|, find <ACB.

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Asked on June 3, 2026Mathematics

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.

Accepted Answer · 2026-06-03T06:08:42.814Z

Here's the solution to your question. 10. (a) Given that m * n = 2m - n, where m and n are real numbers, find the value of (i) 5 (4 5) Step 1: Calculate the expression inside the parenthesis, 4 * 5. Using the definition m * n = 2m - n, with m=4 and n=5: 4 * 5 = 2(4) - 5 = 8 - 5 = 3 Step 2: Substitute the result back into the original expression and calculate 5 * 3. Using the definition m * n = 2m - n, with m=5 and n=3: 5 * 3 = 2(5) - 3 = 10 - 3 = 7 The value is 7. (ii) n if n (3 n) = 6. Step 1: Calculate the expression inside the parenthesis, 3 * n. Using the definition m * n = 2m - n, with m=3 and the second operand as n: 3 * n = 2(3) - n = 6 - n Step 2: Substitute this result back into the given equation: n * (6 - n) = 6. Now, apply the definition m * n = 2m - n again, where the first operand is n and the second operand is (6 - n): 2(n) - (6 - n) = 6 Step 3: Solve the equation for n. 2n - 6 + n = 6 3n - 6 = 6 3n = 6 + 6 3n = 12 n = (12)/(3) n = 4 The value of n is 4. 10. (b) In the diagram, ABD is a straight line. O is the centre of the circle ABC, OBC = 20^, ADC = 32^ and |CB| = |BD|, find ACB. Step 1: Use the properties of OBC. Since O is the center of the circle, OB = OC (radii). Therefore, OBC is an isosceles triangle. The base angles are equal: OCB = OBC = 20^. The angle at the center BOC = 180^ - ( OBC + OCB) = 180^ - (20^ + 20^) = 180^ - 40^ = 140^. Step 2: Relate the central angle to the inscribed angle. The angle subtended by an arc at the center is twice the angle subtended by the same arc at any point on the remaining part of the circle. For arc BC, BAC is the inscribed angle and BOC is the central angle. BAC = (1)/(2) BOC = (1)/(2) (140^) = 70^ Step 3: Use the properties of CBD. Given that |CB| = |BD|, CBD is an isosceles triangle. The base angles are equal: BCD = BDC. We are given ADC = 32^, which is the same as BDC. So, BCD = 32^. The angle CBD = 180^ - ( BCD + BDC) = 180^ - (32^ + 32^) = 180^ - 64^ = 116^. Step 4: Use the straight line ABD. Since ABD is a straight line, ABC and CBD are supplementary angles. ABC + CBD = 180^ ABC + 116^ = 180^ ABC = 180^ - 116^ = 64^ Step 5: Find ACB using the sum of angles in ABC. In ABC, the sum of angles is 180^. ACB = 180^ - ( BAC + ABC) ACB = 180^ - (70^ + 64^) ACB = 180^ - 134^ ACB = 46^ The value of ACB is 46^. That's 2 down. 3 left today — send the next one.

Accepted Answer · 2026-06-03T06:08:42.814Z

Here's the solution to your question. 10. (a) Given that m * n = 2m - n, where m and n are real numbers, find the value of (i) 5 (4 5) Step 1: Calculate the expression inside the parenthesis, 4 * 5. Using the definition m * n = 2m - n, with m=4 and n=5: 4 * 5 = 2(4) - 5 = 8 - 5 = 3 Step 2: Substitute the result back into the original expression and calculate 5 * 3. Using the definition m * n = 2m - n, with m=5 and n=3: 5 * 3 = 2(5) - 3 = 10 - 3 = 7 The value is 7. (ii) n if n (3 n) = 6. Step 1: Calculate the expression inside the parenthesis, 3 * n. Using the definition m * n = 2m - n, with m=3 and the second operand as n: 3 * n = 2(3) - n = 6 - n Step 2: Substitute this result back into the given equation: n * (6 - n) = 6. Now, apply the definition m * n = 2m - n again, where the first operand is n and the second operand is (6 - n): 2(n) - (6 - n) = 6 Step 3: Solve the equation for n. 2n - 6 + n = 6 3n - 6 = 6 3n = 6 + 6 3n = 12 n = (12)/(3) n = 4 The value of n is 4. 10. (b) In the diagram, ABD is a straight line. O is the centre of the circle ABC, OBC = 20^, ADC = 32^ and |CB| = |BD|, find ACB. Step 1: Use the properties of OBC. Since O is the center of the circle, OB = OC (radii). Therefore, OBC is an isosceles triangle. The base angles are equal: OCB = OBC = 20^. The angle at the center BOC = 180^ - ( OBC + OCB) = 180^ - (20^ + 20^) = 180^ - 40^ = 140^. Step 2: Relate the central angle to the inscribed angle. The angle subtended by an arc at the center is twice the angle subtended by the same arc at any point on the remaining part of the circle. For arc BC, BAC is the inscribed angle and BOC is the central angle. BAC = (1)/(2) BOC = (1)/(2) (140^) = 70^ Step 3: Use the properties of CBD. Given that |CB| = |BD|, CBD is an isosceles triangle. The base angles are equal: BCD = BDC. We are given ADC = 32^, which is the same as BDC. So, BCD = 32^. The angle CBD = 180^ - ( BCD + BDC) = 180^ - (32^ + 32^) = 180^ - 64^ = 116^. Step 4: Use the straight line ABD. Since ABD is a straight line, ABC and CBD are supplementary angles. ABC + CBD = 180^ ABC + 116^ = 180^ ABC = 180^ - 116^ = 64^ Step 5: Find ACB using the sum of angles in ABC. In ABC, the sum of angles is 180^. ACB = 180^ - ( BAC + ABC) ACB = 180^ - (70^ + 64^) ACB = 180^ - 134^ ACB = 46^ The value of ACB is 46^. That's 2 down. 3 left today — send the next one.

Given that mn = 2m - n, where m and n are real numbers, find the value of (i) 5(45), (ii) n if n(3*n) = 6. (b) In the diagram, ABD is a straight line. O is the centre of the circle ABC, <OBC = 20°, <ADC = 32° and |CB| = |BD|, find <ACB.

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Given that mn = 2m - n, where m and n are real numbers, find the value of (i) 5(45), (ii) n if n(3*n) = 6. (b) In the diagram, ABD is a straight line. O is the centre of the circle ABC, <OBC = 20°, <ADC = 32° and |CB| = |BD|, find <ACB.

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