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. Problem 36: Find the quadratic equation whose roots are x = -3 and x = (1)/(2). Step 1: Calculate the sum and product of the roots. Let the roots be r_1 = -3 and r_2 = (1)/(2). Sum of roots: S = r_1 + r_2 = -3 + (1)/(2) = -(6)/(2) + (1)/(2) = -(5)/(2). Product of roots: P = r_1 × r_2 = (-3) × (1)/(2) = -(3)/(2). Step 2: Form the quadratic equation using the formula x^2 - Sx + P = 0. x^2 - (-(5)/(2))x + (-(3)/(2)) = 0 x^2 + (5)/(2)x - (3)/(2) = 0 Step 3: Multiply the entire equation by 2 to clear the denominators. 2(x^2) + 2((5)/(2)x) - 2((3)/(2)) = 2(0) 2x^2 + 5x - 3 = 0 The correct option is B. B. 2x^2 + 5x - 3 = 0 Problem 37: Calculate b in the figure below if B=60^, c=9cm and a=11cm. Step 1: Identify the given values and the appropriate formula. We are given two sides (a=11cm, c=9cm) and the included angle (B=60^). We need to find the third side b. The Cosine Rule is suitable here: b^2 = a^2 + c^2 - 2ac B Step 2: Substitute the values into the Cosine Rule. b^2 = (11)^2 + (9)^2 - 2(11)(9) 60^ Recall that 60^ = (1)/(2). b^2 = 121 + 81 - 2(11)(9)((1)/(2)) b^2 = 202 - (11)(9) b^2 = 202 - 99 b^2 = 103 Step 3: Solve for b. b = sqrt(103) b ≈ 10.1488 cm Rounding to two decimal places, b ≈ 10.15 cm. The correct option is B. B. 10.15cm Problem 38: Calculate the length of a chord of a circle of radius 5cm if the perpendicular distance from the center to the chord is 3cm. Step 1: Visualize the geometry and identify the relevant right-angled triangle. Let the radius be r=5cm. Let the perpendicular distance from the center to the chord be d=3cm. Let half the length of the chord be x. The radius, the perpendicular distance, and half the chord form a right-angled triangle, with the radius as the hypotenuse. Step 2: Apply the Pythagorean theorem. x^2 + d^2 = r^2 x^2 + (3)^2 = (5)^2 x^2 + 9 = 25 x^2 = 25 - 9 x^2 = 16 Step 3: Solve for x and then find the full length of the chord. x = sqrt(16) x = 4 cm The length of the full chord is 2x. Chord length = 2 × 4 = 8 cm The correct option is A. A. 8cm Problem 39: A sector of a circle of 8cm has 90^ at the centre. Calculate its area. Step 1: Identify the given values. Radius r = 8cm. Central angle = 90^. Step 2: Use the formula for the area of a sector. The area of a sector is given by A = ()/(360^) × r^2. A = (90^)/(360^) × (8cm)^2 A = (1)/(4) × (64cm^2) A = 16 cm^2 Step 3: Calculate the numerical value and round. Using ≈ 3.14159: A = 16 × 3.14159 cm^2 A ≈ 50.26544 cm^2 Rounding to two decimal places, A ≈ 50.27 cm^2. The correct option is D. D. 50.27cm^2 Problem 40: In the figure above, AD//GE, /BF/ = /CF/ and FBC = 55^. Find BFC. Step 1: Identify the type of triangle formed by B, F, and C. Given that /BF/ = /CF/, triangle BFC is an isosceles triangle. Step 2: Use the property of isosceles triangles. In an isosceles triangle, the angles opposite the equal sides are equal. Therefore, FBC = FCB. Given FBC = 55^, so FCB = 55^. Step 3: Use the sum of angles in a triangle. The sum of angles in BFC is 180^. BFC + FBC + FCB = 180^ BFC + 55^ + 55^ = 180^ BFC + 110^ = 180^ BFC = 180^ - 110^ BFC = 70^ The information AD//GE is not needed to solve for BFC. The correct option is B. B. 70^ Problem 41: The radii of two spheres are in the ratio 2:3. If the area of the smaller sphere is 32cm^2, calculate the area of the bigger sphere. Step 1: Set up the ratio of the radii. Let r_1 be the radius of the smaller sphere and r_2 be the radius of the bigger sphere. (r_1)/(r_2) = (2)/(3) Step 2: Relate the ratio of radii to the ratio of surface areas. The surface area of a sphere is given by A = 4 r^2. Let A_1 be the area of the smaller sphere and A_2 be the area of the bigger sphere. The ratio of their areas is: (A_1)/(A_2) = (4 r_1^2)/(4 r_2^2) = (r_1^2)/(r_2^2) = ((r_1)/(r_2))^2 Step 3: Substitute the given ratio and area to find the unknown area. (A_1)/(A_2) = ((2)/(3))^2 = (4)/(9) Given A_1 = 32 cm^2. (32)/(A_2) = (4)/(9) Cross-multiply to solve for A_2: 4 A_2 = 32 × 9 A_2 = (32 × 9)/(4) A_2 = 8 × 9 A_2 = 72 cm^2 The correct option is A. A. 72cm^2 3 done, 2 left today. You're making progress.