Simplify sqrt(27) x sqrt(15)

Here are the solutions to the problems: Problem 8: Simplify sqrt(27) × sqrt(15) Step 1: Simplify each square root. sqrt(27) = sqrt(9 × 3) = sqrt(9) × sqrt(3) = 3sqrt(3) sqrt(15) = sqrt(3 × 5) Step 2: Multiply the simplified terms. sqrt(27) × sqrt(15) = 3sqrt(3) × sqrt(3 × 5) = 3 × sqrt(3 × 3 × 5) = 3 × sqrt(9 × 5) = 3 × sqrt(9) × sqrt(5) = 3 × 3 × sqrt(5) = 9sqrt(5) The correct option is D. D. 9sqrt(5) Problem 9: Find the radius of a circle whose area is the same as that of a rectangle of sides 4cm by 11cm. (take = (22)/(7)) Step 1: Calculate the area of the rectangle. Area of rectangle = length × width = 11 cm × 4 cm = 44 cm^2 Step 2: Set the area of the circle equal to the area of the rectangle. Area of circle = r^2 r^2 = 44 cm^2 Step 3: Substitute the value of and solve for r. (22)/(7) r^2 = 44 r^2 = 44 × (7)/(22) r^2 = 2 × 7 r^2 = 14 r = sqrt(14) cm Step 4: Approximate the value of r and compare with options. sqrt(14) ≈ 3.74 cm The closest option is B. B. 3.7 cm Problem 10: Find an expression for the nth term, U_n, of the sequence -2, 3, -4, 5, ... Step 1: Analyze the sequence. The absolute values of the terms are 2, 3, 4, 5, ..., which corresponds to (n+1) for n=1, 2, 3, 4, .... The signs alternate: negative, positive, negative, positive, ... For n=1, the term is negative. For n=2, the term is positive. This pattern of signs can be represented by (-1)^n or (-1)^n+1. If n=1, (-1)^1 = -1 (negative). If n=2, (-1)^2 = 1 (positive). So, the sign factor is (-1)^n. Step 2: Combine the absolute value and sign factor. The nth term U_n is (-1)^n × (n+1). The correct option is C. C. (-1)^n(n+1) Problem 11: Find the area of an equilateral triangle whose sides are 6cm. Step 1: Recall the formula for the area of an equilateral triangle. The area (A) of an equilateral triangle with side length s is given by: A = sqrt(3)4 s^2 Step 2: Substitute the given side length s = 6 cm. A = sqrt(3)4 (6 cm)^2 A = sqrt(3)4 × 36 cm^2 A = 9sqrt(3) cm^2 The correct option is D. D. 9sqrt(3) cm^2 Problem 12: How many terms are in an Arithmetic Progression whose third term is 6 and the common difference and last term are 2 and 100 respectively? Step 1: Identify the given information. Third term (U_3) = 6 Common difference (d) = 2 Last term (U_n) = 100 Step 2: Use the formula for the nth term of an AP, U_n = a + (n-1)d, to find the first term (a). For the third term: U_3 = a + (3-1)d 6 = a + 2(2) 6 = a + 4 a = 6 - 4 a = 2 Step 3: Use the first term, common difference, and last term to find the number of terms (n). U_n = a + (n-1)d 100 = 2 + (n-1)2 100 - 2 = (n-1)2 98 = 2(n-1) (98)/(2) = n-1 49 = n-1 n = 49 + 1 n = 50 The correct option is A. A. 50 Problem 13: Find the perimeter of a parallelogram whose adjacent sides are of lengths 8cm and 12cm. Step 1: Recall the formula for the perimeter of a parallelogram. A parallelogram has two pairs of equal adjacent sides. If the lengths of the adjacent sides are a and b, the perimeter (P) is given by: P = 2(a+b) Step 2: Substitute the given side lengths. Given a = 8 cm and b = 12 cm. P = 2(8 cm + 12 cm) P = 2(20 cm) P = 40 cm The correct option is D. D. 40 cm Problem 14: Let P and Q represent the statements: P = Olu is brilliant, Q = Sola is lazy. Which of the following does p q represent? Step 1: Understand the logical connective . The symbol represents the logical operator "AND". Step 2: Combine the statements P and Q using "AND". P: Olu is brilliant Q: Sola is lazy p q means "Olu is brilliant AND Sola is lazy". The correct option is C. C. Olu is brilliant and Sola is lazy That's 2 down. 3 left today — send the next one.