The matrix m = 1 & 5 2 & 7 and matrix n = 2 & 0 6 & 1 find:

Okay, I see matrix n now! Let's continue with the remaining questions. 4. The matrix m = 1 & 5 \\ 2 & 7 and matrix n = 2 & 0 \\ 6 & 1 find: a) m^T (already solved in previous message) b) nm b) Find nm. Step 1: Multiply matrix n by matrix m. nm = 2 & 0 \\ 6 & 1 1 & 5 \\ 2 & 7 Step 2: Perform the matrix multiplication. nm = (2 × 1) + (0 × 2) & (2 × 5) + (0 × 7) \\ (6 × 1) + (1 × 2) & (6 × 5) + (1 × 7) Step 3: Calculate the values for each element. nm = 2 + 0 & 10 + 0 \\ 6 + 2 & 30 + 7 nm = 2 & 10 \\ 8 & 37 The product nm is 2 & 10 \\ 8 & 37 . 5. y varies directly as the square of x and y=96 when x=4. Find the value of: a) the constant k b) y where x=5 c) x where y=24 The relationship is y = kx^2. a) Find the constant k. Step 1: Substitute the given values y=96 and x=4 into the variation equation. 96 = k(4^2) 96 = 16k Step 2: Solve for k. k = (96)/(16) k = 6 The constant k is 6. b) Find y where x=5. Step 1: Use the constant k=6 in the variation equation y = 6x^2. Step 2: Substitute x=5 into the equation. y = 6(5^2) y = 6(25) y = 150 The value of y is 150. c) Find x where y=24. Step 1: Use the constant k=6 in the variation equation y = 6x^2. Step 2: Substitute y=24 into the equation. 24 = 6x^2 Step 3: Solve for x^2. x^2 = (24)/(6) x^2 = 4 Step 4: Take the square root of both sides. x = ±sqrt(4) x = ± 2 The values of x are 2 or -2. 6. Solve the equation 2x^2 = 5 Step 1: Isolate x^2 by dividing both sides by 2. x^2 = (5)/(2) Step 2: Take the square root of both sides. x = ±sqrt((5)/(2)) Step 3: Rationalize the denominator by multiplying the numerator and denominator by sqrt(2). x = ±sqrt(5)sqrt(2) × sqrt(2)sqrt(2) x = ±sqrt(10)2 The solutions for x are ±sqrt(10)2. 7. P varies inversely as the square of n and directly as m. And P=6 when n=2 and m=27. Find the value of: a) The constant k b) P when n=-6 and m=-2 c) n when m=8 and P=1 The relationship is P = k (m)/(n^2). a) Find the constant k. Step 1: Substitute the given values P=6, n=2, and m=27 into the variation equation. 6 = k (27)/(2^2) 6 = k (27)/(4) Step 2: Solve for k. k = (6 × 4)/(27) k = (24)/(27) Step 3: Simplify the fraction. k = (8)/(9) The constant k is (8)/(9). b) Find P when n=-6 and m=-2. Step 1: Use the constant k=(8)/(9) in the variation equation P = (8)/(9) (m)/(n^2). Step 2: Substitute n=-6 and m=-2 into the equation. P = (8)/(9) (-2)/((-6)^2) P = (8)/(9) (-2)/(36) Step 3: Simplify the expression. P = (8)/(9) × (-(1)/(18)) P = -(8)/(162) P = -(4)/(81) The value of P is -(4)/(81). c) Find n when m=8 and P=1. Step 1: Use the constant k=(8)/(9) in the variation equation P = (8)/(9) (m)/(n^2). Step 2: Substitute m=8 and P=1 into the equation. 1 = (8)/(9) (8)/(n^2) 1 = (64)/(9n^2) Step 3: Solve for n^2. 9n^2 = 64 n^2 = (64)/(9) Step 4: Take the square root of both sides. n = ±sqrt((64)/(9)) n = ±(8)/(3) The values of n are (8)/(3) or -(8)/(3). 8. The sketch shows the graph of y = x^2 - 2x - 8. Find the coordinates of: a) A and B b) the minimum point on the graph a) Find the coordinates of A and B. Step 1: A and B are the x-intercepts, where y=0. Set the equation to 0. x^2 - 2x - 8 = 0 Step 2: Factor the quadratic equation. (x-4)(x+2) = 0 Step 3: Solve for x. x-4=0 x=4 x+2=0 x=-2 Step 4: Assign the coordinates based on the sketch (A is negative, B is positive). The coordinates of A are (-2, 0) and B are (4, 0). b) Find the minimum point on the graph. Step 1: The x-coordinate of the minimum point (vertex) of a parabola y = ax^2 + bx + c is given by x = -(b)/(2a). For y = x^2 - 2x - 8, a=1 and b=-2. x = -(-2)/(2(1)) x = (2)/(2) x = 1 Step 2: Substitute the x-coordinate back into the equation to find the y-coordinate. y = (1)^2 - 2(1) - 8 y = 1 - 2 - 8 y = -9 The minimum point on the graph is (1, -9). Last free one today — make it count tomorrow, or type /upgrade for unlimited.