Expand (2-x)7 using the binomial theorem (a+b)^n = sum_k=0^n nk a^n-k b^k.

Step 1: Expand (2-x)^7 using the binomial theorem (a+b)^n = _k=0^n nk a^n-k b^k. Here, a=2, b=-x, and n=7. We need terms up to x^3 (i.e., for k=0, 1, 2, 3). For k=0: 70 (2)^7-0 (-x)^0 = 1 · 2^7 · 1 = 128 For k=1: 71 (2)^7-1 (-x)^1 = 7 · 2^6 · (-x) = 7 · 64 · (-x) = -448x For k=2: 72 (2)^7-2 (-x)^2 = (7 · 6)/(2 · 1) · 2^5 · x^2 = 21 · 32 · x^2 = 672x^2 For k=3: 73 (2)^7-3 (-x)^3 = (7 · 6 · 5)/(3 · 2 · 1) · 2^4 · (-x^3) = 35 · 16 · (-x^3) = -560x^3 Step 2: Combine the terms. (2-x)^7 = 128 - 448x + 672x^2 - 560x^3 + The expansion up to and including the term in x^3 is: 128 - 448x + 672x^2 - 560x^3 b) Step 1: Relate (1.98)^7 to the expansion of (2-x)^7. We need 2-x = 1.98. Solving for x: x = 2 - 1.98 = 0.02 Step 2: Substitute x=0.02 into the expansion obtained in part (a). (1.98)^7 ≈ 128 - 448(0.02) + 672(0.02)^2 - 560(0.02)^3 Step 3: Calculate each term. 128 -448(0.02) = -8.96 672(0.02)^2 = 672(0.0004) = 0.2688 -560(0.02)^3 = -560(0.000008) = -0.00448 Step 4: Sum the terms. (1.98)^7 ≈ 128 - 8.96 + 0.2688 - 0.00448 = 119.30432 Step 5: Round the result to four significant figures. The first four significant figures are 1, 1, 9, 3. The next digit is 0, so we round down. (1.98)^7 ≈ 119.3 The approximate value of (1.98)^7 to four significant figures is: 119.3 a) Step 1: Identify the given lengths and apply the Power of a Point Theorem for intersecting secants. The secants WQ and YQ intersect externally at Q. The theorem states that QX · QW = QZ · QY. From the problem statement and diagram: WX = 5 cm XQ = 6 cm WQ = WX + XQ = 5 + 6 = 11 cm ZQ = 4 cm Let YZ = y. Then QY = QZ + YZ = 4 + y. Step 2: Substitute the values into the theorem equation. QX · QW = QZ · QY 6 · 11 = 4 · (4 + y) 66 = 16 + 4y Step 3: Solve for y (the length of chord YZ). 66 - 16 = 4y 50 = 4y y = (50)/(4) = (25)/(2) = 12.5 The length of chord YZ is: 12.5 cm b) Step 1: Apply the Power of a Point Theorem for a tangent and a secant from an external point Q. The theorem states that SQ^2 = QZ · QY. From part (a), we found YZ = 12.5 cm. We are given ZQ = 4 cm. So, QY = QZ + YZ = 4 + 12.5 = 16.5 cm. Step 2: Substitute the values into the theorem equation. SQ^2 = 4 · 16.5 SQ^2 = 66 Step 3: Solve for SQ. SQ = sqrt(66) The length of the tangent SQ is: sqrt(66) cm Step 1: The given formula is S = (a(1-r^n))/(1-r). We need to make a the subject of the formula. Step 2: Multiply both sides of the equation by (1-r) to clear the denominator. S(1-r) = a(1-r^n) Step 3: Divide both sides by (1-r^n) to isolate a. a = (S(1-r))/(1-r^n) The formula with a as the subject is: a = (S(1-r))/(1-r^n) Send me the next one 📸