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 questions 7, 8, and 9. Question 7: Given P = 2i - 3j + k, Q = 3i - 4j - 3k, and R = 3P + 2Q. Find |R| correct to 2 decimal places. Step 1: Express vectors P and Q in component form. P = 2 \\ -3 \\ 1 Q = 3 \\ -4 \\ -3 Step 2: Calculate 3P. 3P = 3 2 \\ -3 \\ 1 = 3 × 2 \\ 3 × (-3) \\ 3 × 1 = 6 \\ -9 \\ 3 Step 3: Calculate 2Q. 2Q = 2 3 \\ -4 \\ -3 = 2 × 3 \\ 2 × (-4) \\ 2 × (-3) = 6 \\ -8 \\ -6 Step 4: Calculate vector R by adding 3P and 2Q. R = 3P + 2Q = 6 \\ -9 \\ 3 + 6 \\ -8 \\ -6 = 6+6 \\ -9+(-8) \\ 3+(-6) = 12 \\ -17 \\ -3 Step 5: Calculate the magnitude of R, |R|. The magnitude of a vector x \\ y \\ z is given by sqrt(x^2 + y^2 + z^2). |R| = sqrt(12^2 + (-17)^2 + (-3)^2) |R| = sqrt(144 + 289 + 9) |R| = sqrt(442) Step 6: Calculate the numerical value and round to 2 decimal places. |R| ≈ 21.023796... Rounded to 2 decimal places: |R| ≈ 21.02 Question 8: Simplify: (2y^2 - 3xy - 2x^2)/(4y^2 - x^2) Step 1: Factor the numerator 2y^2 - 3xy - 2x^2. We can factor this quadratic expression by finding two terms that multiply to 2y^2 × (-2x^2) = -4x^2y^2 and add to -3xy. These terms are -4xy and xy. 2y^2 - 3xy - 2x^2 = 2y^2 - 4xy + xy - 2x^2 Factor by grouping: = 2y(y - 2x) + x(y - 2x) = (2y + x)(y - 2x) Step 2: Factor the denominator 4y^2 - x^2. This is a difference of squares, a^2 - b^2 = (a-b)(a+b), where a = 2y and b = x. 4y^2 - x^2 = (2y - x)(2y + x) Step 3: Substitute the factored expressions back into the fraction and simplify. ((2y + x)(y - 2x))/((2y - x)(2y + x)) Cancel out the common factor (2y + x): = (y - 2x)/(2y - x) The simplified expression is (y - 2x)/(2y - x). Question 9: Find x if 3^2x+3 + 1 = 28. Step 1: Isolate the exponential term. 3^2x+3 = 28 - 1 3^2x+3 = 27 Step 2: Express the right side of the equation with the same base as the left side. We know that 27 = 3^3. 3^2x+3 = 3^3 Step 3: Equate the exponents. Since the bases are the same, the exponents must be equal. 2x + 3 = 3 Step 4: Solve for x. 2x = 3 - 3 2x = 0 x = (0)/(2) x = 0 3 done, 2 left today. You're making progress.