ML=(-2, -3) and LN=(-6, -1). Find (i) MN, (ii) |ML|. (b) The diagram shows vectors PT = a, PS = b and PS is divided into 3 equal parts at Q and R. U is the midpoint of TR. Express, in simplest form, in terms of a and/or b. (i) TS (ii) TR (c) Using the information in (b), show that PT = 2QU.

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Asked on September 13, 2026Mathematics

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.

Accepted Answer · 2026-09-13T13:05:29.502Z

Here are the solutions to the questions. Question 5 (a) (i): Find MN Given: ML = -2 \\ 3 and LN = -6 \\ -1 . Step 1: Use vector addition to find MN. MN = ML + LN MN = -2 \\ 3 + -6 \\ -1 Step 2: Add the corresponding components. MN = -2 + (-6) \\ 3 + (-1) MN = -8 \\ 2 The vector MN is -8 \\ 2 . Question 5 (a) (ii): Find |ML| Given: ML = -2 \\ 3 . Step 1: Use the formula for the magnitude of a vector x \\ y , which is sqrt(x^2 + y^2). |ML| = sqrt((-2)^2 + (3)^2) Step 2: Calculate the squares and sum them. |ML| = sqrt(4 + 9) |ML| = sqrt(13) The magnitude of ML is sqrt(13) units. Question 5 (b) (i): Express TS in terms of a and/or b Given: PT = a, PS = b. Step 1: Express TS using the path TP + PS. Since PT = a, then TP = -a. TS = TP + PS TS = -a + b The vector TS is b - a. Question 5 (b) (ii): Express TR in terms of a and/or b Given: PT = a, PS = b. PS is divided into 3 equal parts at Q and R. Step 1: Express TR using the path TP + PR. We know TP = -a. Since PS is divided into 3 equal parts at Q and R, PR = (2)/(3)PS. PR = (2)/(3)b Step 2: Substitute the expressions for TP and PR into the equation for TR. TR = -a + (2)/(3)b The vector TR is (2)/(3)b - a. Question 5 (c): Using the information in (b), show that PT = 2QU Given: PT = a, PS = b. PS is divided into 3 equal parts at Q and R. U is the midpoint of TR. Step 1: Express QU in terms of a and b. QU = QR + RU From part (b), QR = (1)/(3)PS = (1)/(3)b. Since U is the midpoint of TR, RU = (1)/(2)RT. From part (b)(ii), TR = (2)/(3)b - a. So, RT = -TR = -((2)/(3)b - a) = a - (2)/(3)b. RU = (1)/(2)(a - (2)/(3)b) = (1)/(2)a - (1)/(3)b Step 2: Substitute QR and RU into the expression for QU. QU = (1)/(3)b + ((1)/(2)a - (1)/(3)b) QU = (1)/(2)a Step 3: Compare 2QU with PT. 2QU = 2((1)/(2)a) = a We are given PT = a. Therefore, PT = 2QU. (Shown) Question 6 (a) (i): Find g(-6)-3 Given: g(x) = x^2 + 1. Step 1: Calculate g(-6). g(-6) = (-6)^2 + 1 g(-6) = 36 + 1 g(-6) = 37 Step 2: Subtract 3 from g(-6). g(-6) - 3 = 37 - 3 g(-6) - 3 = 34 The value is 34. Question 6 (a) (ii): Find h^-1g(x), in its simplest form. Given: h(x) = 2x - 5 and g(x) = x^2 + 1. Step 1: Find the inverse function h^-1(x). Let y = h(x). y = 2x - 5 Swap x and y: x = 2y - 5 Solve for y: x + 5 = 2y y = (x + 5)/(2) So, h^-1(x) = (x + 5)/(2). Step 2: Substitute g(x) into h^-1(x). h^-1g(x) = h^-1(x^2 + 1) h^-1g(x) = ((x^2 + 1) + 5)/(2) h^-1g(x) = (x^2 + 6)/(2) The expression for h^-1g(x) is (x^2 + 6)/(2). Question 6 (b): 2g(x) - h(x) can be expressed in the form px^2 + qx + 7. Find the value of p and q. Given: h(x) = 2x - 5 and g(x) = x^2 + 1. Step 1: Substitute g(x) and h(x) into the expression 2g(x) - h(x). 2g(x) - h(x) = 2(x^2 + 1) - (2x - 5) Step 2: Expand and simplify the expression. 2g(x) - h(x) = 2x^2 + 2 - 2x + 5 2g(x) - h(x) = 2x^2 - 2x + 7 Step 3: Compare the simplified expression with px^2 + qx + 7 to find p and q. By comparing 2x^2 - 2x + 7 with px^2 + qx + 7: p = 2 q = -2 The values are p = 2 and q = -2. Question 6 (c): Find the value of x for which xh(x) - 2g(x) = -12 Given: h(x) = 2x - 5 and g(x) = x^2 + 1. Step 1: Substitute h(x) and g(x) into the given equation. x(2x - 5) - 2(x^2 + 1) = -12 Step 2: Expand and simplify the equation. 2x^2 - 5x - (2x^2 + 2) = -12 2x^2 - 5x - 2x^2 - 2 = -12 -5x - 2 = -12 Step 3: Solve for x. -5x = -12 + 2 -5x = -10 x = (-10)/(-5) x = 2 The value of x is 2. Send me the next one 📸

Accepted Answer · 2026-09-13T13:05:29.502Z

Here are the solutions to the questions. Question 5 (a) (i): Find MN Given: ML = -2 \\ 3 and LN = -6 \\ -1 . Step 1: Use vector addition to find MN. MN = ML + LN MN = -2 \\ 3 + -6 \\ -1 Step 2: Add the corresponding components. MN = -2 + (-6) \\ 3 + (-1) MN = -8 \\ 2 The vector MN is -8 \\ 2 . Question 5 (a) (ii): Find |ML| Given: ML = -2 \\ 3 . Step 1: Use the formula for the magnitude of a vector x \\ y , which is sqrt(x^2 + y^2). |ML| = sqrt((-2)^2 + (3)^2) Step 2: Calculate the squares and sum them. |ML| = sqrt(4 + 9) |ML| = sqrt(13) The magnitude of ML is sqrt(13) units. Question 5 (b) (i): Express TS in terms of a and/or b Given: PT = a, PS = b. Step 1: Express TS using the path TP + PS. Since PT = a, then TP = -a. TS = TP + PS TS = -a + b The vector TS is b - a. Question 5 (b) (ii): Express TR in terms of a and/or b Given: PT = a, PS = b. PS is divided into 3 equal parts at Q and R. Step 1: Express TR using the path TP + PR. We know TP = -a. Since PS is divided into 3 equal parts at Q and R, PR = (2)/(3)PS. PR = (2)/(3)b Step 2: Substitute the expressions for TP and PR into the equation for TR. TR = -a + (2)/(3)b The vector TR is (2)/(3)b - a. Question 5 (c): Using the information in (b), show that PT = 2QU Given: PT = a, PS = b. PS is divided into 3 equal parts at Q and R. U is the midpoint of TR. Step 1: Express QU in terms of a and b. QU = QR + RU From part (b), QR = (1)/(3)PS = (1)/(3)b. Since U is the midpoint of TR, RU = (1)/(2)RT. From part (b)(ii), TR = (2)/(3)b - a. So, RT = -TR = -((2)/(3)b - a) = a - (2)/(3)b. RU = (1)/(2)(a - (2)/(3)b) = (1)/(2)a - (1)/(3)b Step 2: Substitute QR and RU into the expression for QU. QU = (1)/(3)b + ((1)/(2)a - (1)/(3)b) QU = (1)/(2)a Step 3: Compare 2QU with PT. 2QU = 2((1)/(2)a) = a We are given PT = a. Therefore, PT = 2QU. (Shown) Question 6 (a) (i): Find g(-6)-3 Given: g(x) = x^2 + 1. Step 1: Calculate g(-6). g(-6) = (-6)^2 + 1 g(-6) = 36 + 1 g(-6) = 37 Step 2: Subtract 3 from g(-6). g(-6) - 3 = 37 - 3 g(-6) - 3 = 34 The value is 34. Question 6 (a) (ii): Find h^-1g(x), in its simplest form. Given: h(x) = 2x - 5 and g(x) = x^2 + 1. Step 1: Find the inverse function h^-1(x). Let y = h(x). y = 2x - 5 Swap x and y: x = 2y - 5 Solve for y: x + 5 = 2y y = (x + 5)/(2) So, h^-1(x) = (x + 5)/(2). Step 2: Substitute g(x) into h^-1(x). h^-1g(x) = h^-1(x^2 + 1) h^-1g(x) = ((x^2 + 1) + 5)/(2) h^-1g(x) = (x^2 + 6)/(2) The expression for h^-1g(x) is (x^2 + 6)/(2). Question 6 (b): 2g(x) - h(x) can be expressed in the form px^2 + qx + 7. Find the value of p and q. Given: h(x) = 2x - 5 and g(x) = x^2 + 1. Step 1: Substitute g(x) and h(x) into the expression 2g(x) - h(x). 2g(x) - h(x) = 2(x^2 + 1) - (2x - 5) Step 2: Expand and simplify the expression. 2g(x) - h(x) = 2x^2 + 2 - 2x + 5 2g(x) - h(x) = 2x^2 - 2x + 7 Step 3: Compare the simplified expression with px^2 + qx + 7 to find p and q. By comparing 2x^2 - 2x + 7 with px^2 + qx + 7: p = 2 q = -2 The values are p = 2 and q = -2. Question 6 (c): Find the value of x for which xh(x) - 2g(x) = -12 Given: h(x) = 2x - 5 and g(x) = x^2 + 1. Step 1: Substitute h(x) and g(x) into the given equation. x(2x - 5) - 2(x^2 + 1) = -12 Step 2: Expand and simplify the equation. 2x^2 - 5x - (2x^2 + 2) = -12 2x^2 - 5x - 2x^2 - 2 = -12 -5x - 2 = -12 Step 3: Solve for x. -5x = -12 + 2 -5x = -10 x = (-10)/(-5) x = 2 The value of x is 2. Send me the next one 📸

ML=(-2, -3) and LN=(-6, -1). Find (i) MN, (ii) |ML|. (b) The diagram shows vectors PT = a, PS = b and PS is divided into 3 equal parts at Q and R. U is the midpoint of TR. Express, in simplest form, in terms of a and/or b. (i) TS (ii) TR (c) Using the information in (b), show that PT = 2QU.

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ML=(-2, -3) and LN=(-6, -1). Find (i) MN, (ii) |ML|. (b) The diagram shows vectors PT = a, PS = b and PS is divided into 3 equal parts at Q and R. U is the midpoint of TR. Express, in simplest form, in terms of a and/or b. (i) TS (ii) TR (c) Using the information in (b), show that PT = 2QU.

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