University:
Stanford University
Course:
Introduction to Mathematical Thinking
Academic year:

2023
Views:

223

Pages:

2
Author:

MAlphen

4 In ∆𝑂𝑅𝑆, the point A on OR is such that 𝑂𝐴 = 2𝐴𝑅. B is the midpoint of OS, X is the midpoint of AB. In the diagram, ⃗𝑂⃗⃗⃗𝐴⃗→ = 2p, ⃗𝑂⃗⃗⃗𝐵⃗→ = 2q and 𝑆𝑌 = 4 𝑌𝑅. 3 (a) Express as simply as possible, in terms of p and q. ⃗𝐴⃗⃗⃗𝑋⃗→ , (i) ⃗𝑂⃗⃗⃗𝑋⃗→ , (ii) (iii) ⃗𝑅⃗⃗⃗⃗𝑆→ , (iv) ⃗𝑂⃗⃗⃗𝑌⃗→ . What do the results in (ii) and (iv) tell you about O, X and Y? (b) Find the numerical value of (i) (ii) Area of ∆𝑂𝐴𝑋 , Area of ∆𝑂𝐴𝑌 Area of ∆𝑂𝐴𝑋 , Area of ∆𝑂𝑅𝑆 Ans: (a)(i)q – p (ii)p + q (iii)4q – 3p (iv) 12 (p + q) 7 ⃗𝑂⃗⃗⃗𝑌⃗→ = 12 ⃗𝑂⃗⃗⃗𝑋⃗→ ; O, X and Y are collinear. (b)(i) 7 (ii)1 7 12 6 5 In the diagram ⃗𝑂⃗⃗⃗𝐴⃗→ = 𝐚, ⃗𝑂⃗⃗⃗𝐵⃗→ = 𝐛. R lies on AB such that AR : RB = 3 : 2. Q lies on OB produced such that BQ = 2OB. ⃗𝑄⃗⃗⃗𝑃⃗→ = 2 ⃗𝑂⃗⃗⃗𝐴⃗→ . (i) (ii) (iii) Express, as simply as possible, in terms of a and/or b, ⃗𝑂⃗⃗⃗𝑅⃗→ , (a) (b) ⃗𝑂⃗⃗⃗𝑃⃗→ . Make two statements about the points O, R and Find P. Area of ∆𝑂𝐴𝑅, Area of ∆𝑂𝐴𝐵 (b) Area of ∆𝑂𝐴𝑅, Area of ∆𝐴𝑅𝑃 Area of ∆𝑂𝐴𝑃 , (c) 𝐴𝑟𝑒𝑎 𝑜𝑓 ∆𝑂𝑄𝑃 𝐴𝑟𝑒𝑎 𝑜𝑓 ∆𝑂𝐴𝑃 , (d) 𝐴𝑟𝑒𝑎 𝑜𝑓 ∆𝐵𝑅𝑃𝑄 2 Ans: (i)(a) 𝐚 + 3 𝐛 (ii) O, R and P are collinear points, magnitude of 𝑂𝑅 = 1 𝑂𝑃 5 5 5 (iii)(a) 3 (b) 1 (c) 1 (d) 15 5 4 2 28 (a)