2026 Pure Mtc Practice Items With Solutions Tr. Jk Begumisa

• SET ONE: Page 1 of 21 JK BEGUMISA – 2026 Pure Mathematics Items : +256786108453/+256700798389 UGANDA ADVAN CED LEVEL CERTIFICATE OF EDUC ATION PURE MATHEMATICS ITEMS WITH POSSIBLE RESPONSES: © JK BEGUMISA – JAN, 2026 “Revise, Reflect, Achiev e” For orders of ; “Pure Mathem atics Item Bank – 2025 ”, contact ; JK BEGUMISA . PART I: 3D VECTORS: ITEM ONE: In Ntake village, O is the trading center. A and B are two homesteads. The position vectors of two homesteads are such that; 𝑶𝑨=𝒂 and 𝑶𝑩=𝒃. A community notice board G is placed on the line OA such that; 𝑂𝐺⃗⃗⃗⃗⃗ =2𝑂𝐴⃗⃗⃗⃗⃗ . Point M is the midpoint of AB. The village chief’s palace is along the line OB such that a straight footpath from G to M connects to it at N. Tasks: a). Find; GM⃗⃗⃗⃗⃗⃗ , in terms of; 𝒂 and 𝒃. b). Show that the position vector of the village chief’s palace is given by ; 𝑂𝑁⃗⃗⃗⃗⃗⃗ =(2−3 2𝛽)𝒂+1 2𝛽𝒃 c). Hence, show that; 𝑶𝑵:𝑵𝑩=2:1 Possible Responses: a). AM⃗⃗⃗⃗⃗⃗ :MB⃗⃗⃗⃗⃗⃗ =1:1 ⇒AM⃗⃗⃗⃗⃗⃗ =1 2(𝒃−𝒂) OM⃗⃗⃗⃗⃗⃗ =OA⃗⃗⃗⃗⃗ +AM⃗⃗⃗⃗⃗⃗ =𝒂+1 2(𝒃−𝒂) ⇒OM⃗⃗⃗⃗⃗⃗ =1 2(𝒂+𝒃) 𝑇ℎ𝑢𝑠; GM⃗⃗⃗⃗⃗⃗ =OM⃗⃗⃗⃗⃗⃗ −0G⃗⃗⃗⃗⃗ =1 2(𝒂+𝒃)−2𝒂 ∴GM⃗⃗⃗⃗⃗⃗ =1 2(𝒃−3𝒂) b). Let;GN⃗⃗⃗⃗⃗ =β GM⃗⃗⃗⃗⃗⃗ ON⃗⃗⃗⃗⃗ =OG⃗⃗⃗⃗⃗ +GN⃗⃗⃗⃗⃗ =OG⃗⃗⃗⃗⃗ +β GM⃗⃗⃗⃗⃗⃗ ON⃗⃗⃗⃗⃗ =2𝒂+𝛽[1 2(𝒃−𝟑𝒂)] ∴ON⃗⃗⃗⃗⃗ =(2−3 2𝛽)𝒂+1 2𝛽𝒃 𝑯𝒆𝒏𝒄𝒆 𝒑𝒓𝒐𝒗𝒆𝒅. • SET ONE: Page 2 of 21 JK BEGUMISA – 2026 Pure Mathematics Items : +256786108453/+256700798389 c). Let;ON⃗⃗⃗⃗⃗ =λOB⃗⃗⃗⃗⃗ ⇒ON⃗⃗⃗⃗⃗ =λ𝒃 ………………………………………...(𝑖) W𝑒 𝑡ℎ𝑢𝑠 𝑒𝑞𝑢𝑎𝑡𝑒 𝑡ℎ𝑒 𝑡𝑤𝑜 𝑒𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛𝑠 𝑜𝑓 ON⃗⃗⃗⃗⃗ : ⇒(2−3 2𝛽)𝒂+1 2𝛽𝒃=𝜆𝒃…………………………(𝑖𝑖) 𝐸𝑞𝑢𝑎𝑡𝑖𝑛𝑔 𝑐𝑜𝑟𝑟𝑒𝑠𝑝𝑜 𝑛𝑑𝑖𝑛𝑔 𝑐𝑜−𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡𝑠 𝑜𝑓;𝒂 𝑎𝑛𝑑 𝒃 𝑖𝑛 𝑒𝑞𝑛 (𝑖𝑖) For 𝒂; (2−3 2𝛽)=0 ⇒𝛽=4 3 For 𝒃; 1 2𝛽=𝜆 1 2(4 3)=𝜆 ⇒λ=2 3 𝑇ℎ𝑢𝑠; ⇒ON⃗⃗⃗⃗⃗ =2 3𝒃 [𝐹𝑟𝑜𝑚 𝑒𝑞𝑛 (𝑖)] 𝐴𝑙𝑠𝑜; NB⃗⃗⃗⃗⃗ =−ON⃗⃗⃗⃗⃗ +OA⃗⃗⃗⃗⃗ +AB⃗⃗⃗⃗⃗ =−2 3𝒃+𝒂+(𝒃−𝒂) ⇒NB⃗⃗⃗⃗⃗ =1 3𝒃 Henc e; ON⃗⃗⃗⃗⃗ :NB⃗⃗⃗⃗⃗ =(2 3𝒃) (1 3𝒃)=2:1 ∴ON⃗⃗⃗⃗⃗ …