Advanced level 2025 Adamawa regional mock pure maths 2

Advanced level 2025 Adamawa regional mock pure maths 2

Advanced level 2025 Adamawa regional mock pure maths 2

1 (i) Given that the quadratic function 𝒇(𝒙) = (𝒂 – 𝟐)𝒙𝟐 – 𝟐𝒂𝒙 + 𝟗, touches the x-axis at only one point, find two
possible functions f1(x) and f2(x) that satisfy this condition and determine the coordinates of the single point where each
of these functions touch the x-axis. 6 marks
(ii) Given that the polynomial 𝑷(𝒙) = 𝒂𝒙𝟑 + (𝟏 + 𝟒𝒂)𝒙𝟐 – (𝒃 – 𝟏𝟎)𝒙 – 𝟐𝒃 , is exactly divisible by the quadratic
polynomial 𝟐𝒙𝟐 + 𝟓𝒙 – 𝟑, find the values of the constants a and b 4 marks
2 (i) The position vectors of the points 𝑨, 𝑩, 𝑪 are = 𝒊 + 𝟒𝒋 + 𝟐𝒌 , 𝒃 = -𝟐𝒊 + 𝟐𝒋 + 𝟔𝒌 and 𝒄 = 𝟑𝒊 – 𝟑𝒋 – 𝟐𝒌
respectively, find
a) the vector, 𝒗, where 𝒗 = 𝒂 + 𝒃 + 𝒄 2 marks
b) the unit vector 𝒗̂ 2 marks
c) the angle, to the nearest degree which 𝒗 makes with the 𝑦-axis 2 marks
(ii) Given that the mean value theorem is satisfied, determine the value of 𝒄, 𝟐 < 𝒄 < 𝟒, for the function
𝒇(𝒙) = 𝒙𝟐 + 𝟒𝒙 , which is differentiable over ]𝟐, 𝟒[ 4 marks
3 (i) A farmer has an 80𝑚 length of fencing. He wants to use it to form 3 sides of a rectangular enclosure against an
existing wall which provides the 4th side. Determine the maximum Area he can enclose and give its dimensions.
5 marks
(ii) Two quantities 𝑥 and 𝑦 are known to be connected by a law of the form 𝒚 = 𝒂(𝒙 + 𝟐)𝒏 , where 𝑎 and 𝑛 are real
constants. Linearize 𝒚 = 𝒂(𝒙 + 𝟐)𝒏 , stating clearly the gradient and intercept. 3 marks
(iii) Given the statements 𝑷: 𝐴𝑑𝑎𝑚𝑜𝑢 𝑖𝑠 𝑎 𝑆𝑡𝑢𝑑𝑒𝑛𝑡 , 𝑸: 𝐴𝑑𝑎𝑚𝑜𝑢 𝑤𝑖𝑙𝑙 𝑤𝑟𝑖𝑡𝑒 𝑡ℎ𝑒 𝑅𝑒𝑔𝑖𝑜𝑛𝑎𝑙 𝑀𝑜𝑐𝑘, write statement
that represent
a) 𝑷 ⟹ 𝑸 1 mark
b) ~(𝑷 ⇒ 𝑸) 1 mark
4 (i) Given the real-valued function 𝒇(𝒙) = 𝟏
𝒙𝟐-𝟗
a) State the domain of definition, 𝐷𝑓 , of 𝑓 1 mark
b) Evaluate 1 mark
𝐥𝐢𝐦
𝒙→-∞
𝒇(𝒙) 𝐚𝐧𝐝 𝐥𝐢𝐦
𝒙→+∞
𝒇(𝒙)
c) State all the equations of the asymptotes of the curve, 𝐶𝑓 , of 𝑓 1 mark
d) Given that 𝒇′(𝒙) = -𝟐𝒙
(𝒙𝟐-𝟗)𝟐 , draw a table of variation for 𝑓(𝑥) 2 marks
(ii) Solve in the form 𝒚 = 𝒇(𝒙), (𝟏 – 𝒙𝟐) 𝒅𝒚 𝒅𝒙 – 𝟐𝒙𝒚 = 𝟎 , 𝑥 = 0 and 𝑦 = 1 5 marks
5(i) Given the matrix
𝐓 = (- -𝟑𝟏 𝟏 – -𝟒𝟐 𝟏 𝟏 𝟓 𝟎)