NJC 2023 JC1 Promos 9649

2 1 Let A be an nn matrix such that 2 AO . If u is a vector in n such that Au 0 , show that u and Au are linearly independent. [3] 2 The Lemniscate of Bernoulli is a curve with polar equation 22 cos 2ra  , where a is a positive constant. (i) Find the total area of the regions bounded by the lemniscate. [3] (ii) Find an expression for the total length of the lemniscate, leaving your answer in the form  π 4 0 4f da   , where  f  is a function of  to be determined. Simplify your answer. [3] 3 A sequence of real numbers 123,,, . . .uu u satisfies the recurrence relation 2 21 1 4 nn nuu c u   where c is a real number. (i) Describe the behaviour of the sequence in the case where (a) 1 2c , [1] (b) 0c as n becomes very large. [3] (ii) Determine the range of values of c for which the sequence is convergent for any values of 1u and 2u , justifying your answer. [4] 4 The set nnM  denotes the vector spa ce consisting of all real nn matrices and with the usual operations of addition and scalar multiplication. Let v be a nonzero constant vector in n and  null space of nn nS  XM v X  . (i) Show that nS is a subspace of nnM  . [3] (ii) In the case where 2n and 1 2  v , find a basis for 2S . [3] (iii) Explain why the dimension of nS is 2nn . [2] (iv) Show that the set   column space of nnXM v X  is not a subspace of nnM  . [ 2 ]

3 5 A patient undergoing medical treatment is required to take a dose of 60 mg of a particular drug once a day at the start of each day. It has b een found that the amount of drug in the patient’s bloodstream at the end of a day is 1 16 of the amount present at the start of the day. Let na denote the amount of drug, in mg, in the patient’s bloodstream at the end of the thn day of taking the drug. There is no drug present in the patient's bloodstream before the first day of taking the drug. (i) By formulating a recurrence relation for na , find an expression for na in terms of n for 0n . [4] (ii) State the value of lim nn a  . [1] After a long period of time, the patient visits his doctor for a follow up session. During the session, the doctor tells the patient that he needs to increase the daily dosage of the drug intake from 60 mg to d mg starting from the next day. An overdose is said to have occurred once the amount of the drug present in the bloodstream exceeds 80 mg. (iii) Find the maximum value of d such that an overdose will never occur. [5] 6 A curve C is defined parametrically by x t , cos 2yt t for 0 πt . The region below the x-axis bounded by C and the x-axis is denoted by R. A solid is obtained by rotating R completely about the y-axis. Find (i) the surface area of this solid, and [3] (ii) the exact volume of this solid. [4] Use Simpson’s Rule for four st