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 strips to estimate the volume of the solid, correct to 5 decimal places. Comment on the accuracy of your estimate with reference to your answer from part (ii). [ 3 ]

4 7 Do not use a calculator in answering this question. The ancient Greek mathematician, Theon of Sm yrna, discovered a method to approximate the value of 2 . He found that if a b is a fraction that approximates 2 , then the fraction 2ab ab   provides an even better approximation to 2 . Starting with an initial approximation 0 0 a b , successive approximations 312 123 , , , . . . aaa bb b can be generated by the relations 11 11 2, . nn n nn n aa b ba b     (i) Let n n n a b  v . State the 22 matrix A such that 1nn vA v . Express nv in the form 0Bv , where B is a matrix to be determined in terms of A. [2] (ii) Express nA in the form 1PDP where P is an invertible matrix and D is a diagonal matrix. [5] (iii) Hence, using 0 1 1  v , show that n n a b tends to 2 as n tends to infinity. [4] 8 Let   2 2 ,,ax bx c a b c  P  denote the vector space of polynomials in x with degree at most 2, with real coefficients. The function 22T : PP is defined by       Tf f + fx xx  . (i) Show that T is a linear transformation. [2] (ii) Find a basis for the null space of T. Hence, state the rank of T. [3] (iii) Show that   22 2T4 2 2ax bx c ax a b x a b c    and find   32T ax bx c . Hence, formulate a conjecture for   2Tn ax bx c , where n is a positive integer, and prove your conjecture by induction. [7] (iv) Show that if 0a and the equation 2 0ax bx c  has real roots, then the equation   2T0n ax bx c has real and distinct r oots for any positive integer n. [2]

5 9 It is given that the equation f0 x  , where  2 2fe 2 xx xx  , has exactly two real roots  and  with   . (i) Without finding the values of  and  , show that there exists an integer N such that 12NN N   , where the value of N is to be determined. [2] (ii) A student attempts to use the iterative formula 2 2 1 e nx nn nx xx    to find the values of  and  . With the aid of a sketch, explain why this iterative formula is suitable for finding an approximation only for  , but not for  . [4] (iii) Use the iterative formula in part (ii) with a suitable initial approximation to obtain an approximation to  , giving your answer to 2 decimal places. [3] (iv) Use one iteration of the Newton-Raphs on method with initial approximation 0 1.5x  to obtain an approximation to  , giving your answer to 4 decimal places. [2] (v) Prove that f0 x  for all non-zero real values of x. Hence, determine the range of values of k such that one iteration of the Newton-Raphson with initial approximation 0x k will produce an overestimate of  . [4]

6 10 Figure 1 shows a photograph of a hollow, cylindric al wall light at an MRT station in Singapore. The axis of the cylinder is vertical and parallel to the wall. Figure 1 (i) Give a reason why the outline of the light on the wall is expected to be a hyperbola. [1] For this question, it may be assumed that:  the light source is a single point at the centre of the cylinder,  the outline of light on the wall above and below the wall light are from the same hyperbola,  any light ray that is reflected inside the cy linder at least once and reaches the wall will not produce any noticeable effect on the wall,  the thickness of the cylinder is negligible. The radius and height of the cylindrical wall light are 4 cm and 12 cm respectively. Figures 2 and 3 below show the front view and si de view of the wall light respectively. The cartesian plane is superimposed onto the front view with the light source at the origin, while the cylinder is touching the wall as seen in the side view. Figure 2: Front view Figure 3: Side view Take 1 unit on the cartesian plane to represent 1 cm. (ii) Show that the equation of the outline of the light on the wall is 22 136 16 yx . [4] Wall Light source Outline of light on the wall Wall light

7 An MRT station staff member de cided to install a supporting ro d of length 2 cm at the same horizontal level as the light source as shown in Figure 4 below. As a result, a new outline of light on the wall is obtained, giving rise to a new hyperbola on the wall. Figure 4 (iii) Did the installation of the supporting rod affect the eccentricity of the hyperbola? Justify your answer. [2] (iv) Find the equation of the new outline of the light on the wall. [2] The photographer observed that in the photograph in Figure 1, the bott