RVHS ACJC EJC NJC 2022 FM Prelim P1

1 9649/01/2022 2022 RVHS_ACJC_EJC_NJC Prelim Paper 1_Modified 1. It is given that the function f(x) = e1 e1 x x − + . (i) By substituting exu= or otherwise , show that 1 0 e1f ( ) d ln 2x x a b +=+  , where a and b are constants to be determined. [3] (ii) Use Simpson’s rule with 5 ordinates to find an approximate value of 1 0 f ( ) dxx , leaving your answer to 5 decimal places and hence find an approximate value for e1ln 2 +  . [3] 2. The sequence  nP is given by 0 0P = , 1 7 5P = and ( ) 1 11 1 5 2 33 n n n nP P P − +−= + + for 1n . By considering the sequence  nQ , where ( ) 11 310 n nnPQ +=+ for 0n , find an expression for nP as a function of n. [7] 3. A curve C has polar equation 2 cos5r =+ for 4π0 5 . (i) Find the polar coordinates of the points on C for which the value of r is a maximum or a minimum. [3] (ii) Sketch C, stating the line of symmetry and indicating clearly the polar coordinates of the points found in part (i). [3] (iii) Find the area of the region bounded by C, the line yx= and the initial line. [2] 4. (a) Show by integration that ( ) 3 2 22 43 11 d 3 xx xcxx ++ =− +  . [2] (b) (i) Solve the differential equation 24 d1 d1 y xy x x x+=+ for 0x , given that 0y= when 1x= . [4] (ii) Use one step of the Euler method to calculate an approximation to the value of y when 1.5x= . [1] (iii) Show that there is a large discrepancy between the approximation found in part (ii) and the actual value of y when 1.5x= . Give a reason why the discrepancy occurred. [3]

2 9649/01/2022 5. The linear transformation 44T: →  is represented by the matrix 2 3 6 2 2 3 3 3 4 6 9 5 2 3 3 a − − − −= −  − M . The range space of T is denoted by R. Suppose that 3a . (i) Explain why 262 2 3 3,,4 9 5 23 a            − − −                          is a basis of R. [4] (ii) Given that w x y z       is an element of R, show that w x y=+ . [3] It is now given that 3a= . (iii) State a basis for the null space of T. Hence, find the set of vectors x such that 1 1 2 1  −=   Mx . [3] 6. The equation 3e 2 1 0 x x− + = has two positive roots. Show that the value of one of the roots lies between 1 and 6 while that of the other root lies between 6 and 9. [2] An attempt to estimate the larger root α is made by using the Newton-Raphson method applied to the equation 3e 2 1 0 x x− + = with the initial approximation 0 6x = . Calculate the next two approximations, 1x and 2x , and explain why 1x is a worse approximation to α than 0x is. [4] Two iterative formulae are proposed to find the value of α: 3 1 e1 2 nx nx + += and 1 3ln(2 1)nnxx+ =− . Only one of the formulae is feasible. For the formula that is not feasible, d emonstrate graphically why the iterative process fails. For the formula that is feasible, use it with first iteration 1 7.5x = to obtain α correct to 3 decimal places. [4]

3 9649/01/2022 7. The urban development authority of a metropolitan region is studying the population movements between three cities within the region, namely A, B and C. Every year, it is observed that the populations of these cities move concurrently in the following way: • 20% of the population of A moves to C; • 10% of the population of B moves to A while another 20% moves to C; • 30% of the population of C moves to A while another 10% moves to B. Let na , nb and nc denote the population (in millions) of A, B and C respectively at the end of year n. The population movements may be modelled by the recurrence relation 1nn+ =v Mv , where 341 5 10 10 7 1 10 10 11 55 p q  = M and n nn n a b c  = v . Assume that the total population of the three cities remains constant at any time. Initially, the populations of A, B and C are 7 600 000 , 4 800 000 and 5 600 000 respectively. (i) State the values of p and q. [1] (ii) With the values of p and q found in part (i), find the eigenvalues and eigenvectors of the matrix M. [5] (iii) Hence by diagonalizing the matrix M, determine the steady-state populations of A, B and C in the long run. [4]

4 9649/01/2022 8. A company manufactures double -walled glass mugs that contain an air gap between two layers of glass. The diagram below shows the cross -section of a double -walled glass mug with height 10 cm. The inner glass layer is represented by the curve C1 with equation 24 21 24 xxy    = + +       . The outer glass layer is represented by the curve C2 with equation 42 114 10 xy   + − =       . The shaded region indicates the air gap between the inner and outer glass layers. The glass mug is formed by rotating the shaded region  radians about y-axis. (i)Using the substitution 4 sinx = where π0, 2 find 4 1d 4 xxx −    . [4] (ii)Hence, find the exact volume of the air gap. [4] (iii)Find the maximum volume of liquid that the mug can hold. [2] (iv)Find the surface area of the inner glass layer. [2] Cross section of a double-walled glass mug Air gap y x 10

5 9649/01/2022 9. The Solow model of economic growth was developed in 1956 by economist and Nobel Memorial Prize in Economic Sciences winner Robert Solow. A differential equation central to this model is given by d d k sq kt =− () where • k denotes the amount of capital per unit of labour where 0k  , • t denotes time in years, • s denotes the savings rate where 01 s , • q denotes the production function per unit of labour and is a function of k, and •  denotes the constant depreciation rate where 0  . One common choice for the production function in this model is the Cobb -Douglas production function which gives rise to q Ak = , where  and A are positive constants where 01  and A represents total factor productivity. For the rest of this question, take q Ak = where 01  and 0A . Suppose that s is constant. (i) Find the equilibrium value for the differential equation () . [1] (ii) By using the substitution 1u Ak −= , show that the differential equation () can be reduced to ( )( ) 2d 1d u sA ut = − − . Hence, given that 0kk= initially where 0 0k  , find k in the form ( ) 1 11 e tsA h   −− + , where h is constant in terms of 0k ,  , s, A and  to be determined. [6] (iii) Hence, or otherwise, determine whether the equilibrium value found from part (i) is stable or unstable. [2] Now, suppose instead that s varies with t. (iv) In this model , consumption per capita is given by the difference between production per unit of labour, q, and the amount of investment, sq. Also, a steady state is said to have been achieved when sq k= . Consider the Solow diagram below which illustrates consumption per capita at the steady state. y k O Steady-state consumption per capita

6 9649/01/2022 9. [Continued] As consumption per capita is an indicator of the economic well-being of members of a society, it is generally in the interest of a policymaker to determine the savings rate, s, that maximises the steady -state consumption per capita c. The value of s that achieves this, Gs , is known as the Golden Rule savings rate. Write down an expression for c and find Gs . [4] 10. A logistics company set up an online platform providing delivery services to users on a monthly paid subscription basis. The company’s sales manager models the number of subscribers that the company has at the end of each month. She notes that approximately 10% of the existing subscribers leave each month, and that there will be a constant number k of new subscribers in each subsequ