HCI 9758 2023 Prelim P1

1 The variables x and y are related by the differential equation 2 2 2 d 2ed xy x −= . Solve the differential equation. Hence find a possible solution curve that has an oblique asymptote, and state the equation of the oblique asymptote. [4] 2 In the triangle PQR, angle π 3PQR= radians and angle π 3PRQ θ= − radians, where θ is small enough for 3θ and higher powers of θ to be negligible. Show that 21PR PQ αθ βθ≈+ + , where α and β are exact real constants to be determined. [5] 3 A curve C has equation 24 5xy y+= . (a) Find d d y x in terms of x and y. [2] (b) Find the exact distance between the two tangents to C that are parallel to the x- axis. [3] 4 Without the use of a calculator, show that 2 22 2 d ln 3 π24 x xp qxx− − = +−+∫ , where p and q are exact real constants to be determined. [6] P Q R

2 © HCI 2023 5 A curve G with equation f (3 2)yx a= −+ , where a is a positive real constant, is transformed onto the curve with equation f( )yx= by a sequence of transformations. (a) Describe fully, in terms of a , the sequence of transformations. [3] A point on G with coordinates ( ,0)b is mapped onto the point with coordinates (, )cd on the curve with equation 1 f( )y x= . (b) Find c and d in terms of a and b . [4] 6 (a) By writing 1 (2 1)(2 3)rr++ in partial fractions , find an expression for 1 1 (2 1)(2 3)r n rr= ++∑ in terms of n . [4] (b) Explain why ( ) 1 11 3 (2 1)(2 3) r r n rr= −+ ++∑ is a convergent series, and state its sum to infinity. [4] 7 It is given that πtan for ,4f( ) πsin for 3 ,2 x axaax x ax aa   −<≤ =  <≤  and that f( ) f( 4 )x xa= − for all real values of x , where a is a positive real constant. (a) Sketch the graph of f( )yx= for 47ax a− ≤≤ . [3] (b) By using the sketch in part (a), write down an equation relating f( )x and f( ) x− . [1] (c) Show that 4 2 f ( ) d (1 ln )π a a kaxx m − = +∫ , where k and m are constants to be determined. [5]

3 © HCI 2023 8 Relative to the origin O , two points A and B have position vectors given by 2= −a ij and α=bk respectively, where α is a negative real constant. (a) Find the angle between OA and OB . [1] Let OBA θ= , where θ is in degrees. It is given that 1sin 2θ = . (b) Find the two possible values of ,θ and justify which of the two values is the correct value of θ. [2] (c) By considering a suitable scalar product, find the exact value of α . [3] A point C has position vector c such that = −cba . (d) State the shape of the quadrilateral OABC , justifying your answer. Find the exact area of the quadrilateral OABC . [3] 9 The function f is defined by 2f : ln ( 1) 2xx ++ , , 0xx∈≥ . (a) Show that f has an inverse. [2] (b) Find 1f () x− and state the domain of 1f− . [3] The functi