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Instructions Use black ink or ball-point pen. If pencil is used for diagrams/sketches/graphs it must be dark (HB or B). Fill in the boxes at the top of this page with your name, centre number and candidate number. Answer all the questions and ensure that your answers to parts of questions are clearly labelled. Answer the questions in the spaces provided – there may be more space than you need. You should show sufficient working to make your methods clear. Answers without working may not gain full credit. Inexact answers should be given to three significant figures unless otherwise stated. Information A booklet ‘Mathematical Formulae and Statistical Tables’ is provided. There are 11 questions in this question paper. The total mark for this paper is 100. The marks for each question are shown in brackets – use this as a guide as to how much time to spend on each question. Calculators must not be used for questions marked with a * sign. Advice Read each question carefully before you start to answer it. Try to answer every question. Check your answers if you have time at the end. If you change your mind about an answer, cross it out and put your new answer and any working underneath.
1. The curve C has equation y = f ( x ) where , (a) Show that (3) Given that P is a point on C such that f ʹ( x ) = –1, (b) find the coordinates of P. **(3) (Total 6 marks)
2.** The curve C has equation y = (2 x − 3)^5 The point P lies on C and has coordinates ( w , –32). Find ( a ) the value of w , (2) ( b ) the equation of the tangent to C at the point P in the form y = mx + c , where m and c are constants. (5) (Total 7 marks) ___________________________________________________________________________
5. The point P lies on the curve with equation x = (4 y – sin 2 y )^2. Given that P has ( x , y ) coordinates , where p is a constant, (a) find the exact value of p. (1) The tangent to the curve at P cuts the y- axis at the point A. (b) Use calculus to find the coordinates of A. **(6) (Total 7 marks)
6.** (i) Find, using calculus, the x coordinate of the turning point of the curve with equation y = e^3 x^ cos 4 x ,. Give your answer to 4 decimal places. (5) (ii) Given x = sin^2 2 y , 0 < y < , find as a function of y. Write your answer in the form = p cosec( qy ), 0 < y < , where p and q are constants to be determined. (5) (Total 10 marks) ___________________________________________________________________________
7. (i) Given y = 2 x ( x^2 – 1)^5 , show that (a) = g( x )( x^2 – 1)^4 where g( x ) is a function to be determined. (4) (b) Hence find the set of values of x for which ⩾ 0 (2) (ii) Given x = ln(sec2 y ), 0 < y < find as a function of x in its simplest form. **(4) (Total 10 marks)
8.** f( x ) = ( 2 1 )( 3 )
x x x
x^2 x , x 3, x – 2
( a ) Show that f( x ) = ( 2 1 )( 3 )
x x
The curve C has equation y = f ( x ). The point P
1 , (^) lies on C. ( b ) Find an equation of the normal to C at P. (8) (Total 13 marks) ___________________________________________________________________________
10. Given that k is a negative constant and that the function f( x ) is defined by f ( x ) = 2 – , x 0, (a) show that f ( x ) =. (3) (b) Hence find f ' ( x ), giving your answer in its simplest form. (3) (c) State, with a reason, whether f ( x ) is an increasing or a decreasing function. Justify your answer. **(2) (Total 8 marks)
11.** f( x ) = , x > 2, x ℝ. (a) Given that , find the values of the constants A and B. (4) (b) Hence or otherwise, using calculus, find an equation of the normal to the curve with equation y = f( x ) at the point where x = 3. (5) (Total 9 marks) TOTAL FOR PAPER: 100 MARKS
