# UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS General Certificate of Education Advanced Level

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2 2 1 The equation of a curve is y = 1 + x 1 + 2x for x > 1. Show that the gradient of the curve is always 2 negative. [3] 2 Solve the equation 2 3 x 1 = 3 x, giving your answers correct to 3 significant figures. [4] 3 Find the exact value of 1 4 ln x x dx. [5] 4 The parametric equations of a curve are x = e t cos t, y = e t sin t. Show that dy dx = tan t 1 4. [6] 5 (i) Prove that cot + tan 2 cosec 2. [3] (ii) Hence show that 1 3 cosec 2 d = ln 3. [4] 6 O B r C A In the diagram, A is a point on the circumference of a circle with centre O and radius r. A circular arc with centre A meets the circumference at B and C. The angle OAB is radians. The shaded region is bounded by the circumference of the circle and the arc with centre A joining B and C. The area of the shaded region is equal to half the area of the circle. (i) Show that cos 2 = 2 sin 2. [5] 4 (ii) Use the iterative formula 2 sin n+1 = 1 2 n 2 cos 1, 4 n with initial value 1 = 1, to determine correct to 2 decimal places, showing the result of each iteration to 4 decimal places. [3] UCLES /31/O/N/13

3 3 7 Let f x = 2x2 7x 1 x 2 x (i) Express f x in partial fractions. [5] (ii) Hence obtain the expansion of f x in ascending powers of x, up to and including the term in x 2. [5] 8 Throughout this question the use of a calculator is not permitted. (a) The complex numbers u and v satisfy the equations u + 2v = 2i and iu + v = 3. Solve the equations for u and v, giving both answers in the form x + iy, where x and y are real. [5] (b) On an Argand diagram, sketch the locus representing complex numbers satisfying + i = 1 and the locus representing complex numbers w satisfying arg w 2 = 3. Find the least value 4 of w for points on these loci. [5] 9 C D B A O The diagram shows three points A, B and C whose position vectors with respect to the origin O are given by 2 OA = 1, 0 OB = 3 and 3 OC = 0. The point D lies on BC, between B and C, and is such that CD = 2DB. (i) Find the equation of the plane ABC, giving your answer in the form ax + by + c = d. [6] (ii) Find the position vector of D. [1] (iii) Show that the length of the perpendicular from A to OD is [4] [Question 10 is printed on the next page.] UCLES /31/O/N/13 [Turn over

4 4 10 h 60 C A tank containing water is in the form of a cone with vertex C. The axis is vertical and the semivertical angle is 60, as shown in the diagram. At time t = 0, the tank is full and the depth of water is H. At this instant, a tap at C is opened and water begins to flow out. The volume of water in the tank decreases at a rate proportional to h, where h is the depth of water at time t. The tank becomes empty when t = 60. (i) Show that h and t satisfy a differential equation of the form dh dt = Ah 3 2, where A is a positive constant. [4] (ii) Solve the differential equation given in part (i) and obtain an expression for t in terms of h and H. [6] (iii) Find the time at which the depth reaches 2 1 H. [1] [The volume V of a cone of vertical height h and base radius r is given by V = 1 3 r2 h.] Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. UCLES /31/O/N/13

6 2 1 The equation of a curve is y = 1 + x 1 + 2x for x > 1. Show that the gradient of the curve is always 2 negative. [3] 2 Solve the equation 2 3 x 1 = 3 x, giving your answers correct to 3 significant figures. [4] 3 Find the exact value of 1 4 ln x x dx. [5] 4 The parametric equations of a curve are x = e t cos t, y = e t sin t. Show that dy dx = tan t 1 4. [6] 5 (i) Prove that cot + tan 2cosec2. [3] (ii) Hence show that 1 3 cosec 2 d = ln 3. [4] 6 O B r C A In the diagram, A is a point on the circumference of a circle with centre O and radius r. A circular arc with centre A meets the circumference at B and C. The angle OAB is radians. The shaded region is bounded by the circumference of the circle and the arc with centre A joining B and C. The area of the shaded region is equal to half the area of the circle. (i) Show that cos 2 = 2sin2. [5] 4 (ii) Use the iterative formula n+1 = 1 2sin2 n 2 cos 1, 4 n with initial value 1 = 1, to determine correct to 2 decimal places, showing the result of each iteration to 4 decimal places. [3] UCLES /32/O/N/13

7 3 7 Let f x = 2x2 7x 1 x 2 x (i) Express f x in partial fractions. [5] (ii) Hence obtain the expansion of f x in ascending powers of x, up to and including the term in x 2. [5] 8 Throughout this question the use of a calculator is not permitted. (a) The complex numbers u and v satisfy the equations u + 2v = 2i and iu + v = 3. Solve the equations for u and v, giving both answers in the form x + iy,wherex and y are real. [5] (b) On an Argand diagram, sketch the locus representing complex numbers satisfying + i = 1 and the locus representing complex numbers w satisfying arg w 2 = 3. Find the least value 4 of w for points on these loci. [5] 9 C D B A O The diagram shows three points A, B and C whose position vectors with respect to the origin O are given by 2 OA = 1, 0 OB = 3 and 3 OC = 0. The point D lies on BC, between B and C, andis such that CD = 2DB. (i) Find the equation of the plane ABC, giving your answer in the form ax + by + c = d. [6] (ii) Find the position vector of D. [1] (iii) Show that the length of the perpendicular from A to OD is [4] [Question 10 is printed on the next page.] UCLES /32/O/N/13 [Turn over

8 4 10 h 60 C A tank containing water is in the form of a cone with vertex C. The axis is vertical and the semivertical angle is 60, as shown in the diagram. At time t = 0, the tank is full and the depth of water is H. At this instant, a tap at C is opened and water begins to flow out. The volume of water in the tank decreases at a rate proportional to h,whereh is the depth of water at time t. The tank becomes empty when t = 60. (i) Show that h and t satisfy a differential equation of the form dh dt = Ah 3 2, where A is a positive constant. [4] (ii) Solve the differential equation given in part (i) and obtain an expression for t in terms of h and H. [6] (iii) Find the time at which the depth reaches 1 H. [1] 2 [The volume V of a cone of vertical height h and base radius r is given by V = 1 3 r2 h.] Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. UCLES /32/O/N/13

10 2 1 Given that 2 ln x + 4 ln x = ln x + a, express x in terms of a. [4] 2 Use the substitution u = 3x + 1 to find 3x dx. [4] 3x The polynomial f x is defined by f x = x 3 + ax 2 ax + 14, where a is a constant. It is given that x + 2 is a factor of f x. (i) Find the value of a. [2] (ii) Show that, when a has this value, the equation f x = 0 has only one real root. [3] 4 A curve has equation 3e 2x y + e x y 3 = 14. Find the gradient of the curve at the point 0, 2. [5] p 5 It is given that 4xe 1 2 x dx = 9, where p is a positive constant. 0 8p + 16 (i) Show that p = 2 ln. [5] 7 (ii) Use an iterative process based on the equation in part (i) to find the value of p correct to 3 significant figures. Use a starting value of 3.5 and give the result of each iteration to 5 significant figures. [3] 6 Two planes have equations 3x y + 2 = 9 and x + y 4 = 1. (i) Find the acute angle between the planes. [3] (ii) Find a vector equation of the line of intersection of the planes. [6] 7 (i) Given that sec + 2 cosec = 3 cosec 2, show that 2 sin + 4 cos = 3. [3] (ii) Express 2 sin + 4 cos in the form R sin + where R > 0 and 0 < < 90, giving the value of correct to 2 decimal places. [3] (iii) Hence solve the equation sec + 2 cosec = 3 cosec 2 for 0 < < 360. [4] 8 (i) Express 7x in partial fractions. [5] 1 + x 2 2 3x 7x (ii) Hence expand 1 + x 2 2 3x in ascending powers of x up to and including the term in x2, simplifying the coefficients. [5] UCLES /33/O/N/13

11 3 9 (a) Without using a calculator, use the formula for the solution of a quadratic equation to solve 2 i i = 0. Give your answers in the form a + bi. [5] (b) The complex number w is defined by w = 2e 1 4 i. In an Argand diagram, the points A, B and C represent the complex numbers w, w 3 and w* respectively (where w* denotes the complex conjugate of w). Draw the Argand diagram showing the points A, B and C, and calculate the area of triangle ABC. [5] 10 y B O A x A particular solution of the differential equation 3y 2 dy dx = 4 y3 + 1 cos 2 x is such that y = 2 when x = 0. The diagram shows a sketch of the graph of this solution for 0 x 2 ; the graph has stationary points at A and B. Find the y-coordinates of A and B, giving each coordinate correct to 1 decimal place. [10] UCLES /33/O/N/13

12 4 BLANK PAGE Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. 9709/33/O/N/13

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