Cambridge A Level Mathematics 9709 — 2019 Feb/March Paper 3 · Variant 2

2 The sequence of values given by the iterative formula 2x6 n 12xn + , xn+1 = 3x5 8 n + with initial value x1 2, converges to = !. (i) Use the formula to calculate correct to 4 decimal places. Give the result of each iteration to ! 6 decimal places. [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (ii) State an equation satisfied by and hence find the exact value of [2] ! !. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 2(i) Use the iterative formula correctly at least once M1 Obtain answer 1.3195 A1 Show sufficient iterations to 6 d.p. to justify 1.3195 to 4 d.p., or show there is a sign change in (1.31945, 1.31955) A1 3 2(ii) State x = 6 5 2 12 3 8 + + x x x , or equivalent B1 State answer 5 4 , or exact equivalent B1 2

More questions on Numerical solution of equations

4 4 Show that 2 4. [5] 2 ln x dx = −ln Ó 1 x−3 ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................

Mark scheme: 4 Integrate by parts and reach 1 1 2 2 1 ln . d − − + ∫ ax x b x x x Obtain 1 1 2 2 1 2 ln 2 . d − − − + ∫ x x x x x , or equivalent A1 Complete the integration, obtaining 1 1 2 2 2 ln 4 − − − − x x x , or equivalent A1 Substitute limits correctly, having integrated twice M1(dep*) Obtain the given answer following full and correct working A1 5

More questions on Integration

6 The variables x and y satisfy the differential equation dy = ky3e−x, dx where k is a constant. It is given that y 1 when x 0, and that y e when x 1. Solve the = = = = differential equation, obtaining an expression for y in terms of x. [7] ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................

Mark scheme: 6 Separate variables correctly and attempt integration of at least one side B1 Obtain term 2 1 2 − y , or equivalent B1 Obtain term – k e−x B1 Use a pair of limits, e.g. x = 0, y = 1 to obtain an equation in k and an arbitrary constant c M1 Use a second pair of limits, e.g. x = 1, y = e , to obtain a second equation and solve for k or for c M1 Obtain k = 1 2 and c = 0 A1 Obtain final answer 1 2 e = x y , or equivalent A1 7

More questions on Differential equations

9 Two planes have equations 2x 3y 1 and x z 3. + −z = −2y + = (i) Find the acute angle between the planes. [4] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (ii) Find a vector equation for the line of intersection of the planes. [6] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 9(i) State or imply a correct normal vector to either plane, e.g. 2i + 3j – k, or i – 2j + k B1 Carry out correct process for evaluating the scalar product of two normal vectors M1 Using the correct process for the moduli, divide the scalar product of the two normal vectors by the product of their moduli and evaluate the inverse cosine of the result M1 Obtain answer 56.9° or 0.994 radians A1 4 9(ii) EITHER: Carry out a complete strategy for finding a point on the line (call the line l) M1 Obtain such a point, e.g. (1, 1, 4) A1 EITHER: State a correct equation for a direction vector ai + bj + ck for l, e.g. 2a + 3b – c = 0 B1 State a second equation, e.g. a – 2b + c = 0, and solve for one ratio, e.g. a : b M1 Obtain a : b : c = 1 : – 3: – 7, or equivalent A1 State a correct answer, e.g. r = i + j + 4k + λ(i – 3j – 7k) A1 OR1: Attempt to calculate the vector product of the two normal vectors M1 Obtain two correct components A1 Obtain i – 3j – 7k, or equivalent A1 State a correct answer, e.g. r = i + j + 4k + λ(i – 3j – 7k), or equivalent A1 Question Answer Marks Guidance 9(ii) OR2: Obtain a second point on l e.g. (0, 4, 11) B1 Subtract position vectors and obtain a direction vector for l M1 Obtain i – 3j – 7k, or equivalent A1 State a correct answer, e.g. r = 4j + 11k + µ(i – 3j – 7k), or equivalent A1 OR3: Express one variable in terms of a second M1 Obtain a correct simplified expression, e.g. 4 3 = − y x A1 Express the third variable in terms of the second M1 Obtain a correct simplified expression, e.g. 11 7 = − z x A1 Form a vector equation for the line M1 State a correct answer, e.g. r = 4j + 11k + λ(i – 3j – 7k), or equivalent A1 6 Question Answer Marks Guidance 9(ii) OR4: Express one variable in terms of a second M1 Obtain a correct simplified expression, e.g. 4 3 3 = −y x A1 Express the same variable in terms of the third M1 Obtain a correct simplified expression, e.g. 11 7 7 = −z x A1 Form a vector equation for the line M1 Obtain a correct answer, e.g. r = 4j + 11k +µ(i – 3j – 7k), or equivalent A1 6

More questions on Vectors

10 y M x O 120 The diagram shows the curve y sin3x cosx for 0 and its maximum point M. = ≤x ≤120, (i) Using the substitution u cosx, find by integration the exact area of the shaded region bounded = by the curve and the x-axis. [6] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (ii) Showing all your working, find the x-coordinate of M, giving your answer correct to 3 decimal places. [6] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 10(i) State or imply du = – sin x dx B1 Using Pythagoras express the integral in terms of u M1 Obtain integrand ( ) 2 1 ± − u u A1 Integrate and obtain 3 7 2 2 2 2 3 7 − + u u , or equivalent A1 Change limits correctly and substitute correctly in an integral of the form 3 7 2 2 + au bu M1 Or substitute original limits correctly in an integral of the form 3 2 (cos ) a x + 7 2 (cos ) b x Obtain answer 8 21 A1 6 10(ii) Use product rule and chain rule at least once M1 Obtain correct derivative in any form A1 + A1 Equate derivative to zero and obtain a horizontal equation in integral powers of sin x and cos x M1 Use correct methods to obtain an equation in one trig function M1 Obtain 2 tan 6 = x , 2 7cos 1 = x or 2 7sin 6 = x , or equivalent, and obtain answer 1.183 A1 6

More questions on Integration