Free until 14 Nov · 80% off for the first 100 · Pricing
Cambridge AS & A Level Mathematics (9709), Paper 2 Pure Mathematics, February/March 2019. Read both free below, or download them. Done the paper? Get it marked by AI against this mark scheme.
9709_m19_qp_229709_m19_ms_229709/22/F/M/19
Each question as it's printed, and the mark scheme's points for it, as text. Diagrams, tables and graphs are in the PDFs above.
Question
Solve the equation sec2 1 + tan2 1 = 5 tan 1 + 4 for 0Å < 1 < 180Å. Show all necessary working. [4]
Mark scheme
1 Use identity sec 2 θ = 1 + tan 2 θ B1 Attempt solution of quadratic equation to find two values of tan θ M1 Obtain tan θ = − 12 , 3 A1 Obtain 71.6 and 153.4 and no others between 0 and 180 A1 4
Question
Given that x satisfies the equation 2x + 3 = 2x − 1, find the value of 4x − 3 − 6x . [4]
Mark scheme
2 Solve non-modular equation (2 x + 3) 2 = (2 x − 1) 2 or linear equation with signs of 2x different M1 Obtain x = − 12 A1 Substitute negative value into expression and show correct evaluation of modulus at least once M1 Obtain 5 − 3 = 2 with no errors seen A1 4
Question
ln y 6, 6.585 1, 2.835 x O The variables x and y satisfy the equation y = Ae px+p , where A and p are constants. The graph of ln y against x is a straight line passing through the points 1, 2.835 and 6, 6.585, as shown in the diagram. Find the values of A and p. [5]
Mark scheme
3 State or imply equation is ln y = ln A + px + p B1 Equate gradient of line to p M1 Obtain p = 0.75 A1 Substitute appropriate values to find ln A M1 Obtain ln A = 1.335... and hence A = 3.8 A1 5
Question
(i) Find the quotient when 4x3 + 8x2 + 11x + 9 is divided by 2x + 1, and show that the remainder is 5. [3] (ii) Show that the equation 4x3 + 8x2 + 11x + 4 = 0 has exactly one real root. [3] The equation of a curve is y = e 2x 4x + 1
Mark scheme
4(i) Carry out division at least as far as 2x 2 + kx M1 Obtain quotient 2 x 2 + 3 x + 4 A1 Confirm remainder is 5 A1 Answer given; necessary detail needed 3 4(ii) State or imply equation is (2 x + 1)(2 x 2 + 3x + 4) = 0 B1 FT their quotient from part (i) Calculate discriminant of 3-term quadratic expression or equivalent M1 Obtain –23 or equiv and conclude appropriately A1 3
Question
and the point P on the curve has y-coordinate 10. (i) Show that the x-coordinate of P satisfies the equation x = 12 ln 40x + 10. [2] (ii) Use the iterative formula xn+1 = 12 ln 40xn + 10 with x1 = 2.3 to find the x-coordinate of P correct to 4 significant figures. Give the result of each iteration to 6 significant figures. [3] (iii) Find the gradient of the curve at P, giving the answer correct to 3 significant figures. [4] 4@ A (a) Show that Ô + dx = ln 48. 2 2 x 2x + 1
Mark scheme
5(i) e2 x M1 Attempt rearrangement of = 10 to x = ... involving ln 4x + 1 Confirm x = 12 ln(40 x + 10) A1 Answer given; necessary detail needed 2 5(ii) Use iteration process correctly at least once M1 Obtain final answer 2.316 A1 Show sufficient iterations to 6 sf to justify answer or show a sign change in the interval [2.3155, 2.3165] A1 3 5(iii) Use quotient rule (or product rule) to find derivative M1 2e 2 x (4 x + 1) − 4e 2 x A1 Obtain or equivalent (4 x + 1)2 Substitute answer from part (ii) (or more accurate value) into attempt at first derivative M1 Obtain 16.1 A1 4
Question
[5] 1 (b) Find Ó sin 2x cot x + 2 cosec x dx. [6]
Mark scheme
6(a) Integrate to obtain form k1 ln x + k 2 ln(2 x + 1) M1 Obtain correct 2ln x + ln(2 x + 1) A1 Use logarithm addition/subtraction property correctly M1 Use logarithm power property correctly M1 Confirm ln 48 with no errors seen A1 Answer given; necessary detail needed 5 6(b) Use identity sin 2 x = 2sin x cos x B1 cos x 2 B1 State or imply cot x + 2cosec x = + sin x sin x Attempt to express integrand in terms of cos 2x and cos x M1 Obtain correct integrand 1 + cos 2 x + 4 cos x A1 Integrate to obtain at least terms k3 sin 2 x and k 4 sin x M1 Obtain correct x + 12 sin 2 x + 4sin x + c A1 6
Question
The parametric equations of a curve are x = 2t − sin 2t, y = 5t + cos 2t, for 0 ≤ t ≤ 12 0. At the point P on the curve, the gradient of the curve is 2. (i) Show that the value of the parameter at P satisfies the equation 2 sin 2t − 4 cos 2t = 1. [4] (ii) By first expressing 2 sin 2t − 4 cos 2t in the form R sin 2t − !, where R > 0 and 0 < ! < 12 0, find the coordinates of P. Give each coordinate correct to 3 significant figures. [7] Additional Page If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
Mark scheme
7(i) Obtain ddxt = 2 − 2cos 2t B1 Obtain ddyt = 5 − 2sin 2t B1 Equate attempt at ddyx to 2 and rearrange M1 Confirm equation 2sin 2t − 4cos 2t = 1 A1 Answer given; necessary detail needed 4 7(ii) State R = 20 or 4.47 B1 Use appropriate trigonometry to find α M1 Obtain α = 1.107 with no errors seen A1 Carry out correct method to find value of t M1 Obtain t = 0.666 A1 Substitute value of t between 0 and 12 π into expressions for x and y M1 Obtain x = 0.361, y = 3.57 A1 7
Checked your answers against these? Let AI mark the whole paper, on your own working.
Mark my 9709/22Examiners follow the generic marking principles at the front of the scheme, then each question's points: method (M), accuracy (A) and independent (B) marks, error carried forward where the scheme allows it, and positive marking, so you're credited for what you got right.
Done this paper? Upload your script and CIE Buddy's AI marker marks it question by question against this exact scheme, puts the ticks on your own working and says what each lost mark needed.
Question papers and mark schemes © Cambridge University Press & Assessment. CIE Buddy is independent and not connected to Cambridge.
On this page, free: the mark scheme for AS & A Level Mathematics 9709/22, February/March 2019 (9709_m19_ms_22), and the question paper (9709_m19_qp_22), to read or download, with each question's marking points written out underneath.
Mathematics 9709/22 for February/March 2019 is out of 50 marks, with 1 hour 15 minutes to answer it.
Yes. Upload your answers to 9709/22/F/M/19 as a PDF or photos, and CIE Buddy marks every question against this paper's official mark scheme, using Cambridge's marking rules. You get your score, each lost mark explained on your own script, and the topics to revise. It's a guide, not an official Cambridge grade.