OCR As Maths Paper 2 MS Maths paper 2 Marking Scheme + Test Feedback Summer 2021

The diagram shows a sector Nt7fi o1“ a circle with centre I) and radius °7.5 cm. The angle Jfi is 25°. (a) Calculate the length of thc straight linc AB The diagram shows a sector AOB of a circle with centre O and radi ...

The diagram shows a sector Nt7fi o1“ a circle with centre I) and radius °7.5 cm. The angle Jfi is 25°. (a) Calculate the length of thc straight linc AB The diagram shows a sector AOB of a circle with centre O and radius 9.5 cm. The angle AOB is 25°. (a) Calculate the length of the straight line AB. (bj Find the area of tlte segment shaded in the diagram. Two curves have equations y —— lnz and y —— p, where 2 is a positive constant. (a) Sketch the curves on a single diagram Two curves have equations y —— lnx and y —— q, where k is a positive constant (a) Sketch the cwves on a sinQe diagram. [3] (b) Explain how yow diagram shows that the equation xlnx —£ = 0 has exactly one real root. In this question you aiiist show detailed reasoning Find the equation of le nomal to the cwve y 4fÑ — 3x+ 1 at the point on the curve a 4. Give yow answer in Te form ex Y by+c 0, where e, b and c axe integers. In this question you must show detailed reasoning. The cubic polynomial 6x3 + H' + 57x— 20 is denoted by f(x). It is given that (2r — l) is a factor of f(.v). (a) Use the factor theorem to show that 1 = — 37. (bJ Using this value of k, factorise f(x) completely. The cubic polynomial 6x" + fir' + 57.x — 20 (c) (i) Hence find the three values of t satis ing the equation 6e °” — 37e*" + 57e *' — 20 = 0. [2] (ii) Express the sum of the three x’a1ues found in part (c)(i) as a single logariihm. A curve has equation y —- n(x + b)2 + r, where o, b and c are constants. The curre has a siationary point at (—3, 2). curve has equation y —- n(x + b)2 + r, where o, b and c are constants. The curre has a siationary point at (—3, 2). (a) State the values of ñ and c. When the curve is translated by (b) Determine the value of u. 4’ the transformed curve passes lhmugh the point (3, —18). 0 The diagram shows the line 3J-+s = 7 which is a tangent to a circle shh centre (3, —2). Find as equation for ih< circle. the diagram shows a model for the roof of a toy building. the roof is in the form of a solid triangular prism AB€!DEF. The base AL!FD of the roof is a horizontal rectangle. and the cross- section ABC. of the roof is an isosceles triannle with AB —- BC The lengths ofAC and CF are lx cm the roof is xcm. The total surface area of the five faci (a) Show’ that K = R(300 — x ), (b) Use differentiation to determine the value ofx for which the volume of the roof is a maximum. [4] (c) Find the maximum vulumc c›f fire ruuf. Give your answer in cm , current (u the nearest integer. (d) Explain why, for this roof, x must be less than a certain value, which you should state. A particle is in equilibrium under the action of the following three forces: (2pi — 4j) N, (—3qi + 5pj) N and (—l3i — 6j) N. Find the values of p and q. A crane lifts a car vertically. The car is inside a crale which is raised by the crane by means of a strong cable. The cable can withstand a maximum tension of 9500 N without breaking. The crate has a mass of 55 kg and the car has a mass of 830kg. (a) Find the maximum acceleration with which the crate and car can be raised. [2j (b) Show on a clearly labelled diagram the forces acting on the crate while it is in motion. Linking Displacement, Velocity and Acceleration A particle P is moving in a straight line. At time i seconds P has velocity vms° ' where v - (2/+ 1)(3 — f). (a) Find the deceleration of P when f = 4. (b) State the positive value of i for which P is instantaneously at rest. (c) Find the total distance that P travels between times f = 0 and i = 4. A ear starts from rest at a set of traffic lights and moves along a straight road with constant acceleration 4 ms**. A motorcycle, travelling parallel to the car with constant speed 16 ms*', passes the same traffie lights exactly 1.5 seconds after the car starts to move. The time after the car starts to move is denoted by f seconds. (a) Determine the two values of t at which the car and motorcycle are the same distance from the traffic lighls. SUVAT Equations