*** OFFICIAL AQA CORE 4 MPC4 JUNE 2015 EXAM DISCUSSION THREAD ***
UNOFFICIAL MARK SCHEME
1. It is given that f(x)=(5−x)(1+6x)19x−2can be expressed as(5−x)A+(1+6x)B, whereA and Bare integers.
(a) Find the values ofAandB. [3 marks]
A=3
B=−1
∴(5−x)3+(1+6x)−1
(b) Hence show that∫04f(x)dx=kln5, wherekis a rational number. [6 marks]
∫04f(x)dx=38ln5
2. (a) Express2cosx−5sinxin the form Rcos(x+α), whereR>0and 0<α<2π, giving your value ofα, in radians, to three significant figures. [3 marks]
2cosx−5sinx=29cos(x+1.19)
(b) (i) Hence find the value of xin the interval0<x<2π for which2cosx−5sinx has its maximum value. Give your value ofxto three significant figures. [2 marks]
x=5.09
(ii) Use your answer to part (a) to solve the equation2cosx−5sinx+1=0in the interval0<x<2π, giving your solutions to three significant figures. [3 marks]
x=0.568,3.34
3. (a) The polynomial f(x)is defined as f(x)=8x3−12x2−2x+d, where dis a constant. When f(x)is divided by (2x+1), the remainder is −2. Use the Remainder Theorem to find the value ofd. [2 marks]
d=1
(b) The polynomial g(x)is defined byg(x)=8x3−12x2−2x+3. (i) Given that x=2−1is a solution of the equationg(x)=0, write g(x)as a product of three linear factors. [3 marks]
g(x)=(2x+1)(2x−3)(2x−1)
(ii) The function his defined by h(x)=g(x)4x2−1for x>2. Simplifyh(x), and hence show thathis a decreasing function. [4 marks]
h(x)=(2x−3)1
EITHER h′(x)=−2(2x−3)−2∴ for x>2, h′(x)<0 therefore h(x) is a decreasing function.
OR
Plot a graph of h(x). h(x)→0 when x→∞.
4. (a) Find the binomial expansion of(1+5x)51up to and including the term inx2. [2 marks]
1+x−2x2
(b) (i) Find the binomial expansion of (8+3x)3−2up to and including the term inx2. [3 marks]
41+161x+2565x2
(ii) Use your expansion from part (b)(i) to find an estimate for3811, giving your answer to four decimal places. [2 marks]
=0.2313
5. A curve is defined by the parametric equationsx=cos2t, y=sint. The pointPon the curve is wheret=6π.
(a) Find the gradient at P. [3 marks]
dxdy=2−1
(b) Find the equation of the normal to the curve atPin the formy=mx+c. [3 marks]
y=2x−21
(c) The normal at Pintersects the curve again at the pointQ(cos2q,sinq). Use the equation of the normal to form a quadratic equation insinqand hence find thex-coordinate ofQ. [5 marks]
=8−1
6. The points Aand Bhave coordinates(3,2,10)and (5,−2,4)respectively. The linelpasses throughAand has equationr=3210+λ31−2
(a) Find the acute angle between the lineland the lineAB. [4 marks]
60o
(b) The pointClies onlsuch that angleABCis90o. Find the coordinates ofC. [4 marks]
C=(15,6,2)
(c) The point Dis such thatBDis parallel toACand angleBCDis90o. The pointElies on the line throughBandDand is such that the length ofDEis half that ofAC. Find the coordinates of the two possible positions ofE. [4 marks]
E=(11,0,0) E=(23,4,−8)
7. A curve has equationy3+2e−3xy−x=k, wherekis a constant. The pointP(ln2,21), lies on this curve.
(a) Show that the exact value ofk is q−ln2, whereqis a rational number. [1 mark]
k=41−ln2
(b) Find the gradient of the curve atP. [6 marks]
dxdy=811
8. (a) A pond is initially empty and is then filled gradually with water. Aftertminutes, the depth of the water,xmetres, satisfies the equation
dtdx=5(1+t)24+5x
Solve this differential equation to findxin terms oft. [7 marks]
x=51[(2−1(1+t)−1+25)2−4]
(b) Another pond is gradually filling with water. Aftertminutes, the surface of the water forms a circle of radiusr metres. The rate of change of the radius is inversely proportional to the area of the surface of the water.
(i) Write down a differential equation, in the variablesrandtand a constant of proportionality, which represents how the radius of the surface of the water is changing with time. (You are not required to solve your differential equation.) [3 marks]
dtdr=πr2k
(ii) When the radius of the pond is1metre, the radius is increasing at a rate of4.5metres per second. Find the radius of the pond when the radius is increasing at a rate of0.5metres per second.
hows your preperation going for c3 and c4? have you finsihed all the content?
Hey! Not bad thanks yeah we've finished the contents for both and I'm about to move onto M2. How about you? Getting around 63-65 in each paper. Desperately want to get that up to about 72 before the exam so I can get at least an A overall
Hey! Not bad thanks yeah we've finished the contents for both and I'm about to move onto M2. How about you? Getting around 63-65 in each paper. Desperately want to get that up to about 72 before the exam so I can get at least an A overall
Hey!!! Yeah I'm abit slower I've finished core 3 and I'm hitting around 69... just abit of c4 and D1 to go... exams are fast approaching I'm getting scared already
Hey!!! Yeah I'm abit slower I've finished core 3 and I'm hitting around 69... just abit of c4 and D1 to go... exams are fast approaching I'm getting scared already
Ah I'm scared too! What do you have left of C4? I only finished it like 2 weeks ago, so we're starting M2 tomorrow I believe because the last few weeks has been mock period! What are you aiming for grade wise?
Vectors, but I've nearly finished.... Some of the harder parametric equations I'm still getting used to.. I've got 10 exams in total :-( I want to get AAA minimum... but need to hurry up so i can start past papaer season lol
Vectors, but I've nearly finished.... Some of the harder parametric equations I'm still getting used to.. I've got 10 exams in total :-( I want to get AAA minimum... but need to hurry up so i can start past papaer season lol
I'm currently learning C4 and M2 simultaneously. C4 is very challenging. I've been getting mid 55-62 on the C3 past papers. Need to raise my marks higher. Anyone have any revision tips?
I'm currently learning C4 and M2 simultaneously. C4 is very challenging. I've been getting mid 55-62 on the C3 past papers. Need to raise my marks higher. Anyone have any revision tips?
Is your issue with the content or timing? Because I'd say nail the content before even worrying about the 90 minutes! Try Solomon papers for a variety of extra practice. Exam solutions and various other videos on YouTube are great. Have you seen Jack Brown's videos?
Is your issue with the content or timing? Because I'd say nail the content before even worrying about the 90 minutes! Try Solomon papers for a variety of extra practice. Exam solutions and various other videos on YouTube are great. Have you seen Jack Brown's videos?
I think it's about the content mostly. Sometimes when I do past papers, I sometimes forget the method of answering a questions. I know of Exam solutions. I haven;t heard of Solomon papers, though. I'll take it into account. Thank you so much!
I think it's about the content mostly. Sometimes when I do past papers, I sometimes forget the method of answering a questions. I know of Exam solutions. I haven;t heard of Solomon papers, though. I'll take it into account. Thank you so much!
Yeah the method varies much more at A2 compared to AS aha. Sure, any time!
Does anyone know what standard integrals we have to know? There's some in my textbook that refer to sin and tan in a fraction but I wasn't sure if this was still relevant to today's syllabus.