数学联合学校计算机科学入学说明.pdf

SOLUTIONSFORADMISSIONSTESTIN

MATHEMATICS,JOINTSCHOOLSANDCOMPUTERSCIENCE

WEDNESDAY3NOVEMBER2010

MarkScheme:

EachpartofQuestion1isworthfourmarkswhichareawardedsolelyforthecorrectanswer.

EachofQuestions2-7isworth15marks

QUESTION1:

2

A.Theliney=kxintersectstheparabolay=(x−1)whentheequation

22

(x−1)=kx⇐⇒x−(k+2)x+1=0

2

hasrealsolutions.Thisquadraticequationhasdiscrimant(k+2)−4whichisnonnegativewhen

k+2
2,i.e.k
0ork+2−2,i.e.k−4.

Theansweris(c).

B.Theoddtermsinthesequence

1111

1,1,2,,4,,8,,16,,...,

24816

fn−1

romamongstthefirst2nterms,are1,2,4,...,2andtherelevanteventermsaretheirreciprocals.

So,recognisingtheseasgeometricseries,weneedtosum

n−11++···+11

1+2+4+...+2+1+

242n−1

n−n

1(2−1)1(2−1)

=+

(2−1)(1/2−1)

n
1−n

=(2−1)+2−2

n1−n

=2+1−2.

Theansweris(a).

C.Ifxsolvestheequation

22

sinx+3sinxcosx+2cosx=0

2

thencosx=0,sothatwecandividebycosxtofind

2

tanx+3tanx+2=0.

Th