big questions practice.doc

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Big Questions Practice

1. Clara organizes cans in triangular piles, where each row has one less can than the rowbelow. For example, the pile of 15 cans shown has 5 cans in the bottom row and 4cans in the row above it.

(a ! pile has "# cans in the bottom row. $how that the pile contains "1# cans.

(b %here are &"4# cans in a pile. 'ow man cans are in the bottom row)

(c (i %here are Scans and the are organized in a triangular pile with ncans in

the bottom row. $how that n"* n+ "S #.

(ii Clara has "1## cans. -xplain wh she cannot organize them in a triangular pile.

2. et Snbe the sum of the first nterms of the arithmetic series " * 4 * / * 0.

(a Find

(i S4 (ii S1##.

etM

1#

"1.

(b (i FindM". (ii $how thatM

&

1#

/1.

2t ma now be assumed thatMn

1#

"1 n, for n4. %he sum Tnis defined b

TnM1*M

"*M

&* ... *M

n.

(c (i 3rite downM4. (ii Find T4.

(d sing our results from part (a (ii, find T1##.

&. ! cit is concerned about pollution, and decides to loo at the number of people using taxis. !t the end of theear "###, there were "6# taxis in the cit. !fter nears the number of taxis, T, in the cit is given b

T "6# 1.1"n.

(a (i Find the number of taxis in the cit at the end of "##5.

(ii Find the ear in which the number of taxis is double the number of taxis there were at theend of "###.


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(b !t the end of "### there were "5 /## people in the cit who used taxis. !fter nears the numberof people,P, in the cit who used taxis is given b

P .e7#1#

###5/#"#.18 n+

(i Find the value ofPat the end of "##5, giving our answer to the nearest whole number.

(ii !fter seven complete ears, will the value ofPbe double its value at the end of "###)9ustif our answer.

(c etRbe the ratio of the number of people using taxis in the cit to the number of taxis. %he cit

will reduce the number of taxis ifR (x. etAbe the area of the region enclosed b the graph of g and thex=axis, between

x # andx a, where a>#. ?iven thatA ", find the value of a.


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5. ! Ferris wheel with centre @ and a radius of 15 metres is represented in the diagram below. 2nitiall seat ! is

at ground level. %he next seat is A, where A@B! /

C.

(a Find the length of the arc !A.

(b Find the area of the sector !@A.

(c %he wheel turns clocwise through an angle

of&

". Find the height of ! above the

ground.

%he height, hmetres, of seat C above the groundafter tminutes, can be modelled b the function

h(t 15 + 15 cos

+

4C"t .

(d (i Find the height of seat C when t4

C.

(ii Find the initial height of seat C.

(iii Find the time at which seat C first reaches its highest point.

(e Find h> (t.

(f For # t,

(i setch the graph of h>

(ii find the time at which the height is changing most rapidl.

/. %he following is the cumulative fre


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