METH - Polynomials, Logs, Exp, Infinite Solutions

7/25/2019 METH - Polynomials, Logs, Exp, Infinite Solutions

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PAST EXAM QUESTIONS: Polynomials, Logarithms,

Exponentials, Innite

sol!tions

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Question 2

)7(log)2(log3 55 +is closest to

A. 0.08

B. 0.22

C. 1.39

D. 2.32

E. 2.5

Question 3

The solution/s to the equation

xx ee =22is/are

A.

( )2log e

B. 0

C. ( )2log,0 e

D.

2

1log,1 e

E.

22e

Question 5

The complete set o linear actors o the pol!nomial

xxxx 107" 23" +

are

A.

10,1,1 ++ xxx

B.

5,1,2 ++ xxx

C.

5,1,2, ++ xxxx

D.

2,1,5, ++ xxxx

E.10,1,1, ++ xxxx


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Question 2

Solve the following equations forx.

a)

( ) #2 1 =+xe.

*)( ) ( ) 0,13loglog2 55 >= xxx

1 + 2 = 3 marks

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Question 3

( )2log 3is closest in $alue to

A. % 0."1

B. 0."1

C. 0.#3

D. 0.#7

E. 1.58

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Question 4

The simultaneous linear equations

5x + (a 3)y =1ax +2y =a

&herea R

, &ill ha$e no solutions or

A.

a = 2

B.

a = 5

C.

a R '(2)

D.

a R '2,5

E.

a (2,5)

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Q!estion &

Consider the system of simultaneous linear equations given by

mx +y = 22x + (m 1)y =m

ind the value!s" of mfor whi#h there is no solution. 3 marks

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Question 2

*or the unctionf(x) =ax 3 + bx 2 +cx + d,

3#)3(an+

")1(

10)2(

")1(

====

'f

'f

f

f

matri- equationAX=B is to e use+ to in+ the $alues o a, b, can+ d&here

=

=

3#

"

10

"

an+B

d

c

b

a

X

.

The matri- Ais

A.

3000

0100

0020

0001

B.

13927

1111

12"8

1111

C.

3003

0101

0022

3121

D.

11#81

1123

12"8

1111


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E.

01#27

0123

12"8

1111

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Q!estion $

Solvelog e (3) +2log e (x) = log e ("x)

forx. 3 marks

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Question 5

The simultaneous linear equations

2x + (k2)y = 2(k+1) x +2y = 1

ha$e no solutions or

A.

k= 2

B.

k= 3

C.

{ }3,2k

D.Rk

E.

k R '2,3

Question 6

$he gra%h of

1an+3 2 +== k xxyxy

do notinterse#t for

A)

k=12 2

-)

12 2


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1 mark

& marks

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2 + 2 = & marks

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2 marks

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Question 3

a) 'ithout using #al#ulator solve02112 = ++ eee xx

forx. 'rite answer ine(a#t form.

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solve2 2 "x xe e = .

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Q!estion $% $he solution!s" of ( ) ( ) 07log7log =++ xxee

is)are

*. ,- , . / C.25

-25

0.25

E.25

ANS5E6S

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2 E 3 * C

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Q!estion $

a)

( ) #2 1 =+xe


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( )

( )

( ) 13log

13log

31

=+=

=+

e

e

x

x

x

e

7( mar89

*)

( ) ( )

( ) ( )

15

35

0since,13

log

13

log

13loglog

0,13loglog2

1

5

2

5

52

5

55

=

=

=

=

=

>=

x

x

xx

x

x

xx

xxx

7( mar89

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3 C

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Q!estion &

mx +y = 2 (1)2x + (m 1)y =m (2)

or no solutions or innite solutions the determinant of the matri(

121

m

m

equals 4ero.

$hat is-

m 1

2 m 1 = 0

7(


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7( mar89

m(m 1)2 = 0

m2 m 2 = 0

(m 2)(m +1)= 0

m = 2 or m = 1

7( mar89

m = 2,

in (1) 2x +y = 2in (2) 2x +y = 2

$hey are the same equation hen#e there are an innite number of solutions.

5fm = 1

-

in (1) x +y = 2 (3)in (2) 2x 2y = 1 (")

(") 2 x +y = 12

(5)

!3" and !" des#ribe %arallel lines with di6erenty7inter#e%ts so there are no

%oints of interse#tion and hen#e no solutions.

So form = 1

there is no solution.

7( mar89

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2 E

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Q!estion $


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0)"3(

0"3

"3

)"(log)3(log

)"(log)(log)3(log

)"(log)(log2)3(log

2

2

2

2

==

=

=

=+

=+

xx

xx

xx

xx

xx

xx

ee

eee

eee

x = 0 or x = "3

7( mar89

butlog e (x)

is not dened forx = 0

so

x ="

3

7( mar89

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8 C

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3 C C 1/ 11 E

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19/20

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3a02112 = ++ eee xx

-( ) 022 = xx eee

-

( )( 022 = xx eee

-( )( ) 012 =+

xx eee. Sin#e

0eand

01 +xe-

02 = xe-

2=xe-

2log ex =.

3il*aha $%(% Q2

( )1

log 2 52

ex = +


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:ES$5;< 9

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2/

( ) ( ) 07log7log =++ xx ee-

( ) ( ) 077log =+ xxe-

( )( ) 077 exx =+-

1"92 =x-

502 =x. Sin#e

07 >x- i.e.

7>x-

2550 ==x E

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METH - Polynomials, Logs, Exp, Infinite Solutions