Solution to Mathematics Utme 2010_0

8/8/2019 Solution to Mathematics Utme 2010_0

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This solution is provided by Mathsvillage.com Mr, Simeon AbiolaTawose(+2348085573332)SOLUTION TO MATHEMATICS UTME 2010

Question 1Which of the mathematic Question paper type is givento you.Solution : Type BI will be using the type B to solve the questionsQuestion 2Find r , if 6 r 78 = 511 9Solution

8 9

2 1 2 1

6 7 511

converting both sides to decimal system

6 8 8 7 5 9 1 9 1

384 8 7 405 9 1

391 8 415

8 24

3

r

r

r

r

r

r

=

+ + = + ++ + = + ++ =

==

Question 3Simplify ( )3 54 14 9 2 19of 9 1

Solution3 4 2 19 14 9 19 24 4

=

Question 4A student measures a piece of rope and found that itwas 1.26m long. If the actual length of the rope was1.25m, what was the percentage error in themeasurement?SolutionActual size = 1.25 mMeasured size = 1.26 mError = (1.26 1.25) m = 0.01 m

% 100% 0.01

% 100 0.80%1.25

error Error actual size

error

=

= =

Question 5At what rate will the interest on N400 increase to N24in 3 years reckoning in simple interestSolution

100 100 242%

400 3 I

R P T

= = = Question 6If 523 6: : p q = and 3 14 2: :q r = find : : p q r

Solution2352 4

3 6 556

45

343 31 2 2

4 2 4 1 312

4 25 3

4 25 3

: :

( )

: :

2( )

3: : : :

: : :1:

multiply through by 1515 :15 :15 12 :15 :10

option B

pq

qr

p q

p q i

q r

r q ii

p q r q q q

p q r

p q r

= = =

=

= = = =

=

==

=

Question 7

Evaluate14

181 216

Solution11 144 44 ( )4

1 1 14

11

81 3 32 2 2

16 22

3 2 1 122 3 2 3

= =

= = = Question 8Given that log 2 =0.3010, log7 = 0.8451, Evaluatelog112Solution

4log112 log(16 7) log16 log 7 log 2 log 7

4 log 2 log 7 4(0.3010) 0.8457

1.204 0.8451 2.0491

= = + = += + = += + =

Question 9

Rationalize2 3 5

5 3

+

Solution

2 3 5 2 3 5 5 3

5 3 5 3 5 3

2 15 5 6 155 3

3 15 112

+ + += +

+ + += +=

Question 10Express the product 0.21 and 0.34 in standard formSolution

20.21 0.34 0.0714 7.14 10 = =

Question 11Which of the venn diagram below represent

1 1 P Q R Solution

Option A

Question 12In a survey of 50 newspaper readers, 40 read championand 30 read Guardian. How many read both papers

Solutionlet be the set of those read Champion newspapers

let be the set of those read Guardian newspapers

( ) (50), also ( ) 50

( ) ( ) ( ) ( )

C

G

n n n C G

n C G n C n G n C G

= = = +

This material is downloaded at www.mathsvillage.com owned by Simeon AbiolaTawose 1

P `Q

R
http://www.mathsvillage.com/http://www.mathsvillage.com/


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This solution is provided by Mathsvillage.com Mr, Simeon AbiolaTawose(+2348085573332)50 30 40 ( )

( ) 20

20 read both newspaper

n C G

n C G= + =

Question 13Make Q the subject of the formula if

( ) 15M

P X Q= + +

Solution

( ) 15

1 ( )5

5 5

5 5

5 5

M P X Q

M P X Q

P MX MQ

P MX MQ

P MX Q

M

= + +

= +

= + =

=

Question 14

If 2 29 6 4 x xy y+ + is a factor of 3 327 8 x y , find theother factor

Solution3 3 2 2

3 3 3 3

2 2

2 2

note: ( )( )

27 8 (3 ) (2 )

(3 2 )[(3 ) (3 )(2 ) (2 ) ]

(3 2 )(9 6 4 )

The other factor is (3 2 )

a b a b a ab b

x y x y

x y x x y y

x y x xy y

x y

= + + =

= + += + +

Question 15

Factorize completely3 2

23 102 8 x x x

x+ Solution

3 2 2

2 2

3 10 ( 3 10)2 8 2( 4)

( 5)( 2) ( 5)2( 2)( 2) 2( 2)

x x x x x x x x

x x x x x x x x

+ + = + += = + +

Question 16Solve for x and y if x y = 2 and x2 y2 = 8Solution

2 2

2 ( )

8

( )( ) 8 ( )

2( ) 8

4 ( )

Adding ( ) and ( ) together

2 6

6

Substitute 3 for in ( )

x y a

x y

x y x y b

x y

x y c

a c

x

x

x a

= = + = + =

+ =

==

1

( , ) (3,1)

y

x y=

=

Question 17If y varies directly as the square root of x and y =3 and

x =16. Calculate y when x = 64

Solution

when 16, 3

3 16

34

when 64

3 364 8 6

4 4

y x

y k x

x y

k

k

x

y k x

== =

=

==

= = = =

Question 18If x is inversely proportional to y and x = 2 when y =2, find x if y = 4Solution

12

14

1

when 2 , 2

52 2

5

when 4

51

4

x yk

x y

x y

k

k

y

k x

y

=

= =

=

==

= = =

Question 19

For what range of values of x is 1 1 1 12 4 3 2 x x+ > +Solution

1 1 1 12 4 3 2

1 1 1 12 3 2 4

3 2 2 16 4

16 4

32

x x

x

x x

x x

x

+ > + >

>>>

Question 20Solve the inequalities 6 4 2 5 x x <

Solution6 4 2 5

6 4 2 5 4

10 2 1

10 2 2 1

x x

x x

x x

x x x

< < < < 5 1

5 1

1 5

option B

x x

x x

x

< >

<

Question 21Find the sum to infinity 0.5 + 0.05 + 0.005 +0.0005 +.Solution



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This solution is provided by Mathsvillage.com Mr, Simeon AbiolaTawose(+2348085573332)0.05

0.5, 0.10.50.5 0.5 5

1 1 0.1 0.9 9

a r

aS

r

= = =

= = = =

Question 22The 3 rd term of an arithmetic progression is 9 and the

7th

term is 29. Find the 10th

term of the progressionSolution

3

7

3

7

10

9

29

( 1)

(3 1) 9

2 9 ( )

(7 1) 29

6 29 ( )

subtract (2) from (1)

4 20

5

substitute 5 for in equation (i)

2( 5) 9

1

1 (10 1)( 5) 1 45 44

option c

n

T

T

T a n d

T a d

a d i

T a d

a d ii

d

d

d

a

a

T

= = = + = + =

+ = = + =

+ =

== + = =

= + = =

Question23If 2 x y x y = +, find the value of (2 3) 5

Solution2

2

(2 3) 2 3 11(2 3) 5 11 5

(2 3) 5 11 5 11 25 36

= + = = = + = + =

Question 24If p and q are two non zero numbers and 18( p + q ) =(18 + p)q. which if the following must be true(a) q =18 (b) p =18 (c) p < 1 (d) q < 1

Solution18( ) (18 )

18 18 1818

18

p q p q

p q q pq p pq

q

+ = +

+ = +==

Option A

Question 25

If 3

152 7

x = find the value of x

Solution3

15

2 77 6 15

7 21

3

x

x

x

x

=

==

=

Question 26

Evaluate

2 0 5

4 6 3

8 9 1

Solution2 0 5

4 6 3 2(6 27) 0(4 24) 5(36 48)

8 9 1

42 60 102

= +

= =

Question 27

If P =2 3

1 1

what is P -1

Solution

1

for a 2 2 matrix

1

a b A

c d

d b Ac aad cd

=

= 31

1 5 515 1 2

5 5

1 3 1 311 2 1 22 3

P

= = = +

Question 28

From the diagram above, find xSolution

90 (angle in a semicircle)

180 (90 25 ) (sum of s in a )

65

65 (alternate segment)

RT S

RST

RST

x R S T

=

= +

=

= =

o

o o o

o

o

Question 29The interior angles of a quadrilateral are ( x +15) o, (2 x 45) o, ( x 30) o and ( x +10) o. Find the value of the leastinterior angle

Solution( 15) (2 45) ( 30) ( 10) 360

5 50 360

5 410

82

The least angle will ( 30) (82 30) 52

x x x x

x

x

x

x

+ + + + + = ==

= = =

o

o o o o o

o

o

o

o o


R

2 5 o

x

T

U

S

O


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This solution is provided by Mathsvillage.com Mr, Simeon AbiolaTawose(+2348085573332)

Question 30

From the cyclic quadrilateral TUVW above, find thevalue of xSolution

180 (opp. s of cyclic quad.)

88 (3 20) 180

108 (3 ) 180

3 72

24

T W V T UV

x

x

x

x

+ =

+ + =+ =

==

o

o o o

o o o

o

o

Question 31If the two smaller size of a right angled triangle are 4cmand 5cm, find its area.

Solution

2 21 1 5 4 102 2

A bh cm cm= = =

Question 32An arc subtends an angle of 50 o at the centre of circle of radius 6cm. Calculate the area of sector formed

Solution

2

2 2 2

50 , 6

Area of sector 36050 22 110

Area of sector 67 7360

r cm

r

cm cm

= =

=

= =

o

o

o

o

Question 33A cylindrical pipe 50m long with radius 7 m has oneend open. What is the total surface area of the pipe?Solution

2

2

2

2

( 2 )7 (7 2 50) 7 (107) 749

749

A r rh

A r r h A m

A m

= +

= += + = ==

Question 34What is the locus of the point that is equidistant from

P (1,3) and Q (3,5)

SolutionBefore moving on to solve this question, I need to makeyou understand that the locus of point which isequidistant from two fix point two is the perpendicular

bisector of the fixed two points.

Let the bisector of PQ = R

Mid-point of PQ is1 3 3 5

, (2, 4)2 2 R+ +

= = 2 1

12 1

5 3 2of 1

3 1 2 y y

m slope PQ x x

= = = = = Perpendicular bisector to PQ is RX For two lines to be perpendicular 1 2 1m m =

2

11

1m

= = Using point R (2,4)Using one point slope form equation of a line

1 2 1( ) ( )

Take as (2,4)

4 1( 2)

4 2

6

y y m x x

R

y x

y x

y x

=

= = +=

Option C

Question 35Find the distance between the points 1 12 2( , ) and

1 12 2( , )

Solution2 2

2 1 2 1

2 21 1 1 12 2 2 2

2 2 2 21 1 1 12 2 2 2

( ) ( )

( ( ) ( ( )

( ) ( ) 1 1 2

D x x y y

D

D

= + = +

= + + + = + =

Question 36Find the gradient of the line passing through P (1,1) andQ(2,5)Solution

2 1

2 1

5 1 44

2 1 1

y ym

x x

= = = =

Question 37


88 o

(3 x + 20) oT

U

V

W

5cm

4cm

A

B C

6cm 6cm

A B

O

50 o

P (1,3)

Q(3,5)

R

X


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This solution is provided by Mathsvillage.com Mr, Simeon AbiolaTawose(+2348085573332)Find the equation of a line parallel to y = 4 x +2

passing through (2, 3)

Solution

1

1 2

2

1 1

1 1

4 2 ( )

4

For parallel lines4

Using one-point form equation ( , ) (2,3)

( )

3 4( 2)

3 4 8

4 11 0

y x y mx c

m

m mm

x y

y y m x x

y x

y x

y x

= + = +=

===

= = = + =

Question 38If 815cot , where is acute, find sin =Solution

2 2 2

cot

15 8

225 64 289 17

15sin

17

adjopp

AC

AC

opphyp

=

= += + = =

= =

Question 39

If the area of PQR above is 12 3 cm 2, find the valueof q?

Solution12

12

sin

12 3 8 sin 0

312 3 4

26

A rq

q

q

q cm

== 6

=

=

o

Question 40If 3(2 1) , find

dy y x

dx= +

Solution

Using the generalized power rule which state that given

[ ] [ ]1 1( ) then ( ) ( )n ndy y f x n f x f xdx

= = 3

3 1

2 2

(2 1)

3 (2 1) (2 1)

3 (2 1) (2) 6(2 1)

d dx

y x

dy x x

dxdy

x xdx

= +

= + +

= + = +

Question 41

If sin , f ind dydx y x x=Solution

(sin ) sin ( )

cos sin

sin cos

dy dv duu v

dx dx dxdy d d

x x x xdx dx dxdy

x x xdxdy

x x xdx

= +

= +

= +

= +

Question 42At what value of x does the function 23 2 y x x= +Solution

2

2

2

3 2

2 2

At turning point 0

2 2 0

1

To test for min (max) point

2 0

1 is minimium is a point of the function

y x x

dy x

dxdydx

x

x

d ydx

x

= +

= +

=

+ ==

= >

=

Question 43

Evaluate2 3 2

0( ) x x dx+

Solution4 3 22 3 2 16 8 20 2

4 3 4 3 3 30 0( ) ( ) (0 0) 6 x x x x dx + = + = + + = =

Question 44

Find (sin 2) x dx+ Solution

(sin 2) cos 2 x dx x x k + = + +

Question 45Marks 2 3 4 5 6 7 8

No of students 3 1 5 2 4 2 3

From the table above, if the pas mark is 5, how manystudents failed the test?

SolutionStudents that failed the test = 3 +1+5 = 9


15

8

A

B C

R

P

Q

8cm

60 oq


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Solution to Mathematics Utme 2010_0

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