REMAINDER THEOREM and QUADRATIC THEORY...Pegasys 2008 Higher Mathematics - Unit 2 REMAINDER THEOREM...

Pegasys 2008 Higher Mathematics - Unit 2

REMAINDER THEOREM and QUADRATIC THEORY

1. For the polynomial 432 323 xxxx , its degree is and coefficient of x3 is

2. The remainder when 492 3 xx is divided by x + 3 is

A. 49

B. 31

C. 5

D. 23

3. The quotient when 253 23 xxx 3 is divided by x – 2 is

A. 1973 2 xx

B. 1553 2 xx

C. 1973 2 xx

D. 1553 2 xx

4. When 29 23 xkxx is divided by x + 2 the remainder is 0. The value of k is

A. 18

B. 18

C. 17

D. 0

5. When 11534 xpxx is divided by 3x the remainder is 4 . The value of p is

A. 0

B. 1

C. 4

D. 3

Degree Coefficient

A 4 1

B 1 4

C 4 1

D 1 1


6. If x – 1 is a factor of axax 2223 , the value(s) of a is/are

A. 1

B. 1 and 1

C. 1

D. undefined

7. The remainder when 1 – x3 is divided by x – ½ is

. A. 8

7

B. 2

1

C. 8

9

D. 2

3

8. The solution to the inequality 0822 xx is

A. 2x and 4x

B. 4x and 2x

C. 24 x

D. 42 x

9. 249 xx expressed in the form 2)( qxp equals

A. 2)2(5 x

B. 2)2(13 x

C. 2)2(13 x

D. 2)2(5 x

10. 542 xx expressed in the form bax 2)( equals

A. 5)2( 2 x

B. 9)2( 2 x

C. 20)4( 2 x

D. 11)4( 2 x


11. The solution of the inequality 1572 2 xx is

A. 2

3x and 5x

B. 52

3 x

C. 5x and 2

3x

D. 2

35 x

12. When qxxx 93 23 is divided by 2x the remainder is 5. The value of q is

A. 3

B. 17

C. 27

D. 9

13. Here are 4 terms used to describe the roots of a quadratic equation

I real

II unequal

III equal

IV non-real

Which of them describe(s) the roots of 0543 2 xx ?

A. I and II

B. I and III

C. IV

D. some other combination


I real

II unequal

III equal

IV non-real


A. I and II

B. I and III

C. IV



15. The equation xpx 102 has equal roots. The value(s) of p is

A. 25

B. 25

C. 25

D. undefined

16. The equation 33 2 pxx has equal roots. The value(s) of p is/are

A. 6

B. 6

C. 6

D. undefined

17. The equation 043 2 kxx has real roots. The range of values of k is

A. 3

4k

B. 3

4k

C. 3

4k

D. 3

4k

18. The range of values of k for which the equation 014 2 kxx has non-real roots is

A. 4x and 4x

B. 4x and 4x

C. 44 x

D. 44 x

19. The remainder when 34511 xx is divided by x + 4 is

A. -33

B. 247

C. -73

D. -225


20. The remainder when 53 kxx is divided by x – 2 is 3. The value of k is

A. 5

B. 3

C. 5

D. 3

21. Given that 2x is a root of the equation 01144 23 pxxx , the value of p is

A. 6

B. 26

C. 6

D. 26

22. The equation 0842 xax has equal roots. The value of a is

A. 2

1

B. 2

1

C. 2

D. 2

23. The remainder when 244664 23 xxx is divided by x – 3 is

A. 0

B. 30

C. 30

D. 276

24. How many real roots does the equation 0)7)(2( 2 xxx have?

A. none

B. 1

C. 2

D. 3

25. The remainder when 532 23 xx is divided by x – 4 is

A. 39

B. 15

C. 203

D. 75



I real

II unequal

III equal

IV non-real


A. I and II

B. I and III

C. IV


27. When 13 24 kxx is divided by x – 2 the remainder is 7. The value of k is

A. 9

B. 3

C. 510

D. 14

28. The solution to the inequality 0)3)(2( xx is

A. 2x and 3x

B. 32 x

C. 23 x

D. 3x and 2x

29. The remainder when 468 3 xx is divided by 12 x is

A. 2

B. 9

C. 6

D. 2

30. When 1043 xx is divided by (x – 2) the quotient and remainder are

A. xx 22 remainder 10

B. xx 22 remainder 10

C. xx 22 remainder 6

D. xx 62 remainder 22


INTEGRATION

1. 124)(' 3 xxxf ; )(xf equals

A. Cxxx 24

B. Cx 212 2

C. Cx 22 2

D. Cxx 24

2. xxdx

dy 23 , an expression for y given that y = 3 and x = 2 is

A. 23

2

1xx

B. 32

1 23 xx

C. 32

1 23 xx

D. 5222

1 23 xx

3. dxx 2)32( equals

A. xxx 963

4 23

B. Cxxx 963

4 23

C. Cx 128

D. 3

3

4x

4. Given dx

dy= 12 x and y = 5 when x = 2, an expression for y in terms of x is

A. 12 x

B. 12 xx

C. 282 xx

D. xx 2


5. dxx equals

A. 16

B. 8

C. 0

D. 1

6. dxx2 can be illustrated by which of these graphs?

A. B.

C. D.

7. The shaded area in the diagram equals

A. 1

B. 5

C. 3

7

D. 3

-4

4

-1

2

x

y

0 -1 2 x

y

0 -2 1

x

y

0 x

y

0

-1 2 -2 1

x

y

0 1 2

y = x2


8.

The shaded area is represented by

A. dxxgxf )()(

B. dxxgxf )()(

C. dxxgxf )()(

D. dxxgxf )()(

9. dxxx )2( equals

A. 3

1

B. 0

C. 6

D. 6

10. dxxx 48 2 equals

A. Cxx )13

4(2 2

B. Cx )14(4

C. Cxx )1(4 2

D. Cx 4

2

1

-3

4

-3

4

-3

4

-3

4

-1

2

x

y

0 -3 4

y = f(x)

y = g(x)


11. x

1 dx equals

A. Cx

2

1

B. Cx 2

C. C

x

33

2

D. C

x

33

2

12. The curve for which xxdx

dy29 2 passes through the point )2,1( . y expressed in terms of x

is

A. Cxxy 233

B. 93 23 xxy

C. 23 23 xxy

D. 233 xxy

13. ttdt

ds23 2 . When s = 4 and t = 1 an expression for s in terms of t is

A. 26 t

B. 4723 tt

C. 23 tt

D. 423 tt


14.

The shaded area in the diagram can be found from

A. dxxgxf )()(

B. dxxgxf )()(

C. dxxgxf )()(

D. dxxgxf )()(

15. Given that a > 1 and ,24)32( dxx the value of a is

A. 4

B. 7

C. 4

D. 7

16. dxx3 2 equals

A. Cx 33

3

1

B. Cx 5

5

2

C. Cx 3 5

5

3

D. Cx

33

2

-3

5

-3

5

-2

3

-2

3

1

a

x

y

0

(-3, -2)

(5, 3)

y = g(x)

y = f(x)


17. The area given by 3x dx can be represented by which of the following graphs?

A. B.

C. D.

18. dxx 21

equals

A. 3

16

B. 4

1

C. 3

14

D. 2

1

19. The value of dxx is

A. 1

B. 42

C. 8

1

D. 18

1

-2

2

-2

2 x

y

o

-2

2

x

y

o

-2 2

x

y

o

-2

2 x

y

o

0

4

1

16


20.

The shaded area in this diagram can be represented by

A. dxxfxg )]()([

B. dxxfxg )]()([

C. dxxfxg )]()([

D. dxxfxg )]()([

21. An expression for the area enclosed by the parabola with equation )4)(3( xxy and the x-

axis is

A. dxxx )12( 2

B. dxxx )12( 2

C. dxxx )12( 2

D. dxxx )12( 2

-b

a

-b

a

b

a

b

a

-4

3

-4

3

-3

4

-3

4

x

y

0 -b a

y = g(x)

y = f(x)


22. The graph of )(xfy crosses the x-axis at (6, 0). Given that )(,23)(' xfxxf

A. 4222

3 2 xx

B. 622

3 2 xx

C. xx 22

3 2

D. 4222

3 2 xx

23. 53)13( dxx , where a > 1. The value of the constant a is

A. 3

2

B. 4

C. 2

D. 3

24. The shaded area in the diagram

can be represented by

A. dxx )4( 2

B. 4 dx

C. dxx )4( 2

D. dxx2

1

a

x

y

0

y = x2 - 2

y =2

-1

1

-2

0

-2

2

-2

2


25. dxx23

2 equals

A. Cx

39

2

B. Cx

3

2

C. Cx

6

2

D. Cx

33

4


TRIGONOMETRY

1. The solution of the equation 2

3)16sin( x ; 3600 x is

A. 44o and 224

o

B. 44o and 104

o

C. 14o and 134

o

D. 224o and 316

o

2. The solution of the equation xxx sincossin2 for x0 is

A.

,3

,0

B. 3

C.

,6

,0

D. 6

3. The value of 22 )cos3(sin)cossin3( xxxx is

A. 0

B. 4

C. 10

D. 9

4. Given that a is an acute angle and that 5

1tan a , the exact value of sin2a is

A. 13

1

B. 12

5

C. 13

5

D. 12

1


5. When )sin()2cos()cos()2sin( xyyxxyyx is simplified it equals

A. )2cos( yx

B. )2sin( yx

C. y3cos

D. y3sin

6. The exact value of 75sin is

A. 22

13

B. 22

31

C. 22

31

D. 2

1

7. The solution set of the equation 2

32sin x for 3600 x is

A. 30o and 60

o

B. 60o and 120

o

C. 240o and 300

o

D. 120o and 150

o

8. A is an acute angle with 2

1tan A . The value of A2cos is

A. 5

3

B. 5

3

C. 3

1

D. 6

1


9. When xxxx 5sin7sin5cos7cos is simplified, the answer is

A. x2cos

B. x12cos

C. x2sin

D. x12sin

10. When xx 5sin5cos 22 is simplified, the answer is

A. 0

B. x10cos

C. x10sin

D. 1

11. xx sin2sin written in factorised form is

A. x3sin

B. x3

C. )1cos2(sin xx

D. )1cos2(sin xx

12. The solution of 0sin2sin xx ; 3600 x is

A. 60o, 180

o, 300

o

B. 120o, 180

o, 240

o

C. 30o, 180

o, 330

o

D. 150o, 180

o, 210

o

13. When )60sin()60sin( xx is simplified the answer is

A. xx sincos3

B. 0

C. xsin

D. xcos3

14. The value of 22 )cos2sin5()cos5sin2( yyyy is

A. 0

B. 7

C. 21

D. 29



1)50cos( x ; 3600 x are

A. {200o, 260

o}

B. {20o, 80

o}

C. {170o, 290

o}

D. {110o, 300

o}

16. The solution set of the equation 1sin4 2 x ; 3600 x is

A. {30o, 150

o}

B. {60o, 120

o}

C. {30o, 150

o, 210

o, 330

o}

D. {60o, 120

o, 240

o, 330

o}


330sincos30cossin xx ; 1800 x

A. {0o, 150

o}

B. {30o, 150

o}

C. {60o, 150

o}

D. {90o, 150

o}

18. If 5

4cos x , 900 x , the exact value of x2sin is

A. 25

12

B. 25

24

C. 25

418

D. 25

414


19. If the exact value of 5

1sin x , 900 x , the exact value of x2cos is

A. 5

2

B. 5

3

C. 5

4

D. 5

7

20. The exact value of 105cos is

A. 22

13

B. 22

31

C. 22

31

D. 2

1


THE CIRCLE

1. The point )2,( h lies on the circle with equation 1322 yx . h equals

A. 3

B. 3

C. 17

D. 17

2. P )6,4( and Q(12, 14) are diametrically opposite points on the circumference of a circle. Which

line of the table correctly identifies its centre and radius?

Centre Radius

A. (8, 4) 116

B. )10,4( 232

C. (8, 4) 232

D. )10,4( 116

3. When the circle with equation 9)2()1( 22 yx is reflected in the x-axis its equation

becomes

A. 9)2()1( 22 yx

B. 9)2()1( 22 yx

C. 9)2()1( 22 yx

D. 9)2()1( 22 yx

4. P )4,3( is a point on the circle 2522 yx . The gradient of the tangent to the circle at P is

A. 3

4

B. 3

4

C. 4

3

D. 4

3


5. AB is a diameter of the circle centre ).4,2( If A ),1,0( the coordinates of B are

A. )4,4(

B. )8,5(

C. )2,2(

D. )7,4(

6. A )2,3( and B(5, 4) are opposite ends of a diameter of a circle. Which line of the table

correctly identifies its centre and radius?

Centre Radius

A. (1, 1) 25

B. (4, 3) 5

C. (1, 1) 5

D. (4, 3) 25

7. The circle with equation 1622 yx is moved 4 units right parallel to the x-axis and 2 units

down parallel to the y-axis. The equation of the image is

A. 16)2()4( 22 yx

B. 16)2()4( 22 yx

C. 16)2()4( 22 yx

D. 16)2()4( 22 yx

8. The coordinates of the centre and length of the radius of the circle with equation

034222 yxyx are

Centre Radius

A. )2,1( 22

B. )2,1( 2

C. )2,1( 22

D. )2,1( 2


9. The equation of the circle with centre )1,3( and radius 5 is

A. 25)1()3( 22 yx

B. 5)1()3( 22 yx

C. 25)1()3( 22 yx

D. 5)1()3( 22 yx

10. The gradient of the tangent to the circle 1322 yx at the point )3,2( is

A. 2

3

B. 3

2

C. 2

3

D. 3

2

11. The centre and radius of the circle with equation 034622 yxyx are

Centre Radius

A. )2,3( 4

B. )2,3( 10

C. )2,3( 4

D. )2,3( 10

12. The circle 0107322 yxyx cuts the y-axis at P and Q. The midpoint of PQ is

A. (0, 35)

B. (0, 15)

C. ( )53,0

D. ( )51,0


13. The equation of the circle 044222 yxyx when it has been reflected in the x - axis is

A. 044222 yxyx

B. 044222 yxyx

C. 044222 yxyx

D. 044222 yxyx

14. A circle has equation 01912633 22 yxyx . Its centre is

A. )6,3(

B. )6,3(

C. )2,1(

D. )2,1(

15. The gradient of the tangent to the circle 17)1()2( 22 yx at the point (3, 5) is

A. 6

5

B. 4

1

C. 4

D. 5

6

16. P )1,2( and Q(8, 3) are opposite ends of a diameter of a circle. The centre and radius of it are

Centre Radius

A. )2,5( 26

B. )1,5( 26

C. )1,5( 13

D. )2,5( 13


17. A circle with centre (3, 4) passes through the point )1,2( . Its equation is

A. 10)4()3( 22 yx

B. 34)4()3( 22 yx

C. 34)4()3( 22 yx

D. 10)4()3( 22 yx

18. The straight line y = x cuts the circle 0242622 yxyx at the points R and Q.

The x-coordinate of the midpoint of RQ is

A. 2

B. 2

C. 4

D. 4

19. A circle has equation 36222 yxyx . Its centre and radius are

Centre Radius

A. )6,2( 37

B. )6,2( 43

C. )3,1( 13

D. )3,1( 7

20. The gradient of the tangent to the circle 36222 yxyx at the point (3, 2) is

A. 2

5

B. 2

5

C. 5

2

D. 5

2


21. The range of values- of k for which 04622 kyxyx is a circle is

A. 13k

B. 13k

C. 13k

D. 13k

22. When the circle with equation 042422 yxyx is reflected in the y - axis, is equation is

A. 042422 yxyx

B. 042422 yxyx

C. 042422 yxyx

D. 042422 yxyx

23. The point )2,2( lies on the circle with equation 13)4()5( 22 xx . The point which is

diametrically opposite A has coordinates

A. (4, 5)

B. )0,1(

C. )6,8(

D. )1,0(

24. A circle which is congruent to 0204222 yxyx touches it externally at A(5, 5). The

centre of this second circle is

A. (1, 2)

B. (8, 9)

C. )1,3(

D. (9, 8)

25. The gradient of the tangent to the circle 154222 yxyx at the point (3, 4) is

A. 2

B. 2

1

C. 2

1

D. 2


MIXED EXERCISE - UNIT 2

1. The exact value of 6

7sin

is

A. 2

3

B. 2

3

C. 2

1

D. 2

1

2. 121042 23 xxx in its fully factorised form is

A. )1)(2)(3(2 xxx

B. )1)(2)(3(2 xxx

C. )1)(42)(3( xxx

D. )22)(2)(3( xxx

3. The gradient of the tangent to the circle 08622 yxyx at the point )1,7( is

A. 4

3

B. 3

4

C. 3

4

D. 4

3

4. The centre and radius of the circle with equation 06622 xyx are

A. )3,3( and 15

B. (3, 0) and 15

C. (3, 0) and 3

D. )3,3( and 3


5. The shaded region in the diagram is

A. 3

28

B. 3

19

C. 4

D. 8

6. A curve for which xxdx

dy1215 2 passes through the point )1,1( . When y is expressed in

terms of x, y equals

A. 1230 xy

B. 23 65 xxy

C. 165 23 xxy

D. 1030 xy

7. A circle passes through the origin and the points (6, 0) and (0, 8 ). Its equation

A. 25)4()3( 22 yx

B. 5)4()3( 22 yx

C. 5)4()3( 22 yx

D. 25)4()3( 22 yx

8. When 522 xx is expressed in the form bax 2)( the values of a and b are

A. 1a and 4b

B. 1a and 4b

C. 1a and 4b

D. 1a and 4b

x

y

1 3

y = x2


9. The quotient and remainder when 1043 xx is divided by )2( x is

A. xx 22 rem 22

B. xx 22 rem 10

C. xx 22 rem 10

D. xx 22 rem 6


I real

II unequal

III equal

IV non-real


A. I and II

B. I and III

C. IV


11. The value of dxx )1( 2 is

A. 24

B. 3

113

C. 3

210

D. 3

111

12. dxx

xx 43

equals

A. 24

24

xx

B. Cx 43

C. xx

43

3

D. Cxx

43

3

-1

3


13. The solutions of the equation xx 2sinsin for 20 x

A.

2,3

5,,

3,0

B. 6

11,

2

3,

2,

6

C.

2,3

4,,

3

2,0

D. 3

4,

2

3,

3

2,

2


I real

II unequal

III equal

IV non-real

Which of them describe(s) the roots of 3632 xx ?

A. I and II

B. I and III

C. IV


15. is an acute angle such that 5

3cos .The value of 2cos is

A. 25

16

B. 25

43

C. 25

11

D. 25

7

16. 10136 23 xxx in fully factorised form is

A. )1)(5)(2( xxx

B. )54)(2( 2 xxx

C. )1)(5)(2( xxx

D. )3)(2)(2( xxx


17. dxx

x 1 equals

A. Cxx 21

23

23

2

B. Cxx

21

21

C. Cxx

23

21

2

1

2

1

D. Cxx 21

23

23

2

18. The roots of the equation 0102352 23 xxx are

A. 5,2

1,2

B. 5,2

1,2

C. 5,2

1,2

D. 5,2

1,2

19. The exact value of 75cos is

A. 22

1

B. 22

31

C. 22

13

D. 22

13


20.

The area between the curves 2xy and 24 xxy can be represented by the integral

A. dxxx )2(2

B. dxxx )2(2

C. dxxx )2(2

D. dxxx )2(2

x

y

0

y = 4x – x2

y = x2

0

2

-2

0

0

4

0

1

REMAINDER THEOREM and QUADRATIC THEORY...Pegasys 2008 Higher Mathematics - Unit 2 REMAINDER THEOREM...

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Transcript of REMAINDER THEOREM and QUADRATIC THEORY...Pegasys 2008 Higher Mathematics - Unit 2 REMAINDER THEOREM...