topikal matematik tambahan-pengamiran

7/25/2019 Topikal Matematik Tambahan-Pengamiran

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1449/1 2015 Maths Catch Network © www.maths-catch.com [Lihat halaman sebelah]

SULIT

USAHA +DOA+TAWAKALMATHS Catch

PENGAMIRAN (INTEGRATION)1

Find ∫ −4

5 dx.

Cari ∫ −4

5 dx.

[1 mark ][1 markah]

Answer: Jawapan:

∫ −4

5 dx = −

4

5 x + c

2 Find ∫ −12 dx.Cari ∫ −12 dx.

[1 mark ][1 markah]

Answer: Jawapan:

∫ −12 dx = −12 x + c

3 Find ∫ x7 dx.Cari ∫ x7 dx.

[2 marks][2 markah]

Answer: Jawapan:

∫ x7 dx

= x7 + 1

7 + 1 c

=1

8 x8 + c

4 Find ∫ −

17

8 x4 dx.

Cari ∫ −17

8 x4 dx.

[2 marks]

[2 markah]Answer:

Jawapan:

∫ −17

8 x4 dx

= −17

8 ( x4 + 1

4 + 1) + c

= −17

40 x5 + c

5 Find ∫ (

19

10 x4 +

3

13 x11) dx.

Cari ∫ ( 19

10 x4 +

3

13 x11 ) dx.

[2 marks][2 markah]

Answer:

Jawapan:

∫ (19

10 x4 +

3

13 x11) dx

=19

10 ( x4 + 1

4 + 1) +

3

13 (

x11 + 1

11 + 1) + c

=19

50 x5 +

1

52 x12 + c

6 Find (−6 x − 2) dx.

Cari ∫ ( −6x − 2)2 dx.

[2 marks][2 markah]

Answer: Jawapan:

∫ (−6 x − 2)2 dx = ∫ (36 x2 + 24 x + 4) dx = 12 x3 + 12 x2 + 4 x + c

7 Find ∫ (2 x3

+ 6 x2

) dx.

Cari ∫ (2x

1

3

+ 6x

5

2

) dx.

[2 marks][2 markah]

Answer: Jawapan:

∫ (2 x

1

3

+ 6 x

5

2

) dx

= 2(3

4 ) x

4

3

+ 6(2

7 ) x

7

2

+ c

=3

2 x

4

3

+12

7 x

7

2

+ c

8 Given the gradient function

dy

dx = 2 x−25, find the equation

of the curve.

Diberi fungsi kecerunandy

dx = 2x−25 , cari persamaan

lengkung itu.

[2 marks][2 markah]Answer:

Jawapan:

dy

dx = 2 x−25

∴ y = ∫ 2 x−25 dx

=2 x−25 + 1

−25 + 1 c

= −1

12 x24 + c



2


SULIT



dy

dx = −

3

16 x18, find the

equation of the curve.


dx = −

3

16 x18 , cari persamaan

lengkung itu.

[2 marks]

[2 markah]Answer:

Jawapan:

dy

dx = −

3

16 x18

∴ y = ∫ −3

16 x18 dx

= −3

16 (

x18 + 1

18 + 1) + c

= −3

304 x19 + c


dy

dx = (

8

7 x−16 +

10

9 x16), find

the equation of the curve.


dx = (

8

7 x−16 +

10

9 x16 ), cari

ersamaan lengkung itu. [2 marks]

[2 markah]Answer:

Jawapan:

dy

dx = (

8

7 x−16 +

10

9 x16)

∴ y = ∫ (8

7 x−16

+10

9 x16

) dx

=8

7 (

x−16 + 1

−16 + 1) +

10

9 (

x16 + 1

16 + 1) + c

= −8

105 x15 +10

153 x17 + c


dy

dx = (3 x + 1)2, find the



dx = (3x + 1)2 , cari persamaan

lengkung itu.

[2 marks]

[2 markah]

Answer: Jawapan:

dy

dx = (3 x + 1)2

∴ y = ∫ (3 x + 1)2 dx

= ∫ (9 x2 + 6 x + 1) dx = 3 x3 + 3 x2 + x + c


dy

dx = (−4 x 4

+ 9 x−

11

), find the


Diberi fungsi kecerunan dydx

= ( −4x

11

4

+ 9x−

8

11

), cari

ersamaan lengkung itu.

[2 marks][2 markah]

Answer:

Jawapan:

dy

dx = (−4 x

11

4

+ 9 x−

8

11

)

∴ y = ∫ (−4 x

11

4

+ 9 x−

8

11

) dx

= −4(4

15 ) x

15

4

+ 9(11

3 ) x

3

11

+ c

= −16

15 x

15

4

+ 33 x

3

11

+ c

13 Given

dy

dx = (−5 x + 6)2, find the equation of the curve y =

( x) if the curve passes through the point (1, 12

1

3 ).

Diberidy

dx = ( −5x + 6)2 , cari persamaan lengkung y = f(x

ika lengkung itu melalui titik (1, 121

3 ).

[2 marks][2 markah]

Answer: Jawapan:

dy

dx = (−5 x + 6)2

y = ∫ (−5 x + 6)2 dx

y = ∫ (25 x2 − 60 x + 36) dx

=25

3 x3 − 30 x2 + 36 x + c

When x = 1, y =37

3

37

3 =

25

3 (1)3 − 30(1)2 + 36(1) + c

c = −2

∴ y =25

3 x3 − 30 x2 + 36 x − 2

14 Given

dy

dx = −3 x−2 − 6 x, find the equation of the curve y =

( x) if the curve passes through the point (1, 3).

Diberidy

dx = −3x−2 − 6x, cari persamaan lengkung y =

(x jika lengkung itu melalui titik (1, 3). [2 marks]

[2 markah]

Answer:

Jawapan:

dy

dx = −3 x−2

− 6 x

y = ∫ (−3 x−2 − 6 x) dx



3


SULIT


= −3( x−2 + 1

−2 + 1) − 6(

x1 + 1

1 + 1) + c

=3

x − 3 x2 + c

When x = 1, y = 3

3 =3

1(1) − 3(1)2 + c

c = 3

y =3 x

− 3 x2 + 3

15 Given that (−3 x + 2) dx = kx + 2 x + c, where k and c are constants, find

Diberi ∫ ( −3x2 + 2) dx = kx3 + 2x + c, dengan keadaan kdan c ialah pemalar, cari

(a) the value of k

nilai k

(b) the value of c if (−3 x + 2) dx = −1 when x = 1.

nilai c jika ∫ ( −3x2 + 2) dx = −1 apabila x = 1.

[3 marks][3 markah]

Answer:

Jawapan:

(a) (−3 x + 2) dx

= − x3 + 2 x + c

∴ k = −1

(b) When x = 1,

∫ (−3 x2 + 2) dx = −1

−1(1)3 + 2(1) + c = −1c = −2

16 The variables x and y are related by the equation y = x−3, where p is a constant.

Pemboleh ubah x dan y dihubungkan oleh persamaan y

= px−3 , dengan keadaan p ialah pemalar.

(a) Convert the equation y = px−3 to linear form.Tukarkan persamaan y = px−3 kepada bentuk linear.

(b) Diagram 1 shows the straight line obtained by

plotting log10 y against log10 x.

Rajah 1 menunjukkan graf garis lurus yangdiperoleh dengan memplot log 10 y melawan log 10 x.

Diagram 1 Rajah 1

Find the value ofCarikan nilai

(i) log10 p.

(ii) q.

[4 marks][4 markah]

Answer: Jawapan:

(a) log10 y = log10 p − 3(log10 x)= −3(log10 x) + log10 p

(b) (i) log10 p = Y −intercept

= 3

(ii) log10 y = −3(2) + 3= −3

q = log10 y = −3

17 Find ∫ (5 x + 2)9

dx.

Cari ∫ (5x + 2)

4

9

dx.

[2 marks][2 markah]

Answer: Jawapan:

∫ (5 x + 2)

4

9

dx

=

(5 x + 2)4

9 + 1

5(4

9 + 1)

+ c

=9

65 (5 x + 2)

13

9

+ c

18 Find ∫

17

(12 x − 18)12 dx.

Cari ∫ 17

(12x − 18)12 dx.

[2 marks][2 markah]

Answer: Jawapan:

Let u = 12 x − 18du

dx = 12

dx =1

12 du

∫ 17(12 x − 18)12 dx

= ∫ 17

(u)12 (1

12 du)

= ∫ 17

12 u−12 du

= −17

132 u−11 + c

= − 17

132u11 + c

= − 17

132(12 x − 18)11 + c



4


SULIT


19 Given that ∫ 4 x3 dx = 20 when x = 2. Find the value of theconstant c.

Diberi ∫ 4x3 dx = 20 apabila x = 2. Cari nilai pemalar c.

[2 marks][2 markah]

Answer:

Jawapan:

∫ 4 x3 dx = x4 + c

When x = 2,

x4 + c = 20(2)4 + c = 20c = 4

20 Given that ∫ (4 x − 5)2 dx =

50

3 when x = 2. Find the value

of the constant c.

Diberi ∫ (4x − 5)2 dx =50

3 apabila x = 2. Cari nilai

emalar c.

[2 marks]

[2 markah]Answer: Jawapan:

∫ (4 x − 5)2 dx =50

3

y = ∫ (4 x − 5)2 dx

y = ∫ (16 x2 − 40 x + 25) dx

=16

3 x3 − 20 x2 + 25 x + c

When x = 2, y =50

3

50

3 =

16

3 (2)3 − 20(2)2 + 25(2) + c

c = 4

21 Given

d

dx (5 x2 − x − 6) = f ( x), find ∫

6

4 f ( x) dx.

Diberid

dx (5x2 − x − 6) = f(x), cari ∫

6

4 f(x) dx.

[2 marks]

[2 markah]Answer:

Jawapan:

d

dx (5 x2 − x − 6) = f ( x) = [5 x2 − x − 6]

6

4

= 5(6)2 − (6) − 6 − [5(4)2 − (4) − 6]= 168 − (70)

= 98

22 Given that ∫

9

7 f ( x) dx = 2 and ∫

9

7 [7 f ( x) + kx] dx = 110, find

the value of k .

Diberi ∫9

7 f(x) dx = 2 dan ∫

9

7 [7f(x) + kx] dx = 110,

carikan nilai k.[2 marks]

[2 markah]

Answer:

Jawapan:

∫9

7 [7 f ( x) + kx] dx = 110

7∫9

7 f ( x) dx + ∫

9

7 kx dx = 110

7(2) + k [

x2

2 ]

9

7 = 110

14 + k (81

2 −

49

2 ) = 110

k = 6

23 Given that ∫

4

3 f ( x) dx = 1, find

Diberi ∫4

3 f(x) dx = 1, carikan

(a)the value of ∫

3

4 f ( x) dx.

niali ∫3

4 f(x) dx.

(b)the value of k if ∫

4

3 [kx − 5 f ( x)] dx =

39

2 .

nilai k jika ∫43

[kx − 5f(x)] dx =392

.

[4 marks][4 markah]

Answer: Jawapan:

(a)∫3

4 f ( x) dx = −∫

4

3 f ( x) dx

= −1

(b)∫4

3 [kx − 5 f ( x)] dx =

39

2

∫4

3 kx dx − 5∫

4

3 f ( x) dx =

39

2

k [ x2

2 ]

4

3 − 5(1) =

39

2

k (8 − 9

2 ) − 5 =

39

2

k = 7

24 Given that ∫

5

7 f ( x) dx = −9, find

Diberi ∫5

7 f(x) dx = −9, carikan

(a)∫7

5

f ( x) dx

(b)∫5

7 [2 + 3 f ( x)] dx

[4 marks][4 markah]

Answer:

Jawapan:

(a)∫7

5 f ( x) dx = −∫

5

7 f ( x) dx

= 9

(b)∫5

7 [2 + 3 f ( x)] dx

= ∫5

7 2 dx + 3∫

5

7 f ( x) dx



5


SULIT


= [2 x]5

7 + 3(−9)

= (10 − 14) − 27

= −31

25 Given that ∫

b

0 f ( x) dx = 1 and ∫

b

0 f ( x + 4) dx = 9, find the

value of b.

Diberi ∫

b

0 f(x) dx = 1 dan ∫

b

0 f(x + 4) dx = 9, carikan nilaib.

[2 marks]


∫b

0 f ( x + 4) dx = 9

∫b

0 f ( x) dx + ∫

b

04 dx = 9

1 + [4 x]b

0 = 9

4(b) = 8

b = 2b = 2

26 Diagram 2 shows the curve y = f ( x) cutting the x-axis at x

= a and x = b. Rajah 2 menunjukkan lengkung y = f(x) yang memotong

aksi-x di x = a dan x = b.

Diagram 2 Rajah 2

Given that the area of the shaded region is 4 unit2, find

the value of ∫b

a 4 f ( x) dx.

Diberi luas rantau berlorek ialah 4 unit 2 , cari nilai ∫b

a

4f(x) dx. [2 marks]

[2 markah]Answer:

Jawapan:

∫b

a f ( x) dx = −4

∫b

a 4 f ( x) dx = 4∫

b

a f ( x) dx

= 4(−4)= −16


= k and passing the point (20, 20).

Rajah 3 menunjukkan lengkung y = f(x) yang memotongaksi-x di x = k dan melalui titik (20, 20).

Diagram 3 Rajah 3


the value of ∫20

k f ( x) dx.

Diberi luas rantau berlorek ialah 280 unit 2 , cari nilai ∫20

k

(x) dx. [2 marks]


20 × 20 − ∫20

k f ( x) dx = 280

∫20

k f ( x) dx = 20 × 20 − 280

= 120

28 Diagram 4 shows the curve y = f ( x) cutting the x-axis at x = 0, x = a and x = b.

Rajah 4 menunjukkan lengkung y = f(x) yang memotong

aksi-x di x = 0, x = a dan x = b.

Diagram 4 Rajah 4

Given that the area of the shaded region P is 5 unit2 andthe area of the shaded region Q is 4 unit.

Find the value of ∫a0 6 f ( x) dx + ∫ba 4 f ( x) dx.

Diberi luas rantau berlorek P ialah 5 unit 2 dan luasrantau berlorek Q ialah 4 unit.

Cari nilai ∫a

0 6f(x) dx + ∫

b

a 4f(x) dx.

[2 marks][2 markah]

Answer: Jawapan:

∫a

0

6 f ( x) dx + ∫b

a

4 f ( x) dx

= 6∫a

0 f ( x) dx + 4∫

b

a f ( x) dx



6


SULIT


= 6(5) + 4(−4)= 14


= b. Rajah 5 menunjukkan lengkung y = f(x) yang memotong

aksi-x di x = b.

Diagram 5 Rajah 5


the value of ∫b

7 9 f ( x) dx.

Diberi luas rantau berlorek ialah 36 unit 2 , cari nilai ∫b

7

9f(x) dx.

[2 marks][2 markah]

Answer:

Jawapan:

∫b

7 f ( x) dx = 36 − 7 × 4

= 8

∫b

7 9 f ( x) dx = 9∫

b

7 f ( x) dx

= 9(8)= 72

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