Hani Math Variations.pdf

7/31/2019 Hani Math Variations.pdf

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PracticesMakePerfectMathematics_SPM2012hanifharun@SMKTTDI

Variations 313

CHAPTER 5: VARIATIONS

IMPORTANT CONCEPTS

Direct VariationsStatement Using symbol ` , Equation with k as constant of

variation y varies directly as x y x y = kx

P varies directly as Q 2 P Q2 P = k Q2

M is directly proportional to

N M N M = k N

Inverse VariationsStatement Using symbol ` , Equation with k as constant of

variation y varies inversely as x

y x

1y =

x

k

P varies inversely as Q 2P 2

1

QP = 2Q

k

M is inversely proportional to

N M

N

1M =

N

k

Joint VariationsStatement Using symbol ` , Equation with k as constant of

variationw varies directly as x and y w xy w = k xy

P varies directly as q and r 2 P q r 2 P = k q r 2

s varies directly t andinversely as u s

u

t s =

u

kt

d varies directly as e2 and

inversely as f d f

e 2 d =

f

ke2

R varies inversely as M and N

R N M

1 R = N M

k

4 steps to solve problems involving variations: STEP 1 : Change the statement using the symbol . STEP 2 : Write down the equation connecting the variables using k as the constant of

variation. STEP 3 : Find the value of k . STEP 4 : Find the value of variable required.


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Variations 314

A) DIRECT VARIATION

EXAMPLE 1:

Given that P varies directly as Q and P =3

1when Q = 2.

Express P in terms of Q and find the value of P when Q = 18. STEP 1 : Symbol - P Q STEP 2 : Equation - P = k Q , k = constant

STEP 3 : Find the value of k,3

1= k(2)

3

1= 2k

k =6

1

THEN, substitute k = 6

1in the equation P = k Q.

HENCE, P =6

1 Q

STEP 4:Find the value of P when Q = 18, P =6

118

P = 3

Exercises:

1. Given that y varies directly as x and y = 15 when x = 5.a) Express y in terms of x.b) Find the value of y when x = 4.

2. Given that P Q 2 and P = 10 when Q = 2.a) Express P in terms of Q.b) Find the value of P when Q = 4.


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Variations 315

3. Given that y varies directly as x and y = 6 when x = 9.c) Express y in terms of x.d) Find the value of y when x = 25.

4. It is given that m varies directly as n 3 and m = 32 when n = 2. Express m in terms of

n and find the value of m when n =2

1.

EXAMPLE 2 :P varies directly as x 2 where x = 5 + y. Given that P = 72 when y = 1.a) Express P in terms of x. b) Find the values of y when P = 200. STEP 1 : Symbol - P x 2 STEP 2 : Equation - P = k x 2 , k = constant STEP 3 : Find the value of k, substitute and x = 5 + y in the above equation:

P = k(5 + y) 2 Then substitute P=72 and y = 1, 72 = k(5 + 1) 2

72 = 36k

k =6

72

k = 2THEN, substitute k = 2 in the equation P = k x 2

HENCE, P = 2 x2

STEP 4:Find the value of y when P = 200 by substitute x = 5 + y 200 = 2(5 + y) 2 100 = (5 + y) 2

100 = 5 + y 10 = 5 + y y = 10 5 and y = -10 5 y = 5 and y = -15


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Variations 316

Exercises:

1. M varies directly as x 2 where x = 3 + y. Given that M = 64 when y = 1.a) Express M in terms of x.b) Find the values of y when M = 400.

2. Given that T S 2 and S = 2w 3, and T = 12 when W = 1.a)Express T in terms of S b)Find the value of T when W = 5

B) INVERSE VARIATION

EXAMPLE 1:Given that y varies inversely as x and y = 6 when x = 4.Express y in terms of x and find the value of y when x = 3.

STEP 1 : Symbol - y x

1

STEP 2 : Equation - y = xk

, k = constant

STEP 3 : Find the value of k, 6 =4

k

k = 24

THEN, substitute k = 24 in the equation y = xk .

HENCE, y = x

24

STEP 4:Find the value of y when x = 3, y =3

24

y = 8


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Variations 317

Exercises:

1. Given that y varies inversely as x and x = 4 when y = 3.a. Express y in terms of x.b. Find the value of y when x = 6.

2. Given that P varies inversely as x and P = 6 when x =3

1. Express P in terms of x and

find the value of P when x =3

13 .

3. Given that y varies inversely as x 2 and y = 2 when x = 4.a) Express y in terms of x b) Find the value of y when x = 2.

4. Given that S varies inversely as square root of r and S = 5 when r = 16.a) Express S in terms of r .b) Find the value of S when r = 25.


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Variations 318

5. Given that P varies inversely as Q . Complete the following table.

6. The table below shows some values of the variables M and N such that N varies inverselyas the square root of M .

Find the relation between M and N .

C) JOINT VARIATION

EXAMPLE 1: (DIRECT VARIATION & DIRECT VARIATION) y varies directly as x and z. Given that y = 12 when x = 2 and z = 3.a) Express y in terms of x and y,b) Find the value of y when x = 5 and z = 2.

STEP 1 : Symbol - y xz STEP 2 : Equation - y = k xz , k = constant STEP 3 : Find the value of k, 12 = k (2)(3)

12 = 6k

k = 2

THEN, substitute k = 2 in the equation y = k xz.HENCE, y = 2 xz

STEP 4:Find the value of y when x = 5 and z = 2, y = 2(5)(2) y = 20

P 6 2Q 9 4

M 4 36 N 6 2


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Variations 319

Exercises:

1. p varies directly as q and r . Given that p = 36 when q = 4 and r = 3.a) Express p in terms of q and r ,b) Find the value of p when q = 3 and r = 3.

2. Given that y mn and y = 20 when m = 2 and n = 5.a) Express y in terms of m and n b) Find the value of y when m = 3 and n = 4.

EXAMPLE 2: ( INVERSE VARIATION & INVERSE VARIATION)

Given that y varies inversely as x and z. y = 10 when x = 2 and z = 4.Express y in terms of x and z, then find the value of y when x = 4 and z = 5.

STEP 1 : Symbol - y xz

1

STEP 2 : Equation - y = xzk

, k = constant

STEP 3 : Find the value of k, 10 =)4(2

k

k = 80

THEN, substitute k = 80 in the equation y = xzk

.

HENCE, y = xz

80

STEP 4:Find the value of y when x = 4 and z = 5, y =)5(4

80

y =20

80, y = 4.


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Variations 320

Exercises:

1. Given that m y x

1and m = 3 when x = 3 and y = 16. Express m in terms of x and y. Find

the value of y when m = 9 and x = 12.

2. Given that y ed 2

1and y = 2 when d = 3 and e = 4, calculate the value of e when y = 3 and

d = 4.

EXAMPLE 3 (DIRECT VARIATION + INVERSE VARIATION)1. Given that y varies directly as x and varies inversely as v and y = 10 when x = 4 and v=5.

a) Express y in terms of x and v b) Find the value of y when x = 2 and v = 15.

STEP 1 : Symbol - y v x

STEP 2 : Equation - y =vkx

, k = constant

STEP 3 : Find the value of k, 10 =5

4k

50 = 4k

k =2

25

THEN, substitute k =2

25 in the equation y =.vkx

HENCE, y =v x

2

25

STEP 4:Find the value of y when x = 2 and v = 15, y =)15(2)2(25

y =30

50, y =

3

5.


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Variations 321

Exercises:1. M varies directly as N and varies inversely as P and m = 6 when N = 3 and P = 4.

a) Express M in terms of N and P .b) Calculate the value of M when N = 5 and P = 2.

2. Given that w y x

and w = 10 when x = 8 and y = 16. Calculate the value of w when

x= 18 and y = 36.

3. F varies directly as G to the power of two and varies inversely as H . Given that F = 6when G = 3 and H = 2, express F in terms of G and H . Find the value of G when F = 27and H = 4.

4. Given that S 3 M P

and S = 6 when P = 3 and M = 2. Calculate the value of P when S =

2 and M = -4.


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Variations 322

5. Given that p varies directly as x and inversely as the square root of y. If p = 8 when x = 6and y = 9.

a. express p in terms of x and y,b. calculate the value of y when p = 6 and x = 9.

EXAMPLE 4:The table shows some values of the variables d , e and f .

d 12 10e 9 25 f 4 m

Given that d f e

a) express d in terms of e and f b) calculate the value of m.

STEP 1 : Symbol - d f e

STEP 2 : Equation - d = f

ek , k = constant

STEP 3 : Find the value of k, substitute d = 12, e = 9, and f = 4

12 =4

9k

48 = 3k k = 16

THEN, substitute k = 16 in the equation d =. f

ek

HENCE, d = f

e16

STEP 4:Find the value of m (f) when d = 10 and e = 25, 10 =m

2516

10m = 16 5,10m = 80

m = 10


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Variations 325

7. The relation between the variables, P, x and y is represented by P x m y n . If P variesdirectly as the square of x and inversely as the cube of y, then m + n =

A. -1 B. 1 C. 5 D. -6

8. Given that y varies directly as x n . If x is the radius and y is the height of the cyclinder whosevolume is a constant, then the value of n is

A. -2 B. 1 C. 2 D. 3

9. The table shows the corresponding values of d and e. The relation between the variables d ande is represented by

A.e d B. e d

1C. e

d

1D. e 2d

1

10. T varies directly as the square root of p and inversely as the square root of g. This jointvariation can be written as

A. T pg B. Tg

p 2C. T

g p

D. T 2g

p

11. It is given that m varies inversely as s and t. If m = 3 when s = 2 and t = 4, find the value of m when s = 2 and t = 6.

A. 2 B. 3 C. 4 D. 5

12. It is given that G 2 H

1H = 5M -1. If G =3 when M = 2, express G in terms of H.

A. G = 2 H

9B. G = 2

H

27 C. G = 2

H

243D. G =

H

243

d 1 4 9 16

e 30 15 10 7.5


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Variations 326

13.d 6 8

e 3 X

f 2 5

The table shows the relation between the three variables d, e and f. If d f e , calculate the

value of x

A. 6 B. 10 C. 12 D. 13

14. It is given that y x n . If y varies inversely as the square root of x, then the value of n is

A.21

B.21

C. -1 D. 2

15.F G H6 9 2

12 m 3

The table shows the relation between the three variables, F,G and H. If F varies directly as thesquare root of G and inversely as H, then the value of m is

A. 3 B. 9 C. 81 D. 144

16.S 2 3P 3 1M 4 X

The table shows the relation between the variables, S, P and M. If S M P

1, calculate

the value of x

A. 4 B. 8 C. 16 D. 64

17. Given that p y

x2and p = 6 when x = 2 and y = 3, express p in terms of x and y

A. p = y2

x9 2B. p =

y9 x2 2

C. p = y9

x2D. p =

y x9 2


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Variations 327

18. Given that y varies directly as x 2 and that y = 80 when x = 4, express y in terms of x

A. y = x 2 B. y = 5x C. y = 5x 2 D. y = 2 x

1280

19. w 2 3x 12 24y 9 m

The table shows the relation between the three variables, w, x and y. If w y

x, calculate

the value of m

A. 2 B. 4 C. 8 D. 16

20. If M varies directly as the square root of N, the relation between M and N is

A. M N B. M N 2 C. M N 21

D. M21

N

1

PAST YEAR QUESTIONS

1. SPM 2003(Nov)Given that p is directly proportional to n 2 and p = 36, express p in terms of n

A. p = n 2 B. p = 4n 2 C. p = 9n 2 D. p = 12 n 2

2. SPM 2003(Nov)

w 2 3 x 8 18 y 4 n

The table shows some values of the variables, w, x and y which satisfy the relationship w

y

x, calculate the value of n

A. 6 B. 9 C. 12 D. 36


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Variations 328

3. SPM 2004(Nov)P varies directly as the square root of Q . The relation between P and Q is

A. P 21

Q B. P 2Q C. P 2

1

1

Q

D. P 21

Q

4.

SPM 2004(Nov)

P M r

3 8 4

6 w 9

The table shows the relation between the three variables p, m and r . Given that p

r

m, calculate the value of w

A. 16 B. 24 C. 36 D. 81

5. SPM 2004(Jun)

The relation between p, n and r is p r n3

. It is given that p = 4 when n = 8 and r = 6.

Calculate the value of p when n = 64 and r = 3

A. 16 B. 24 C. 32 D. 48

6. SPM 2004(Jun)

It is given that p varies inversely with w and p = 6 when w = 2. Express p in terms of w.

A. p =w

3B.

w

12C. p = 3w D. p = 12 w

7. SPM 2005(Nov)The table shows some values of the variables x and y such that y varies inversely as thesquare root of x.

x 4 16 y 6 3

Find the relation between y and x.

A. y = 3 x B. x

12C. 2

8

3 x D. 2

96

x

8. SPM 2005(Nov)It is given that y varies directly as the square root of x and y = 15 when x = 9. Calculate thevalue of x when y = 30.

A. 5 B. 18 C. 25 D. 36


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Variations 329

9. SPM 2005(Nov)The table shows some values of the variables w, x and y such that w varies directly as thesquare of x and inversely as y.

W x y

40 4 2

M 6 4

Calculate the value of m.A. 90 B. 45 C. 30 D. 15

10. SPM 2005(Jun) Table shows values of the variables x and y.

x 3 m

y 5 15

It is given x varies directly with y. Calculate the value of m. A. 6 B. 9 C. 12 D. 15

11. SPM 2005(Jun)

P varies directly with the square of R and inversely with Q. It is given that P = 2 when Q = 3and R = 4. Express P in terms of R and Q.

A. P =Q

R8

3 2B. 23

32

RQ

C.Q R3

D. P = R

Q

3

4

12. SPM 2006(Jun)It is given that y varies inversely with x and y = 21 when x = 3. Express y in terms of x.

A. y = 7 x B. y =7

xC. y =

x631

D. y = x

63

13. SPM 2006(Jun)Table 2 shows two sets of values of Y , V and W .

Y V W

5

3

3 12

m 5 18

It is given that Y varies directly with the square of V and inversely with W . Find the value of m.

A.3

5B.

9

4C.

9

10D.

25

6


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Variations 330

14. SPM 2007(Nov) Table 1 shows some values of the variables x and y.

x 2 n

y 4 32

It is given that y varies directly as the cube of x. Calculate the value of n.A. 4 B. 8 C. 16 D. 30

15. SPM 2007(Nov)P varies inversely as the square root of M. Given that the constant is k, find the relationbetween P and M.

A. 21

kM P B.2

1

M

k P C. 2kM P D. 2 M

k P

16. SPM 2007(Nov)

The relation between the variables x, y and z is z y x . It is given that x =

45 when y = 2

and z = 8. Calculate the value of z when x =3

5and y = 6.

A. 2 B. 18 C. 32 D. 72

17. SPM 2007(Jun)

It is given that P varies inversely with Q and P =5

2when Q =

2

1. Find the relation between

P and Q.

A. QP5

4B. QP

5

1C.

QP

5

4D.

QP

5

1

18. SPM 2007(Jun)

Table shows some values of the variables F , G and H that satisfy F H G 2

.

F G H

20 2 3

108 6 p

Calculate the value of p.

A. 5 B. 9 C. 10 D. 18


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Variations 331

19. SPM 2008(Nov)

Table shows some values of the variables R and T. It is given that R varies directly as T .

R 54 72

T 36 y

Find the value of y.

A. 24 B. 27 C. 48 D. 64

20. SPM 2008(Nov)

Given y varies inversely as x 3 , and that y = 4 when x = . Calculate the value of x when y =

16

1.

A. 8

1B. C. 2 D. 8

21. SPM 2008(Nov)

It is given that P varies directly as the square root of Q and inversely as the square of R. Find therelation between P, Q and R.

A. R

QP

2

B. 2 R

QP C.

Q

RP

2

D. 2Q

RP

22. SPM 2008 (Jun)

It is given that p varies directly as the square root of w and that p = 5 when w = 4 . Express p interms of w.

A.2

16

5w p B. 2

80

w p C. w p

2

5D.

w p

10

23. SPM 2008(Jun)

Table shows some values of the variables m and n, such that m varies inversely as the cube of n

m

21

x

n 2 3

Calculate the value of x.

A.27

4B.

9

4C.

16

9D.

16

27


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Variations 332

ANSWERS

Chapter 21 Variations

A) DIRECT VARIATION

Example 1 : No. 1. a) y = 3x No. 3. a) y = 2xb) y = 12 b) y = 50

No. 2. a) P = 22

5Q No. 4. m = 3

2

1n , m =

16

1

b) P = 40

Example 2 : No. 1. a) M = 4x 2 b) y = 7 and y = -13

B) INVERSE VARIATION

Example 1 : No. 1. a) y = x12 No. 4. a) S =

r 20

b) y = 2 b) S = 4

No. 2. a) P = x

2No. 5. P = 9 , Q = 81

b) P =5

3N0. 6. k = 12, N =

M

12

No. 3. a) y = 216

x

b) y = 4

C) JOINT VARIATION

Example 1 : No. 1. a) k = 3, p = 3qr

No. 2. a) k = 2, y = mnb) y = 24

Example 2 : No. 1. k = 36, m = y x

36, y = 9

No. 2. k = 72, y = 272de, e =

23

Example 3 : No. 1. a) k = 8, M =P N 8

b) M = 20

No. 2. k = 20, w = y

x20, w = 2


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No. 3. k =3

4, F =

H G

34 2

, G = 9

No. 4. k = 16, S = 316

M P

, P = -8

No. 5. a) k = 4, p = y

x4

b) y = 36

Example 4 : No. 1. a) k = 24, D = EF

24

b) m = 16

No. 2. a) k = 12, w = y

x12

b) K = 81

Objective Questions.

1.D 6. C 11.A 16. C2.B 7. A 12.C 17. A3.A 8. A 13.B 18. C4.C 9. C 14.B 19. D5.B 10. C 15.C 20. C

PAST YEAR QUESTIONS

1.B 8.D 15.B 22.C2.B 9.B 16.B 23. A3.A 10.B 17.D4.B 11.A 18. A5.A 12.A 19. C6.B 13.C 20. C7.B 14. A 21.B

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Hani Math Variations.pdf