CHAPTER 5: VARIATIONS IMPORTANT CONCEPTS Direct Variations Using symbol ` ∝ ’,
Statement
y varies directly as x
y
P varies directly as Q 2
P
M is directly proportional to
M
∝
∝
x
Equation with k as constant of variation y = kx
Q2
P = k Q2
N
∝
M = k N
N
Inverse Variations Using symbol ` ∝ ’,
Statement
y varies inversely as x P varies inversely as Q
1
y
∝
P
∝
M
∝
x 1
2
M is inversely proportional to N
Q
P=
2
1
N
Joint Variations Using symbol ` ∝ ’,
Statement
w varies directly as x and y
w
P varies directly as q and r 2
P
s varies directly t and inversely as u
s
∝
∝
R varies inversely as M and N
•
• •
k N
q r 2
P = k q r 2
t
∝
s =
u e
d ∝
2
d =
f 1
R ∝
R = R =
kt u ke 2 f
k M N
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.
A) DIRECT VARIATION
Variations
Q2
xy
M N 4 steps to solve problems involving variations: •
M=
k
Equation with k as constant of variation w = k xy
2
d varies directly as e and inversely as f
Equation with k as constant of variation k y = x
57
EXAMPLE 1: Given that P varies directly as Q and P =
1
when Q = 2.
3
Express P in terms of Q and find the value of P when Q = 18.
P ∝ Q P = k Q
•
STEP 1 : Symbol STEP 2 : Equation -
•
STEP 3 : Find the value of k,
•
, k = constant
1 3 1 3
= k(2) = 2k
k = THEN, substitute k = HENCE,
•
1 6
1 in the equation P = k Q. 6 1 P = Q 6
STEP 4:Find the value of P when Q = 18,
P =
1 6
P = 3 Exercises:
1.
Given that y varies directly as x and y = 15 when x = 5. x. a) Express y in terms of x b) Find the value of y y when x = 4.
2.
Given that P ∝ Q 2 and P = 10 when Q = 2. a) Express P in terms of Q. P when Q = 4. b) Find the value of P
3. Given th that y varies directly as x and y = 6 when x = 9. c) Express y in terms of x x. y when x = 25. d) Find the value of y Variations
58
× 18
4. It is is give given n that that m varies varies direc directly tly as n 3 and m = 32 when n = 2. Express m in terms of n and find the value of m when n =
1 . 2
EXAMPLE 2 : P varies directly as x 2 where x = 5 + y. Given that P = 72 when y = 1. x. a) Express P in terms of x b) Find the values of y when P = 200. STEP 1 : Symbol P ∝ x 2 • P = k x 2 STEP 2 : Equation , 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 72 k= 6 k=2 THEN, substitute k = 2 in the equation P = k x 2 HENCE, P = 2 x 2 •
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 and y = 10 – 5 y = -10 – 5 and y = 5 y = -15
Exercises: 1.
Variations
M varies directly as x 2 where x = 3 + y. Given that M = 64 when y = 1. x. a) Express M in terms of x 59
y when M = 400. b) Find the values of y
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
2.
B) INVERSE VARIATION EXAMPLE 1: Given that y varies inversely as x and y = 6 when x = 4. Express y in terms of x x and find the value of y when x = 3. 1 •
STEP 1 : Symbol -
y
•
STEP 2 : Equation -
y =
•
STEP 3 : Find the value of k,
6=
∝
x k
, k = constant
x k 4
k = 24 THEN, substitute k = 24 in the equation y =
y =
HENCE,
•
k x 24
.
x
STEP 4:Find the value of y when x = 3, y =
24
y = 8 Exercises:
1. Given that y varies inversely as x and x = 4 when y = 3. x. a. Express y in terms of x b. Find the value of y y when x = 6. Variations
60
3
2. Given that P varies inversely as x and P = 6 when x = P when x = 3 find the value of P
1 3
. Express P in terms of x x and
1 . 3
3. Given that y varies inversely as x 2 and y = 2 when x = 4. a) Express y in terms of x x y when x = 2. b) Find the value of y
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.
5. Given that P varies inversely as P Q
Variations
6 9
Q
. Complete the following table. 2
4
61
6. The table below shows some values of the variables M and N such that N varies inversely as the square root of M .
M N
4 6
36 2
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. x and y, a) Express y in terms of x b) Find the value of y when x = 5 and z = 2. • • •
STEP 1 : Symbol STEP 2 : Equation STEP 3 : Find the value of k,
y ∝ xz y = k xz 12 = k (2)(3) 12 = 6k k=2
, k = constant
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
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. Variations
62
2. Given that y ∝ mn and y = 20 when m = 2 and n = 5. a) Express y in terms of m and n y when m = 3 and n = 4. b) Find the value of y
EXAMPLE 2: ( INVERSE VARIATION & INVERSE VARIATION) Given that y varies inversely as x and z . y = 10 when x = 2 and z = 4. x and z , then find the value of y y when x = 4 and z = 5. Express y in terms of x 1 STEP 1 : Symbol y ∝ • xz xz k STEP 2 : Equation , k = constant y = • xz xz k STEP 3 : Find the value of k, 10 = • 2( 4) k = 80
THEN, substitute k = 80 in the equation y =
y =
HENCE,
•
k xz xz
.
80 xz xz
STEP 4:Find the value of y when x = 4 and z = 5, y =
y =
80 20
80 4 ( 5) , y = 4.
Exercises: 1. Given that m
1 ∝
x
y
x and y. Find and m = 3 when x = 3 and y = 16. Express m in terms of x
the value of y y when m = 9 and x = 12.
Variations
63
2. Given that y
1 ∝
2
d e
and 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 x and v b) Find the value of y y when x = 2 and v = 15. x STEP 1 : Symbol y ∝ • v kx y = STEP 2 : Equation , k = constant • v k 4 STEP 3 : Find the value of k, 10 = • 5 50 = 4k 25 k= 2 25 kx THEN, substitute k = in the equation y =. 2 v 25 x HENCE, y = 2v 25 ( 2) STEP 4:Find the value of y when x = 2 and v = 15, y = • 2(15 )
y =
50 , y = 30
5 3
.
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 N and P . b) Calculate the value of M when N = 5 and P = 2.
Variations
64
2. Given that w
∝
x y
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 = 6 when G = 3 and H = 2, express F in terms of G and H . Find the value of G when F = 27 and H = 4.
4. Given that S ∝
P M
3
and S = 6 when P = 3 and M = 2. Calculate the value of P P when S =
2 and M = -4.
5. Given that p varies directly as x and inversely as the square root of y y. If p p = 8 when x = 6 and y = 9. a. express p in terms of x x and y, b. calculate the value of y y when p = 6 and x = 9.
Variations
65
EXAMPLE 4: The table shows some values of the variables d , e and f .
d e f Given that d ∝
12 9 4
10 25 m
e f
a) express d in terms of e and f b) b) calcu calcula late te the the val value ue of of m. e
•
STEP 1 : Symbol -
d ∝
•
STEP 2 : Equation -
d=
•
STEP 3 : Find the value of k, substitute d = 12, e = 9, and f = 4 k 9 12 = 4 48 = 3k k = 16 THEN, substitute k = 16 in the equation
d =.
HENCE,
d =
•
f k e
, k = constant
f
k e f
16 e f
STEP 4:Find the value of m (f) when d = 10 and e = 25,
10 =
16
25 m
10m = 16 × 5, 10m = 80 m = 10 Exercises: 1. The table shows the relation relation between three variables D, D, E and F. D 2
E 3 6
1 2 Given that D
∝
F 4 m
1
EF a) expr expres esss D in in term termss of E and and F b) b) calcu calcula late te the the val value ue of of m. Variations
66
2. The table table shows shows some values values of of the variabl variables es w, x and y, y, such that that w varies varies directly directly as as the square root of x and inversely as y. w 18 3
x 9 K
y 2 36
a) find find the the equati equation on conn connect ecting ing w x and and y. b) calculate the value of K.
Objective Questions
1. The table table shows shows relation between the variables, variables, x and y. y. If y ∝ x , find the value
A.
10 7
X
2
7
Y
5
M
B.
14 5
C. 8
D.
2. Variations
67
35 2
of m.
m n
2 9
X 25
The table shows the relation between the variables, m and n. If m varies inversely as the square root of n, then x = A.
18
B.
25
3. It is given that y
∝
v2
w when y = 25 and w = 3.
A. 5
6
C.
5
28
D.
3
B. 6
15
C. 12.5
B. y = 15x
x
9
, and y = 6 when v = 4 and w = 8, calculate the positive value of v
D. 25
4. Given that y varies inversely inversely as x and y = 5 when x = A. y =
50
C. y =
5 3 x
1 3
, express y in terms of x.
D. y =
3 5 x
5. Given that y varies varies directly as x, x, and y = 6 when when x = 2, express express y in terms of x. A. y =
1
B. y = 3x
3 x
C. y = 12x
6. Given that P varies inversely as A. P =
2 3
12 x
, and P = 2 when Q = 9, express P in terms of Q.
Q
B. P = 6
Q
D. y =
C. P =
Q
6 Q
D. P =
18 Q
7. The relation between the variables, P, x and y is represented by P ∝ x m y n . If P varies directly 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 varies directly directly as x n . If x is the radius and y is the height of the cyclinder whose volume is a constant, then the value of n is A. -2
B. 1
C. 2
D. 3
9. The table shows shows the corresponding corresponding values of d and e. The relation relation between the the variables d and e is represented by
Variations
d
1
4
9
16
e
30
15
1608
7.5
A.e ∝ d
B. e
∝
1
C. e
d
∝
1
D. e
d
1
∝
d
2
10. T varies directly as the square root of p and inversely as the square root of g. This joint variation can be written as A. T
B. T
pg
∝
p 2 g
C. T
p
∝
D. T
g
∝
p g
2
11. It is given that m varies inversely 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
12. It is given that G
A. G =
∝
C. 4
1 H
2
9 H
H = 5M -1. If G =3 when M = 2, express G in terms of H.
B. G =
2
D. 5
27 H
C. G =
2
243 H
D. G =
2
243 H
13. d
6
8
e
3
X
f
2
5
The table shows the relation relation between the three variables variables d, e and f. If d
∝
e f
, calculate the
value of x A. 6
14. It is given that y
Variations
B. 10
∝
C. 12
D. 13
x n . If y varies inversely as the square root of x, then the value of n is
69
A.
1
B.
2
−1
C. -1
2
D. – 2
15. F 6 12
G 9 m
H 2 3
The table shows the relation between the three variables, F,G and H. If F varies directly as the square root of G and inversely as H, then the value of m is A. 3
B. 9
C. 81
D. 144
16. S P M
2 3 4
3 1 X
The table shows the relation relation between the variables, variables, S, P and M. If S
∝
1 P M
, calculate
the value of x A. 4
B. 8
17. Given that p A. p =
∝
x
C. 16
2
y
and p = 6 when x = 2 and y = 3, express p in terms of x and y
9 x 2
B. p =
2 y
18. Given that y varies directly as x A. y = x 2
D. 64
2
2 x
2
C. p =
9 y
x 2
D. p =
9 y
9 x 2 y
and that y = 80 when x = 4, express y in terms of x C. y = 5x 2
B. y = 5x
D. y =
1280 x
2
19. w x y
2 12 9
3 24 m
The table shows the relation between the three variables, w, x and y. If w
x ∝
the value of m A. 2
Variations
B. 4
C. 8
70
D. 16
y
, calculate
20. If M varies directly as the square square root of N, the relation between M and N is is
A. M
∝
N
B. M
∝
N
2
C. M
∝
N
1
1
D. M
2
∝
1
N 2
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
B. p = 4n 2
A. p = n 2
C. p = 9n 2
D. p = 12n 2
2. SPM 2003(Nov)
w x y
2 8 4
3 18 n
The table shows some values of the variables, w, x and y which satisfy the relationship w x y
∝
, calculate the value of n
A. 6
B. 9
C. 12
D. 36
3. SPM 2004(Nov) P varies directly as the square root of Q. The relation between P and Q is 1
A. P ∝ Q 2
1
B. P ∝ Q
2
C. P ∝
1
D. P ∝
Q2
1 Q2
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 m , calculate the value of w r A. 16
B. 24
C. 36
5. SPM 2004(Jun) Variations
71
D. 81
p
∝
3
The relation between p, n and r is p
n
. It is given that p = 4 when n = 8 and r = 6. r Calculate the value of p 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 =
3 w
B.
12
C. p = 3w
w
D. p = 12w
7. SPM 2005(Nov) The table shows some values of the variables x and y such that y varies inversely as the square root of x x.
x y
4 6
16 3
Find the relation between y and x. 12
A. y = 3
B.
x
C.
x
3 8
x
2
D.
96
x 2
8. SPM 2005(Nov) x and y = 15 when x = 9. Calculate the It is given that y varies directly as the the square root of x x when y = 30. value of x
A. 5 B. 18 C. 25 D. 36 9. SPM 2005(Nov) The table shows some values of the variables w, x and y such that w varies directly as the square 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
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. Variations
72
D. 15
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 = 3 and R = 4. Express P in terms of R R and Q. 2 4Q 32 Q 3 R 3 R A. P = B. C. D. P = 2 Q 8Q 3 R 3 R 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 x.
A. y = 7 x
B. y =
x
C. y =
7
1 63 x
D. y =
63 x
13. SPM 2006(Jun) Table 2 shows two sets of values of Y , V and W .
Y
V
W
3
3
12
5
18
5
m
It is given that Y varies directly with the square of V and inversely with W . Find the the value of m. A.
5
4 9
B.
3
C.
10
D.
9
6 25 14. SPM 2007(Nov) Table 1 shows some values of of the variables 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 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 relation between P and M. 1
A.
P = kM
2
B. P =
k 1
C. P
M 2
16. SPM 2007(Nov)
Variations
73
=
kM 2
D. P =
k M
2
y 5 The relation between between the variables variables x, y and z is x∞ . It is given that x = when y = 2 z 4 5
and z = 8. 8. Calculate Calculate the the value value of z when x = A. 2
3
and y = 6.
B. 18
C. 32
D. 72
17. SPM 2007(Jun)
It is given that P varies inversely with Q and P =
2 1 when Q = . Find the relation 5 2
between P and Q. A. P
4 =
5
Q
B. P
1 =
5
Q
C. P =
4
D. P =
5Q
1 5Q
18. SPM 2007(Jun) G
Table shows some values of the variables F , G and H that satisfy F α
F
G
H
20
2
3
108
6
p
2
H
.
Calculate the value of p p. A. 5
B. 9
C. 10
D. 18
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 y. A. 24
B. 27
C. 48
D. 64
20. SPM 2008(Nov)
Given y varies inversely as x 1 16
A.
3
, and that y = 4 when x = ½ . Calculate the value of x when y =
. 1 8
Variations
B. ½
C. 2
74
D. 8
21. SPM 2008(Nov)
It is given that P varies directly directly as the square root of Q and inversely inversely as the square of R. Find the relation between between P, Q and R. A. P α
Q2
B. P α α
R 22. SPM 2008 (Jun)
Q
C. P α
R 2
R 2
α D. P α
Q
R Q
2
It is given that p varies directly directly as the square root of w and that p = 5 when w = 4 . Express p in terms of w. A. p
5 =
16
w
2
B. p
80 =
w
C. p
2
5 =
2
D. p
w
10 =
w
23. SPM 2008(Jun)
Table shows some values of the variables m and n, such that m varies inversely inversely as the cube of n
m
1 2
n
2
x 3
x. Calculate the value of x A.
4
B.
27
4
C.
9
9
D.
16
ANSWERS Chapter 21 Variations A) DIRE DIRECT CT VARI VARIAT ATIO ION N Example 1: No. 1. a) a ) y = 3x b) y = 12
No. 2. a) P =
5
2 b) P = 40
No. 3. a) y = 2x b) y = 50 Q
2
No. 4. m =
1 2
n3 , m =
Example 2: No. No. 1. a) M = 4x 2 b) y = 7 and y = -13 B) INVE INVERS RSE E VARIA VARIATI TION ON Example 1: No. 1. a) y =
12
x
20
No. 4. a) S =
b) y = 2
Variations
b) S = 4
75
r
1 16
27 16
2
No. 2. a) P =
12
3
b) P = No. 3. a) y =
No. 5. P = 9 , Q = 81
x
N0. 6. k = 12, N =
5
M
16
x 2
b) y = 4 C) JOIN JOINT T VARI VARIAT ATIO ION N Example 1: No. 1. a) k = 3, p = 3qr
No. No. 2. a) k = 2, y = mn b) y = 24 Example 2: No. 1. k = 36,
m=
No. 2. k = 72,
y=
Example 3: No. 1. a) k = 8,
M=
36 x 72
de
,
y 2
y=9
,
e=
3 2
8 N
P
b) M = 20 No. 2. k = 20, No. 3. k =
4 3
w= ,
No. 4. k = 16,
F=
S=
20 x , y 4G
2
3 H
,
16 P 3
M
No. 5. a) k = 4,
w=2 G=
,
p=
P = -8 4 x y
b) y = 36 Example 4: No. 1. a) k = 24,
D=
24 EF
b) m = 16 No. 2. a) k = 12,
w=
12
x y
b) K = 81
Objective Questions.
1 .D 2.B 3.A Variations
6. C 7. A 8. A
11.A 1 2. C 1 3. B
16. C 17. A 18. C 76
±9
4.C 5.B
9. C 10. C
1 4. B 1 5. C
19. D 20. C
PAST YEAR QUESTIONS
1 .B 2 .B 3 .A 4 .B 5 .A 6 .B 7 .B
Variations
8 .D 9. B 10.B 11.A 12.A 13.C 14. A
1 5. B 1 6. B 17.D 18. A 19. C 20. C 2 1. B
22.C 23. A
77
