Algebra Chapter 4 Variation
⇔ If the variable x and y are related to each other in such a way that \(\frac{x}{y}\) = k (now zero constant), it is called that x and y are in direct variation and non-zero constant is said to be variation constant, x and y are in Direct variation and it can be written as x y and now zero constant is said to be variation constant.
⇒ If two variables x and y are related to each other in such a way that xy = k (non-zero constant), then it is said x and y are in inverse variation and written as x \(\frac{1}{y}\) and non zero constant is said to be variation constant.
⇒ If a variable in direct variation with the product of two or more variables, the first variable is said to be in joint variation with other variables.
Theorem on joint variation: If three variables x, y, z be such that x ∝ y when z is constant, x ∝ z when y is constant, then x ∝ yz when y and z both vary.
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Algebra Chapter 4 Variation True Or False
Example 1. y ∝ \(\frac{1}{x}\) then \(\frac{y}{x}\) = non zero constant.
Solution: False
Example 2. If x ∝ z and y ∝ z then xy ∝ z.
Solution: True
Class 10 Maths Algebra Chapter 4 Solutions
Example 3. If x ∝ y2 and y = 2a when x = a then y2 = 4ax.
Solution: True
Example 4. If \(\frac{x}{y}\) ∝ z, \(\frac{y}{z}\) ∝ x \(\frac{z}{x}\) ∝ y then xyz = 1.
Solution: False
Example 5. If x ∝ y, y ∝ z then x μ z.
Solution: True
Example 6. If x ∝ \(\frac{1}{y}\), y ∝ \(\frac{1}{z}\) then x ∝ \(\frac{1}{z}\)
Solution: False
Example 7. If A2 ∝ BC, B2 ∝ CA, C2 ∝ AB then the product of three variations constant = 1.
Solution: True
Rational Expressions Class 10 Solutions
Example 8. If x ∝ \(\frac{1}{z}\) when y is constant and x ∝ y when z is constant then x ∝ \(\frac{y}{z}\) then both y and z vary.
Solution: True
Example 9. If x ∝ y then x + y ∝ \(\frac{1}{x-y}\)
Solution: False
Example 10. If x ∝ y then xn ∝ yn.
Solution: True
Example 11. If A2 + B2 ∝ A2 – B2 then A ∝ \(\frac{1}{B}\)
Solution: False
Example 12. If \(\frac{x}{y}\) ∝ x + y and \(\frac{y}{x}\) ∝ x-y then x2 – y = constant.
Solution: True
Class 10 Algebra Chapter 4 Solved Examples
Algebra Chapter 4 Variation Fill In The Blanks
Example 1. If x ∝ \(\frac{1}{y}\) and y ∝ \(\frac{1}{z}\) then x ∝ _____
Solution: z
Example 2. If x ∝ y, xn ∝ ______
Solution: yn
Example 3. If x ∝ y and x ∝ z then (y + z) ∝ _____
Solution: x
Example 4. If \(\frac{x}{y}\) constant, then x and y are in ______ variation.
Solution: Direct
Example 5. If xy = constant then x and y are in _______ variation.
Solution: Inverse
Example 6. If a variable is in direct variation with the product of two or more variables, the first variable is said to be in _______ variation with the other variables.
Solution: Joint
Simplifying Rational Expressions Class 10
Example 7. If V = R\(\frac{T}{P}\) (R = constant) in this relation we can say that V is in _______ variation with T and \(\frac{1}{P}\).
Solution: Joint
Example 8. If \(\frac{x}{y}\) ∝ x + y and \(\frac{y}{x}\) ∝ x -y then x2 – y2 = _______
Solution: Constant
Example 9. If x2 ∝ yz, y2 ∝ zx, z2 ∝ xy then the product of the three non-zero variation constants is _________
Solution: 1
Example 10. If x + y ∝ x- y then x _______
Solution: y
Simplifying Rational Expressions Class 10
Algebra Chapter 4 Variation Short Answer Type Questions
Example 1. If x ∝ y2 and y = 2a when x = a. Find the relation between x and y.
Solution: x = ky2 [x is a non-zero variation constant.]
∴ a = k (2a)2
or, k = \(\frac{1}{4a}\)
x= \(\frac{1}{4a}\)y2
∴ y2 = 4ax.
Example 2. If x ∝ y, y ∝ z and z ∝ x, find the product of three non-zero constants.
Solution: x = k1y, y = k2z, z = k3x [k1, k2, k3 are three non zero constants]
∴ xyz = k1k2k3 xyz
∴ k1k2k3 = 1
∴ Required product is 1.
Example 3. If x ∝ \(\frac{1}{y}\), y ∝ \(\frac{1}{z}\) find if there be any relation direct or inverse variation between x and z.
Solution: \(x=\frac{k_1}{y}, y=\frac{k_2}{z}\)
⇒ [k1, k2 are non-zero constants]
∴ x = \(\frac{k_1}{\frac{k_2}{z}} \quad \text { or, } \quad x=\frac{k_1}{k_2} z\)
∴ x ∝ z [\(\frac{k_1}{k_2}\) constant]
⇒ There is a direct variation.
Example 4. If x ∝ yz and y ∝ zx, show that z is a nonzero constant.
Solution: x = k1yz, y = k2zx [k1k2 are non-zero constant.]
⇒ or, xy = k1k2 yz2x or, 1 = k1k2z2
∴ \(z= \pm \sqrt{\frac{1}{k_1 k_2}}\) = non zero constant.
Class 10 Maths Algebra Important Questions
Example 5. If b ∝ a3 and an increases ratio of 2 : 3, find what ratio b will increase.
Solution: b = ka3 [k is a non zero variation constant]
\(\frac{b_1}{b_2}=\frac{k a_1^3}{k a_2^3}=\left(\frac{2}{3}\right)^3\)
⇒ b1: b2 = 8: 27.
Example 6. If \(\left(a x+\frac{b}{y}\right) \propto\left(c x+\frac{d}{y}\right)\) then show that xy = constant, (where a, b, c, d constant).
Solution: \(\left(a x+\frac{b}{y}\right)\) = k \(\left(c x+\frac{d}{y}\right)\)
⇒ or, x(a-kc) = (kd-b)\(\frac{1}{y}\)
⇒ or, \(x y=\frac{k d-b}{a-k c}=k_1=\text { constant }\)
⇒ when \(k_1=\frac{k d-b}{a-k c}\)
Example 7. If x ∝ y and y ∝ z then show that x + y ∝ z.
Solution: x ∝ y ⇒ x = k1y,
⇒ y ∝ z ⇒ y =k2z [k1 ,k2 are non zero constant]
⇒ Now, \(\frac{x+y}{z}=\frac{k_1 k_2 z+k_2 z}{z}=\frac{\left(k_1 k_2+k_2\right)}{z} \cdot z\)
= (k k k) = constant
∴ x + y ∝ z..
Example 8. If P2– Q2 ∝ PQ then show that (P + Q) ∝ (P- Q)
Solution: P2 + Q2 = 2KPQ [2k is a non-zero variation constant]
⇒ or, \(\frac{\mathrm{P}^2+\mathrm{Q}^2}{2 \mathrm{PQ}}=\mathrm{k}\)
⇒ or, \(\frac{\mathrm{P}^2+\mathrm{Q}^2+2 \mathrm{PQ}}{\mathrm{P}^2+\mathrm{Q}^2-2 \mathrm{PQ}}=\frac{k+1}{k-1}\) [by components dividends]
⇒ or, \(\frac{(\mathrm{P}+\mathrm{Q})^2}{(\mathrm{P}-\mathrm{Q})^2}=\frac{k+1}{k-1}\)
∴ \(\frac{\mathrm{P}+\mathrm{Q}}{\mathrm{P}-\mathrm{Q}}= \pm \sqrt{\frac{k+1}{k-1}}=\text { constant. }\)
⇒ p + Q ∝ P – Q
Example 9. x ∝ y when z constant and x ∝ \(\frac{1}{z}\) when y constant. If y = b when z = c, x = a then find the value of x when y = b2, z = c2.
Solution: x ∝ y (z constant.) x ∝ \(\frac{1}{z}\) (y constant.)
By compound variation, x ∝ \(\frac{y}{z}\) (y, z vary)
∴ x = k\(\frac{y}{z}\) [k is a non zero variation constant.]
⇒ Q = k\(\frac{b}{c}\)
⇒ k = k\(\frac{ac}{b}\)
\(x=\frac{a c}{b} \cdot \frac{y}{z}=\frac{a c}{b} \cdot \frac{b^2 b}{c^2 c}=\frac{a b}{c}\)Class 10 Maths Algebra Important Questions
Example 10. If \(x^3-\frac{1}{y^3} \propto x^3+\frac{1}{y^3}\) then show that x ∝ \(\frac{1}{y}\)
Solution: \(x^3-\frac{1}{y^3}=k\left(x^3+\frac{1}{y^3}\right)\)
⇒ or, \(\frac{x^3-\frac{1}{y^3}}{x^3+\frac{1}{y^3}}=k\)
⇒ or, \(\frac{2 x^3}{-2 \frac{1}{y^3}}=\frac{k+1}{k-1}\)
⇒ or, \(x^3 y^3=\frac{k+1}{1-k}\)
⇒ or, \(x y=\sqrt[3]{\frac{k+1}{1-k}}\) = constant
∴ \(\quad x \propto \frac{1}{y}\)