Relationship Between Two Variables

Here we are going to discuss an important Algebra topic which is nothing but “Relationship between two variables”. Here we will discuss some important formulas which help one to tackle such types of questions. This article is beneficial for SSC, Banking, and for other competitive exam aspirants.

List of important formulas : 

1. (a + b)² = a² + b² + 2ab.

2. (a – b)² = a² + b² – 2ab

3. (a + b)³ = a³ + b³ + 3ab(a + b)

4. (a – b)³ = a³ – b³ – 3ab(a – b)

5. a³ + b³ = (a + b) × (a² + b² – ab)

6. a³ – b³ = (a – b) × (a² + b² + ab) 

7. a² – b² = (a + b)(a – b)

Let’s discuss some questions which help one to understand the uses of the above formulas.

Que 1. If x + y = -3 then find the value of x³ + y³ – 9xy + 17 

1.  -10

2.  -18

3.  -27

4.  -12

Answer : -10

Explanation : 

Here we have x + y = -3

On cubing both sides we get,

 x³ + y³ + 3xy (x + y) = -27

x³ + y³ – 9xy = -27

By adding 17 on both sides we get,

x³ + y³ – 9xy + 17 = -27 + 17 = -10

Hence Option 1 will be the correct choice.

Que 2. If (x + 4y) = 10 and xy = 4 then find the value of x² + 16y² + 24xy

1. 162

2.163

3.164

4.165

Answer : 164

Explanation : 

Here we have x + 4y = 10 and xy = 4.

x + 4y = 10

On squaring both sides we get, 

x² + 16y² + 8xy = 100

x² + 16y² = 100 – 8 × 4

x² + 16y² = 68

x² + 16y² + 24xy = 68 + 24 × 4 = 164

Hence Option 3 will be the correct choice.

Que 3. if x + y = 10 and xy = 5 then find the value of 1/x + 1/y ?

1.  2

2.  3

3.  4

4.  5

Answer : 2

Explanation : 

Here we have x + y = 10 ……….(1)

xy = 5……………(2)

On dividing eq 1 by 2 we get,

⇒ (x + y)/(xy) = 10/5

⇒ x/xy + y/xy = 2

⇒ 1/y+ 1/x = 2.

Hence option 1 will be the correct choice

Que 4. If (x + y)³ + (x – y)³ = Ax³ + Bxy² then find the value of (A + B)

1.  6

2.  7

3.  8

4.  9

Answer : 8

Explanation : 

Here we have (x + y)³ + (x – y)³ = Ax³ + Bxy²

On solving further we get,

⇒ (x + y + x – y) × ( (x + y)² + (x – y)² – (x + y) × (x – y) )

⇒ 2x ( x² + 3y²)

⇒ 2x³ + 6xy² = Ax³ + Bxy²

On comparing further we get,

A = 2 and B = 6

A + B = 8.

Hence option 3 will be the correct choice.

Que 5. If X = 3 + 2√2 and Y = 1/( 3 + 2√2). Find the value of X + Y.

1. 4

2. 5

3. 6 

4. 7

Answer : 6

Explanation : 

X = 3 + 2√2 and Y = 1/(3 + 2√2)

On rationalizing Y we get,

Y = 1/(3 + 2√2) × (3 – 2√2)/(3 – 2√2)

Y = 3 – 2√2

X + Y = 3 + 2√2 + 3 – 2√2

= 6

Hence option 3 will be the correct choice.

Que 6. if x + y = 8 then find the maximum value of xy.

1. 12

2. 15

3. 16

4. 7

Answer : 16

Explanation : 

Here we have, x + y = 8.

to get the maximum value of xy, the value of x and y must be as close as possible.

At x = 4 and y = 4, we get the maximum value of xy without violating any condition.

xy = 4 × 4 = 16

Hence option 3 will be the correct choice.

Que 7. If 3x + 2y = 15 then find the maximum value of x³y².

1. 343

2. 243

3. 443

4. 543

Answer : 243

Explanation :

We know that A.M ≥ G.M

(x + x + x + y + y )/5 ≥ ⁵√x³y²

15/5 ≥ ⁵√x³y²

3⁵ ≥ x³y²

Hence the maximum value will be 243.

Hence option 2 will be the correct choice.

Que 8. If  x + 1/x = 5/x and y + 1/y = 10/y then find the maximum value of x + y

1. 4

2. 5

3. 6

4. 7

Answer : 5

Explanation : 

Here we have, x + 1/x = 5/x

On multiplying by x on both sides we get,

⇒ x² + 1 = 5

⇒ x² = 4

⇒ x = +2 and -2.

y + 1/y = 10/y

On multiplying by ‘y’ on both sides we get,

y² + 1 = 10

y² = 9

y = +3 and -3

The maximum value of x + y = 2 + 3 = 5.

Hence option 2 will be the correct choice.

Que 9. if x + 2y = 8 and xy = 6 then find the value of x³ + 8y³ + x² + 4y² + 26

1. 270

2. 290

3. 310

4. 330

Answer : 290

Explanation : 

Here we have, x + 2y = 8.

On squaring both sides we get,

⇒ x² + 4y² + 4xy = 64

⇒ x² + 4y² = 64 – 24 = 40…………..(1)

x + 2y = 8, On cubing both sides we get,

⇒ x³ + 8y³ + 3 × x × 2y(x + 2y) = 512

⇒ x³ + 8y³ +  36(8) = 512

⇒ x³ + 8y³ = 224……………..(2)

⇒ x³ + 8y³ + x² + 4y² + 26

⇒ 224 + 40 + 26 = 290

Hence option 2 will be the correct choice.

Que 10. If  x + 1/x = 5/x and y + 1/y = 10/y then find the minimum value of x + y

1.   -5

2.   -2

3.   -3

4.   -4

Answer :  -5

Explanation : 

Here we have, x + 1/x = 5/x

On multiplying by x on both sides we get,

⇒ x² + 1 = 5

⇒ x² = 4

⇒ x = +2 and -2.

y + 1/y = 10/y

On multiplying by y on both sides we get,

y² + 1 = 10

y² = 9

y = +3 and -3

The minimum value of x + y = -2 – 3 = -5

Hence option 1 will be the correct choice.