BANK & INSURANCE (QUADRATIC EQUATION PART 2)

Total Questions: 50

1. Directions: In the question, two equations I and II are given. You have to solve both the equations to establish the correct relation between x and y and choose the correct option.

(a) x < y
(b) x ≤ y
(c) x = y or No relation
(d) x > y
(e) x ≥ y

I. x² = 16x

II. 3y² + 24y - 48 = 4y² + 5y

Correct Answer: (c) x = y or No relation
Solution:

From I:
x² = 16x
x² - 16x = 0
x(x - 16) = 0
x = 0 or x = 16

From II:
3y² + 24y - 48 = 4y² + 5y
y² - 19y + 48 = 0

y² - 16y - 3y + 48 = 0

y(y - 16) - 3(y - 16) = 0
(y - 16)(y - 3) = 0
y = 16 or y = 3

2. I. 5x + 4y = 164   

II. 9x + 2y = 212

Correct Answer: (a) x < y
Solution:

From I:
5x + 4y = 164
y = (164 - 5x)/4

From II:
9x + 2y = 212
9x + 2 × {(164 - 5x)/4} = 212
18x + 164 - 5x = 424
13x = 260
x = 20
And, y = (164 - 5 × 20)/4 = 16

3. I. 2x² - 7x = 4x - 14 

II. 6y³ = 1296

Correct Answer: (b) x ≤ y
Solution:

From I:
2x² - 7x = 4x - 14
2x² - 11x + 14 = 0
2x² - 4x - 7x + 14 = 0
2x(x - 2) - 7(x - 2) = 0
(2x - 7)(x - 2) = 0
x = 7/2 or x = 2

From II:
6y³ = 1296
y³ = 216
y = 6

4. I. x² + 2x = 10x + 48 

II. y² + 9y + 20 = 0

Correct Answer: (d) x > y
Solution:

From I:
x² + 2x = 10x + 48
x² - 8x - 48 = 0
x² - 12x + 4x - 48 = 0
x(x - 12) + 4(x - 12) = 0
(x - 12)(x + 4) = 0
x = -4 or x = 12

From II:
y² + 9y + 20 = 0
y² + 5y + 4y + 20 = 0
y(y + 5) + 4(y + 5) = 0
(y + 5)(y + 4) = 0
y = -5 or y = -4

5. I. y³ = 1331     

II. 15x² + 88 = 14x² + 19x

Correct Answer: (e) x ≥ y
Solution:

From I:
y³ = 1331
y = 11

From II:
15x² + 88 = 14x² + 19x
x² - 19x + 88 = 0
x² - 11x - 8x + 88 = 0
x(x - 11) - 8(x - 11) = 0
(x - 8)(x - 11) = 0
x = 8 or x = 11

6. I. x³ - 80% of 150x = 75% of 272x   

II. 32y - 12 × 24 = 32 × 15

Correct Answer: (b) x ≤ y
Solution:

From I:
x³ - 80% of 150x = 75% of 272x
x³ - 0.80 × 150x = 0.75 × 272x
x³ - 120x = 204x
x³ - 324x = 0
x(x² - 324) = 0
x = 0 or x² = 324
x = 0 or x = ±18

From II:
32y - 12 × 24 = 32 × 15
32y = 480 + 288
32y = 768
y = 24

7. I. 35x² + 120 = 145x 

II. 1875y² = 432

Correct Answer: (a) x < y
Solution:

From I:
35x² + 120 = 145x
7x² + 24 = 29x
7x² - 29x + 24 = 0
7x² - 21x - 8x + 24 = 0
7x(x - 3) - 8(x - 3) = 0
(7x - 8)(x - 3) = 0
x = 3 or x = 8/7

From II:
1875y² = 432
625y² = 144
y² = 144/625
y = ±12/25

8. I. 16x² + 22 × 5 = 22 × 9x - 20 × 5 - 8x²   

II. 5y² + 36y = 32

Correct Answer: (a) x < y
Solution:

From I:
16x² + 22 × 5 = 33x - 20 × 5 - 8x²
24x² - 198x + 110 + 100 = 0
24x² - 198x + 210 = 0
4x² - 33x + 35 = 0
4x² - 28x - 5x + 35 = 0
4x(x - 7) - 5(x - 7) = 0
(4x - 5)(x - 7) = 0
x = 5/4 or x = 7

From II:
5y² + 36y = 32
5y² + 36y - 32 = 0
5y² + 40y - 4y - 32 = 0
5y(y + 8) - 4(y + 8) = 0
(5y - 4)(y + 8) = 0
y = 4/5 or y = -8

9. I. 45x² + 36x = 81  

II. 240y + 432 = 0

Correct Answer: (d) x > y
Solution:

From I:
45x² + 36x = 81
5x² + 4x = 9
5x² + 4x - 9 = 0
5x² - 5x + 9x - 9 = 0
5x(x - 1) + 9(x - 1) = 0
(5x + 9)(x - 1) = 0
x = -9/5 or x = 1

From II:
240y + 432 = 0
5y + 9 = 0
y = -9/5

10. I. 15y³ = 1215y   

II. 8x² + 1152 = 192x

Correct Answer: (a) x < y
Solution:

From I:
15y³ = 1215y
y³ = 81y
y³ - 81y = 0
y(y² - 81) = 0
y = 0 or y² = 81
y = 0 or y = 9 or y = -9

From II:
8x² + 1152 = 192x
8x² - 192x + 1152 = 0
x² - 24x + 144 = 0
x² - 12x - 12x + 144 = 0

x(x - 12) - 12(x - 12) = 0
(x - 12)(x - 12) = 0
x = 12