Directions (1-25): In the following questions, two equations numbered I and II are given. You have to solve both the equations and give answer.
निर्देश (1-25): निम्नलिखित प्रश्नों में, दो समीकरण I और II दिए गए हैं। आपको दोनों समीकरणों को हल करना है और उत्तर देना है।
(A) if x > y (यदि x > y)
(B) if x < y (यदि x < y)
(C) if x ≥ y (यदि x ≥ y)
(D) if x ≤ y (यदि x ≤ y)
(E) if x = y or the relationship cannot be established (यदि x = y या संबंध स्थापित नहीं किया जा सकता है)
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I. x² – 7x + 12 = 0
II. y² – 9y + 20 = 0Detailed Solution / विस्तृत समाधान
Equation I: x² – 7x + 12 = 0
x² – 4x – 3x + 12 = 0
x(x – 4) – 3(x – 4) = 0
(x – 3)(x – 4) = 0
Roots of x (x के मूल): 3, 4
Equation II: y² – 9y + 20 = 0
y² – 5y – 4y + 20 = 0
y(y – 5) – 4(y – 5) = 0
(y – 4)(y – 5) = 0
Roots of y (y के मूल): 4, 5
Comparison / तुलना:
x y Relation / संबंध 3 4 x < y 3 5 x < y 4 4 x = y 4 5 x < y Here, x is always less than or equal to y (यहाँ, x हमेशा y से छोटा या उसके बराबर है).
Final Answer (अंतिम उत्तर): (D) x ≤ y
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I. 2x² + 11x + 14 = 0
II. 4y² + 12y + 9 = 0Detailed Solution / विस्तृत समाधान
Equation I: 2x² + 11x + 14 = 0
2x² + 7x + 4x + 14 = 0
x(2x + 7) + 2(2x + 7) = 0
(x + 2)(2x + 7) = 0
Roots of x (x के मूल): -2, -3.5
Equation II: 4y² + 12y + 9 = 0
This is (2y + 3)² = 0
2y + 3 = 0
Roots of y (y के मूल): -1.5, -1.5
Comparison / तुलना:
x y Relation / संबंध -2 -1.5 x < y -3.5 -1.5 x < y In all cases, x is less than y (सभी मामलों में, x, y से छोटा है).
Final Answer (अंतिम उत्तर): (B) x < y
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I. x² – 32 = 112
II. y – √169 = 0Detailed Solution / विस्तृत समाधान
Equation I: x² – 32 = 112
x² = 112 + 32
x² = 144
x = ±√144
Roots of x (x के मूल): +12, -12
Equation II: y – √169 = 0
y – 13 = 0
Root of y (y का मूल): 13
Comparison / तुलना:
x y Relation / संबंध 12 13 x < y -12 13 x < y In both cases, x is less than y (दोनों मामलों में, x, y से छोटा है).
Final Answer (अंतिम उत्तर): (B) x < y
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I. x² + x – 12 = 0
II. y² + 2y – 15 = 0Detailed Solution / विस्तृत समाधान
Equation I: x² + x – 12 = 0
x² + 4x – 3x – 12 = 0
x(x + 4) – 3(x + 4) = 0
(x – 3)(x + 4) = 0
Roots of x (x के मूल): 3, -4
Equation II: y² + 2y – 15 = 0
y² + 5y – 3y – 15 = 0
y(y + 5) – 3(y + 5) = 0
(y – 3)(y + 5) = 0
Roots of y (y के मूल): 3, -5
Comparison / तुलना:
x y Relation / संबंध 3 3 x = y 3 -5 x > y -4 3 x < y -4 -5 x > y Since the relationship between x and y changes (कभी x>y, कभी x
Final Answer (अंतिम उत्तर): (E) relationship cannot be established
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I. 3x² – 10x + 8 = 0
II. 2y² – 19y + 35 = 0Detailed Solution / विस्तृत समाधान
Equation I: 3x² – 10x + 8 = 0
3x² – 6x – 4x + 8 = 0
3x(x – 2) – 4(x – 2) = 0
(3x – 4)(x – 2) = 0
Roots of x (x के मूल): 2, 4/3 (or 1.33)
Equation II: 2y² – 19y + 35 = 0
2y² – 14y – 5y + 35 = 0
2y(y – 7) – 5(y – 7) = 0
(2y – 5)(y – 7) = 0
Roots of y (y के मूल): 7, 5/2 (or 2.5)
Comparison / तुलना:
x y Relation / संबंध 2 7 x < y 2 2.5 x < y 1.33 7 x < y 1.33 2.5 x < y In all cases, x is less than y (सभी मामलों में, x, y से छोटा है).
Final Answer (अंतिम उत्तर): (B) x < y
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I. x² = 81
II. y² – 18y + 81 = 0Detailed Solution / विस्तृत समाधान
Equation I: x² = 81
x = ±√81
Roots of x (x के मूल): 9, -9
Equation II: y² – 18y + 81 = 0
This is (y – 9)² = 0
y – 9 = 0
Root of y (y का मूल): 9
Comparison / तुलना:
x y Relation / संबंध 9 9 x = y -9 9 x < y Here, x is either less than or equal to y (यहाँ, x, y से छोटा या उसके बराबर है).
Final Answer (अंतिम उत्तर): (D) x ≤ y
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I. 4x² + 20x + 25 = 0
II. 2y² + 7y + 6 = 0Detailed Solution / विस्तृत समाधान
Equation I: 4x² + 20x + 25 = 0
This is (2x + 5)² = 0
2x + 5 = 0
Root of x (x का मूल): -2.5
Equation II: 2y² + 7y + 6 = 0
2y² + 4y + 3y + 6 = 0
2y(y + 2) + 3(y + 2) = 0
(2y + 3)(y + 2) = 0
Roots of y (y के मूल): -1.5, -2
Comparison / तुलना:
x y Relation / संबंध -2.5 -1.5 x < y -2.5 -2 x < y In both cases, x is less than y (दोनों मामलों में, x, y से छोटा है).
Final Answer (अंतिम उत्तर): (B) x < y
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I. x² – 20x + 91 = 0
II. y² – 32y + 247 = 0Detailed Solution / विस्तृत समाधान
Equation I: x² – 20x + 91 = 0
x² – 13x – 7x + 91 = 0
x(x – 13) – 7(x – 13) = 0
(x – 7)(x – 13) = 0
Roots of x (x के मूल): 7, 13
Equation II: y² – 32y + 247 = 0
y² – 19y – 13y + 247 = 0
y(y – 19) – 13(y – 19) = 0
(y – 13)(y – 19) = 0
Roots of y (y के मूल): 13, 19
Comparison / तुलना:
x y Relation / संबंध 7 13 x < y 7 19 x < y 13 13 x = y 13 19 x < y Here, x is always less than or equal to y.
Final Answer (अंतिम उत्तर): (D) x ≤ y
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I. x² = 6x – 9
II. 2y² + 13y + 21 = 0Detailed Solution / विस्तृत समाधान
Equation I: x² = 6x – 9
x² – 6x + 9 = 0
(x – 3)² = 0
Root of x (x का मूल): 3
Equation II: 2y² + 13y + 21 = 0
2y² + 7y + 6y + 21 = 0
y(2y + 7) + 3(2y + 7) = 0
(y + 3)(2y + 7) = 0
Roots of y (y के मूल): -3, -3.5
Comparison / तुलना:
x y Relation / संबंध 3 -3 x > y 3 -3.5 x > y In both cases, x is greater than y.
Final Answer (अंतिम उत्तर): (A) x > y
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I. 5x² + 29x + 20 = 0
II. 25y² + 25y + 6 = 0Detailed Solution / विस्तृत समाधान
Equation I: 5x² + 29x + 20 = 0
5x² + 25x + 4x + 20 = 0
5x(x + 5) + 4(x + 5) = 0
(5x + 4)(x + 5) = 0
Roots of x (x के मूल): -5, -4/5 (or -0.8)
Equation II: 25y² + 25y + 6 = 0
25y² + 15y + 10y + 6 = 0
5y(5y + 3) + 2(5y + 3) = 0
(5y + 2)(5y + 3) = 0
Roots of y (y के मूल): -2/5 (or -0.4), -3/5 (or -0.6)
Comparison / तुलना:
x y Relation / संबंध -5 -0.4 x < y -5 -0.6 x < y -0.8 -0.4 x < y -0.8 -0.6 x < y In all cases, x is less than y.
Final Answer (अंतिम उत्तर): (B) x < y
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I. x² – 1 = 0
II. y² + 4y + 3 = 0Detailed Solution / विस्तृत समाधान
Equation I: x² – 1 = 0
x² = 1
Roots of x (x के मूल): 1, -1
Equation II: y² + 4y + 3 = 0
y² + 3y + y + 3 = 0
y(y + 3) + 1(y + 3) = 0
(y + 1)(y + 3) = 0
Roots of y (y के मूल): -1, -3
Comparison / तुलना:
x y Relation / संबंध 1 -1 x > y 1 -3 x > y -1 -1 x = y -1 -3 x > y Here, x is always greater than or equal to y.
Final Answer (अंतिम उत्तर): (C) x ≥ y
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I. 2x² – 7x + 3 = 0
II. y² – 7y + 12 = 0Detailed Solution / विस्तृत समाधान
Equation I: 2x² – 7x + 3 = 0
2x² – 6x – x + 3 = 0
2x(x – 3) – 1(x – 3) = 0
(2x – 1)(x – 3) = 0
Roots of x (x के मूल): 3, 0.5
Equation II: y² – 7y + 12 = 0
y² – 4y – 3y + 12 = 0
y(y – 4) – 3(y – 4) = 0
(y – 3)(y – 4) = 0
Roots of y (y के मूल): 3, 4
Comparison / तुलना:
x y Relation / संबंध 3 3 x = y 3 4 x < y 0.5 3 x < y 0.5 4 x < y Here, x is always less than or equal to y.
Final Answer (अंतिम उत्तर): (D) x ≤ y
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I. x² + 12x + 35 = 0
II. 5y² + 33y + 40 = 0Detailed Solution / विस्तृत समाधान
Equation I: x² + 12x + 35 = 0
x² + 7x + 5x + 35 = 0
x(x + 7) + 5(x + 7) = 0
(x + 5)(x + 7) = 0
Roots of x (x के मूल): -5, -7
Equation II: 5y² + 33y + 40 = 0
5y² + 25y + 8y + 40 = 0
5y(y + 5) + 8(y + 5) = 0
(5y + 8)(y + 5) = 0
Roots of y (y के मूल): -5, -8/5 (or -1.6)
Comparison / तुलना:
x y Relation / संबंध -5 -5 x = y -5 -1.6 x < y -7 -5 x < y -7 -1.6 x < y Here, x is always less than or equal to y.
Final Answer (अंतिम उत्तर): (D) x ≤ y
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I. x = ³√2197
II. 2y² – 54y + 364 = 0Detailed Solution / विस्तृत समाधान
Equation I: x = ³√2197
Since 10³=1000 and 15³=3375, the root is between 10 and 15. The unit digit is 7, so the cube root’s unit digit must be 3. Thus, x=13.
Root of x (x का मूल): 13
Equation II: 2y² – 54y + 364 = 0
Divide by 2: y² – 27y + 182 = 0
y² – 14y – 13y + 182 = 0
y(y – 14) – 13(y – 14) = 0
(y – 13)(y – 14) = 0
Roots of y (y के मूल): 13, 14
Comparison / तुलना:
x y Relation / संबंध 13 13 x = y 13 14 x < y Here, x is either less than or equal to y.
Final Answer (अंतिम उत्तर): (D) x ≤ y
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I. x² + 3x – 28 = 0
II. y² – 11y + 28 = 0Detailed Solution / विस्तृत समाधान
Equation I: x² + 3x – 28 = 0
x² + 7x – 4x – 28 = 0
x(x + 7) – 4(x + 7) = 0
(x – 4)(x + 7) = 0
Roots of x (x के मूल): 4, -7
Equation II: y² – 11y + 28 = 0
y² – 7y – 4y + 28 = 0
y(y – 7) – 4(y – 7) = 0
(y – 4)(y – 7) = 0
Roots of y (y के मूल): 4, 7
Comparison / तुलना:
x y Relation / संबंध 4 4 x = y 4 7 x < y -7 4 x < y -7 7 x < y Here, x is always less than or equal to y.
Final Answer (अंतिम उत्तर): (D) x ≤ y
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I. 6x² + 5x + 1 = 0
II. 15y² + 8y + 1 = 0Detailed Solution / विस्तृत समाधान
Equation I: 6x² + 5x + 1 = 0
6x² + 3x + 2x + 1 = 0
3x(2x + 1) + 1(2x + 1) = 0
(3x + 1)(2x + 1) = 0
Roots of x (x के मूल): -1/3 (≈ -0.33), -1/2 (-0.5)
Equation II: 15y² + 8y + 1 = 0
15y² + 5y + 3y + 1 = 0
5y(3y + 1) + 1(3y + 1) = 0
(5y + 1)(3y + 1) = 0
Roots of y (y के मूल): -1/5 (-0.2), -1/3 (≈ -0.33)
Comparison / तुलना:
x y Relation / संबंध -0.33 -0.2 x < y -0.33 -0.33 x = y -0.5 -0.2 x < y -0.5 -0.33 x < y Here, x is always less than or equal to y.
Final Answer (अंतिम उत्तर): (D) x ≤ y
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I. x² + 5x – 6 = 0
II. 2y² – 11y + 15 = 0Detailed Solution / विस्तृत समाधान
Equation I: x² + 5x – 6 = 0
x² + 6x – x – 6 = 0
x(x + 6) – 1(x + 6) = 0
(x – 1)(x + 6) = 0
Roots of x (x के मूल): 1, -6
Equation II: 2y² – 11y + 15 = 0
2y² – 6y – 5y + 15 = 0
2y(y – 3) – 5(y – 3) = 0
(2y – 5)(y – 3) = 0
Roots of y (y के मूल): 3, 2.5
Comparison / तुलना:
x y Relation / संबंध 1 3 x < y 1 2.5 x < y -6 3 x < y -6 2.5 x < y In all cases, x is less than y.
Final Answer (अंतिम उत्तर): (B) x < y
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I. x² – 11x + 30 = 0
II. y² + y – 30 = 0Detailed Solution / विस्तृत समाधान
Equation I: x² – 11x + 30 = 0
x² – 6x – 5x + 30 = 0
x(x – 6) – 5(x – 6) = 0
(x – 5)(x – 6) = 0
Roots of x (x के मूल): 5, 6
Equation II: y² + y – 30 = 0
y² + 6y – 5y – 30 = 0
y(y + 6) – 5(y + 6) = 0
(y – 5)(y + 6) = 0
Roots of y (y के मूल): 5, -6
Comparison / तुलना:
x y Relation / संबंध 5 5 x = y 5 -6 x > y 6 5 x > y 6 -6 x > y Here, x is always greater than or equal to y.
Final Answer (अंतिम उत्तर): (C) x ≥ y
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I. 2x² – 15x + 28 = 0
II. 2y² – 23y + 66 = 0Detailed Solution / विस्तृत समाधान
Equation I: 2x² – 15x + 28 = 0
2x² – 8x – 7x + 28 = 0
2x(x – 4) – 7(x – 4) = 0
(2x – 7)(x – 4) = 0
Roots of x (x के मूल): 4, 3.5
Equation II: 2y² – 23y + 66 = 0
2y² – 12y – 11y + 66 = 0
2y(y – 6) – 11(y – 6) = 0
(2y – 11)(y – 6) = 0
Roots of y (y के मूल): 6, 5.5
Comparison / तुलना:
x y Relation / संबंध 4 6 x < y 4 5.5 x < y 3.5 6 x < y 3.5 5.5 x < y In all cases, x is less than y.
Final Answer (अंतिम उत्तर): (B) x < y
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I. x² – 9x + 18 = 0
II. 5y² – 22y + 24 = 0Detailed Solution / विस्तृत समाधान
Equation I: x² – 9x + 18 = 0
x² – 6x – 3x + 18 = 0
x(x – 6) – 3(x – 6) = 0
(x – 3)(x – 6) = 0
Roots of x (x के मूल): 3, 6
Equation II: 5y² – 22y + 24 = 0
5y² – 12y – 10y + 24 = 0
y(5y – 12) – 2(5y – 12) = 0
(y – 2)(5y – 12) = 0
Roots of y (y के मूल): 2, 12/5 (or 2.4)
Comparison / तुलना:
x y Relation / संबंध 3 2 x > y 3 2.4 x > y 6 2 x > y 6 2.4 x > y In all cases, x is greater than y.
Final Answer (अंतिम उत्तर): (A) x > y
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I. x² – (1 + √2)x + √2 = 0
II. y² – 3y + 2 = 0Detailed Solution / विस्तृत समाधान
Equation I: x² – (1 + √2)x + √2 = 0
x² – x – √2x + √2 = 0
x(x – 1) – √2(x – 1) = 0
(x – √2)(x – 1) = 0
Roots of x (x के मूल): 1, √2 (≈ 1.414)
Equation II: y² – 3y + 2 = 0
y² – 2y – y + 2 = 0
y(y – 2) – 1(y – 2) = 0
(y – 1)(y – 2) = 0
Roots of y (y के मूल): 1, 2
Comparison / तुलना:
x y Relation / संबंध 1 1 x = y 1 2 x < y 1.414 1 x > y 1.414 2 x < y Since the relationship changes, it cannot be established.
Final Answer (अंतिम उत्तर): (E) relationship cannot be established
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I. 3x² + 4x + 1 = 0
II. y² + 5y + 6 = 0Detailed Solution / विस्तृत समाधान
Equation I: 3x² + 4x + 1 = 0
3x² + 3x + x + 1 = 0
3x(x + 1) + 1(x + 1) = 0
(3x + 1)(x + 1) = 0
Roots of x (x के मूल): -1, -1/3 (≈ -0.33)
Equation II: y² + 5y + 6 = 0
y² + 3y + 2y + 6 = 0
y(y + 3) + 2(y + 3) = 0
(y + 2)(y + 3) = 0
Roots of y (y के मूल): -2, -3
Comparison / तुलना:
x y Relation / संबंध -1 -2 x > y -1 -3 x > y -0.33 -2 x > y -0.33 -3 x > y In all cases, x is greater than y.
Final Answer (अंतिम उत्तर): (A) x > y
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I. 2x² + 5x + 2 = 0
II. 4y² – 1 = 0Detailed Solution / विस्तृत समाधान
Equation I: 2x² + 5x + 2 = 0
2x² + 4x + x + 2 = 0
2x(x + 2) + 1(x + 2) = 0
(2x + 1)(x + 2) = 0
Roots of x (x के मूल): -2, -0.5
Equation II: 4y² – 1 = 0
4y² = 1
y² = 1/4
y = ±√(1/4)
Roots of y (y के मूल): 0.5, -0.5
Comparison / तुलना:
x y Relation / संबंध -2 0.5 x < y -2 -0.5 x < y -0.5 0.5 x < y -0.5 -0.5 x = y Here, x is always less than or equal to y.
Final Answer (अंतिम उत्तर): (D) x ≤ y
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I. x² + 2x – 35 = 0
II. y² + 15y + 56 = 0Detailed Solution / विस्तृत समाधान
Equation I: x² + 2x – 35 = 0
x² + 7x – 5x – 35 = 0
x(x + 7) – 5(x + 7) = 0
(x – 5)(x + 7) = 0
Roots of x (x के मूल): 5, -7
Equation II: y² + 15y + 56 = 0
y² + 8y + 7y + 56 = 0
y(y + 8) + 7(y + 8) = 0
(y + 7)(y + 8) = 0
Roots of y (y के मूल): -7, -8
Comparison / तुलना:
x y Relation / संबंध 5 -7 x > y 5 -8 x > y -7 -7 x = y -7 -8 x > y Here, x is always greater than or equal to y.
Final Answer (अंतिम उत्तर): (C) x ≥ y
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I. 2x² – 9x + 10 = 0
II. 2y² – 13y + 20 = 0Detailed Solution / विस्तृत समाधान
Equation I: 2x² – 9x + 10 = 0
2x² – 5x – 4x + 10 = 0
x(2x – 5) – 2(2x – 5) = 0
(x – 2)(2x – 5) = 0
Roots of x (x के मूल): 2, 2.5
Equation II: 2y² – 13y + 20 = 0
2y² – 8y – 5y + 20 = 0
2y(y – 4) – 5(y – 4) = 0
(2y – 5)(y – 4) = 0
Roots of y (y के मूल): 4, 2.5
Comparison / तुलना:
x y Relation / संबंध 2 4 x < y 2 2.5 x < y 2.5 4 x < y 2.5 2.5 x = y Here, x is always less than or equal to y.
Final Answer (अंतिम उत्तर): (D) x ≤ y
Directions (26-50): In the following questions, two equations numbered I and II are given. You have to solve both the equations and give answer.
निर्देश (26-50): निम्नलिखित प्रश्नों में, दो समीकरण I और II दिए गए हैं। आपको दोनों समीकरणों को हल करना है और उत्तर देना है।
(A) if x > y (यदि x > y)
(B) if x < y (यदि x < y)
(C) if x ≥ y (यदि x ≥ y)
(D) if x ≤ y (यदि x ≤ y)
(E) if x = y or the relationship cannot be established (यदि x = y या संबंध स्थापित नहीं किया जा सकता है)
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I. 2x² – 13x + 21 = 0
II. y² – 7y + 12 = 0Detailed Solution / विस्तृत समाधान
Equation I: 2x² – 13x + 21 = 0
2x² – 7x – 6x + 21 = 0
x(2x – 7) – 3(2x – 7) = 0
(x – 3)(2x – 7) = 0
Roots of x (x के मूल): 3, 3.5
Equation II: y² – 7y + 12 = 0
y² – 4y – 3y + 12 = 0
y(y – 4) – 3(y – 4) = 0
(y – 3)(y – 4) = 0
Roots of y (y के मूल): 3, 4
Comparison / तुलना:
x y Relation / संबंध 3 3 x = y 3 4 x < y 3.5 3 x > y 3.5 4 x < y Since the relationship is not consistent, it cannot be established.
Final Answer (अंतिम उत्तर): (E) relationship cannot be established
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I. x² + x – 56 = 0
II. y² – 17y + 72 = 0Detailed Solution / विस्तृत समाधान
Equation I: x² + x – 56 = 0
x² + 8x – 7x – 56 = 0
x(x + 8) – 7(x + 8) = 0
(x – 7)(x + 8) = 0
Roots of x (x के मूल): 7, -8
Equation II: y² – 17y + 72 = 0
y² – 9y – 8y + 72 = 0
y(y – 9) – 8(y – 9) = 0
(y – 8)(y – 9) = 0
Roots of y (y के मूल): 8, 9
Comparison / तुलना:
x y Relation / संबंध 7 8 x < y 7 9 x < y -8 8 x < y -8 9 x < y In all cases, x is less than y.
Final Answer (अंतिम उत्तर): (B) x < y
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I. x² = 196
II. y = √196Detailed Solution / विस्तृत समाधान
Equation I: x² = 196
x = ±√196
Roots of x (x के मूल): +14, -14
Equation II: y = √196
The square root symbol (√) implies only the positive root.
Root of y (y का मूल): 14
Comparison / तुलना:
x y Relation / संबंध 14 14 x = y -14 14 x < y Here, x is either less than or equal to y.
Final Answer (अंतिम उत्तर): (D) x ≤ y
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I. 3x² + 13x + 12 = 0
II. 2y² + 9y + 10 = 0Detailed Solution / विस्तृत समाधान
Equation I: 3x² + 13x + 12 = 0
3x² + 9x + 4x + 12 = 0
3x(x + 3) + 4(x + 3) = 0
(3x + 4)(x + 3) = 0
Roots of x (x के मूल): -3, -4/3 (≈ -1.33)
Equation II: 2y² + 9y + 10 = 0
2y² + 5y + 4y + 10 = 0
y(2y + 5) + 2(2y + 5) = 0
(y + 2)(2y + 5) = 0
Roots of y (y के मूल): -2, -5/2 (-2.5)
Comparison / तुलना:
x y Relation / संबंध -3 -2 x < y -3 -2.5 x < y -1.33 -2 x > y -1.33 -2.5 x > y Since the relationship is not consistent, it cannot be established.
Final Answer (अंतिम उत्तर): (E) relationship cannot be established
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I. x² – 22x + 120 = 0
II. y² – 26y + 168 = 0Detailed Solution / विस्तृत समाधान
Equation I: x² – 22x + 120 = 0
x² – 12x – 10x + 120 = 0
x(x – 12) – 10(x – 12) = 0
(x – 10)(x – 12) = 0
Roots of x (x के मूल): 10, 12
Equation II: y² – 26y + 168 = 0
y² – 14y – 12y + 168 = 0
y(y – 14) – 12(y – 14) = 0
(y – 12)(y – 14) = 0
Roots of y (y के मूल): 12, 14
Comparison / तुलना:
x y Relation / संबंध 10 12 x < y 10 14 x < y 12 12 x = y 12 14 x < y Here, x is always less than or equal to y.
Final Answer (अंतिम उत्तर): (D) x ≤ y
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I. 5x² – 18x + 9 = 0
II. 3y² + 5y – 2 = 0Detailed Solution / विस्तृत समाधान
Equation I: 5x² – 18x + 9 = 0
5x² – 15x – 3x + 9 = 0
5x(x – 3) – 3(x – 3) = 0
(5x – 3)(x – 3) = 0
Roots of x (x के मूल): 3, 3/5 (or 0.6)
Equation II: 3y² + 5y – 2 = 0
3y² + 6y – y – 2 = 0
3y(y + 2) – 1(y + 2) = 0
(3y – 1)(y + 2) = 0
Roots of y (y के मूल): -2, 1/3 (≈ 0.33)
Comparison / तुलना:
x y Relation / संबंध 3 -2 x > y 3 0.33 x > y 0.6 -2 x > y 0.6 0.33 x > y In all cases, x is greater than y.
Final Answer (अंतिम उत्तर): (A) x > y
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I. x = ³√3375
II. y² = 225Detailed Solution / विस्तृत समाधान
Equation I: x = ³√3375
x = 15
Root of x (x का मूल): 15
Equation II: y² = 225
y = ±√225
Roots of y (y के मूल): +15, -15
Comparison / तुलना:
x y Relation / संबंध 15 15 x = y 15 -15 x > y Here, x is always greater than or equal to y.
Final Answer (अंतिम उत्तर): (C) x ≥ y
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I. x² – 8x + 15 = 0
II. 2y² – 11y + 15 = 0Detailed Solution / विस्तृत समाधान
Equation I: x² – 8x + 15 = 0
(x – 3)(x – 5) = 0
Roots of x (x के मूल): 3, 5
Equation II: 2y² – 11y + 15 = 0
2y² – 6y – 5y + 15 = 0
2y(y – 3) – 5(y – 3) = 0
(2y – 5)(y – 3) = 0
Roots of y (y के मूल): 3, 5/2 (or 2.5)
Comparison / तुलना:
x y Relation / संबंध 3 3 x = y 3 2.5 x > y 5 3 x > y 5 2.5 x > y Here, x is always greater than or equal to y.
Final Answer (अंतिम उत्तर): (C) x ≥ y
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I. x² + 7x – 18 = 0
II. y² – 8y + 15 = 0Detailed Solution / विस्तृत समाधान
Equation I: x² + 7x – 18 = 0
x² + 9x – 2x – 18 = 0
x(x + 9) – 2(x + 9) = 0
(x – 2)(x + 9) = 0
Roots of x (x के मूल): 2, -9
Equation II: y² – 8y + 15 = 0
(y – 3)(y – 5) = 0
Roots of y (y के मूल): 3, 5
Comparison / तुलना:
x y Relation / संबंध 2 3 x < y 2 5 x < y -9 3 x < y -9 5 x < y In all cases, x is less than y.
Final Answer (अंतिम उत्तर): (B) x < y
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I. x² – 4√3x + 9 = 0
II. y² – 5y + 6 = 0Detailed Solution / विस्तृत समाधान
Equation I: x² – 4√3x + 9 = 0
x² – 3√3x – √3x + 9 = 0
x(x – 3√3) – √3(x – 3√3) = 0
(x – √3)(x – 3√3) = 0
Roots of x (x के मूल): √3 (≈ 1.73), 3√3 (≈ 5.19)
Equation II: y² – 5y + 6 = 0
(y – 2)(y – 3) = 0
Roots of y (y के मूल): 2, 3
Comparison / तुलना:
x y Relation / संबंध 1.73 2 x < y 1.73 3 x < y 5.19 2 x > y 5.19 3 x > y Since the relationship is not consistent, it cannot be established.
Final Answer (अंतिम उत्तर): (E) relationship cannot be established
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I. x² + 13x + 42 = 0
II. y² + 16y + 63 = 0Detailed Solution / विस्तृत समाधान
Equation I: x² + 13x + 42 = 0
(x + 6)(x + 7) = 0
Roots of x (x के मूल): -6, -7
Equation II: y² + 16y + 63 = 0
(y + 7)(y + 9) = 0
Roots of y (y के मूल): -7, -9
Comparison / तुलना:
x y Relation / संबंध -6 -7 x > y -6 -9 x > y -7 -7 x = y -7 -9 x > y Here, x is always greater than or equal to y.
Final Answer (अंतिम उत्तर): (C) x ≥ y
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I. x² – x – 42 = 0
II. y² + y – 42 = 0Detailed Solution / विस्तृत समाधान
Equation I: x² – x – 42 = 0
(x – 7)(x + 6) = 0
Roots of x (x के मूल): 7, -6
Equation II: y² + y – 42 = 0
(y + 7)(y – 6) = 0
Roots of y (y के मूल): -7, 6
Comparison / तुलना:
x y Relation / संबंध 7 -7 x > y 7 6 x > y -6 -7 x > y -6 6 x < y Since the relationship is not consistent, it cannot be established.
Final Answer (अंतिम उत्तर): (E) relationship cannot be established
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I. 12x² – 7x + 1 = 0
II. 6y² – 7y + 2 = 0Detailed Solution / विस्तृत समाधान
Equation I: 12x² – 7x + 1 = 0
12x² – 4x – 3x + 1 = 0
4x(3x – 1) – 1(3x – 1) = 0
(4x – 1)(3x – 1) = 0
Roots of x (x के मूल): 1/4 (0.25), 1/3 (≈ 0.33)
Equation II: 6y² – 7y + 2 = 0
6y² – 4y – 3y + 2 = 0
2y(3y – 2) – 1(3y – 2) = 0
(2y – 1)(3y – 2) = 0
Roots of y (y के मूल): 1/2 (0.5), 2/3 (≈ 0.66)
Comparison / तुलना:
x y Relation / संबंध 0.25 0.5 x < y 0.25 0.66 x < y 0.33 0.5 x < y 0.33 0.66 x < y In all cases, x is less than y.
Final Answer (अंतिम उत्तर): (B) x < y
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I. x² + 24 = 10x
II. 2y² + 48 = 20yDetailed Solution / विस्तृत समाधान
Equation I: x² + 24 = 10x
x² – 10x + 24 = 0
(x – 4)(x – 6) = 0
Roots of x (x के मूल): 4, 6
Equation II: 2y² + 48 = 20y
2y² – 20y + 48 = 0
y² – 10y + 24 = 0 (Dividing by 2)
(y – 4)(y – 6) = 0
Roots of y (y के मूल): 4, 6
Comparison / तुलना:
The roots of x are 4, 6.
The roots of y are 4, 6.
The equations are identical. Therefore, x will always be equal to y for corresponding roots.Final Answer (अंतिम उत्तर): (E) x = y or the relationship cannot be established
(Note: When roots are identical like {4, 6} and {4, 6}, the correct answer is ‘relationship cannot be established’ because 4 < 6. However, if both equations are identical, 'x=y' is also a valid interpretation in some contexts. 'E' is the safest choice as it covers both.) -
I. x² = 7x
II. y² + 6y = 0Detailed Solution / विस्तृत समाधान
Equation I: x² = 7x
x² – 7x = 0
x(x – 7) = 0
Roots of x (x के मूल): 0, 7 (Note: Do not cancel x from both sides, you will lose a root)
Equation II: y² + 6y = 0
y(y + 6) = 0
Roots of y (y के मूल): 0, -6
Comparison / तुलना:
x y Relation / संबंध 0 0 x = y 0 -6 x > y 7 0 x > y 7 -6 x > y Here, x is always greater than or equal to y.
Final Answer (अंतिम उत्तर): (C) x ≥ y
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I. 4x² – 4x – 3 = 0
II. 4y² + 12y + 5 = 0Detailed Solution / विस्तृत समाधान
Equation I: 4x² – 4x – 3 = 0
4x² – 6x + 2x – 3 = 0
2x(2x – 3) + 1(2x – 3) = 0
(2x + 1)(2x – 3) = 0
Roots of x (x के मूल): -1/2 (-0.5), 3/2 (1.5)
Equation II: 4y² + 12y + 5 = 0
4y² + 10y + 2y + 5 = 0
2y(2y + 5) + 1(2y + 5) = 0
(2y + 1)(2y + 5) = 0
Roots of y (y के मूल): -1/2 (-0.5), -5/2 (-2.5)
Comparison / तुलना:
x y Relation / संबंध -0.5 -0.5 x = y -0.5 -2.5 x > y 1.5 -0.5 x > y 1.5 -2.5 x > y Here, x is always greater than or equal to y.
Final Answer (अंतिम उत्तर): (C) x ≥ y
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I. x² + 12x + 27 = 0
II. y² + 7y + 10 = 0Detailed Solution / विस्तृत समाधान
Equation I: x² + 12x + 27 = 0
(x + 9)(x + 3) = 0
Roots of x (x के मूल): -9, -3
Equation II: y² + 7y + 10 = 0
(y + 5)(y + 2) = 0
Roots of y (y के मूल): -5, -2
Comparison / तुलना:
x y Relation / संबंध -9 -5 x < y -9 -2 x < y -3 -5 x > y -3 -2 x < y Since the relationship is not consistent, it cannot be established.
Final Answer (अंतिम उत्तर): (E) relationship cannot be established
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I. x = √625
II. y = √676Detailed Solution / विस्तृत समाधान
Equation I: x = √625
x = 25 (Only the positive root)
Root of x (x का मूल): 25
Equation II: y = √676
y = 26 (Only the positive root)
Root of y (y का मूल): 26
Comparison / तुलना:
x = 25 and y = 26. Clearly, x < y.Final Answer (अंतिम उत्तर): (B) x < y
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I. x² – 30x + 221 = 0
II. y² – 31y + 240 = 0Detailed Solution / विस्तृत समाधान
Equation I: x² – 30x + 221 = 0
(221 = 13 * 17; 13 + 17 = 30)
x² – 17x – 13x + 221 = 0
x(x – 17) – 13(x – 17) = 0
(x – 13)(x – 17) = 0
Roots of x (x के मूल): 13, 17
Equation II: y² – 31y + 240 = 0
(240 = 15 * 16; 15 + 16 = 31)
y² – 16y – 15y + 240 = 0
y(y – 16) – 15(y – 16) = 0
(y – 15)(y – 16) = 0
Roots of y (y के मूल): 15, 16
Comparison / तुलना:
x y Relation / संबंध 13 15 x < y 13 16 x < y 17 15 x > y 17 16 x > y Since the relationship is not consistent, it cannot be established.
Final Answer (अंतिम उत्तर): (E) relationship cannot be established
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I. x² + 4x – 12 = 0
II. y² – 5y + 6 = 0Detailed Solution / विस्तृत समाधान
Equation I: x² + 4x – 12 = 0
(x + 6)(x – 2) = 0
Roots of x (x के मूल): -6, 2
Equation II: y² – 5y + 6 = 0
(y – 2)(y – 3) = 0
Roots of y (y के मूल): 2, 3
Comparison / तुलना:
x y Relation / संबंध -6 2 x < y -6 3 x < y 2 2 x = y 2 3 x < y Here, x is always less than or equal to y.
Final Answer (अंतिम उत्तर): (D) x ≤ y
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I. 7x² + 16x + 4 = 0
II. 3y² + 8y + 4 = 0Detailed Solution / विस्तृत समाधान
Equation I: 7x² + 16x + 4 = 0
7x² + 14x + 2x + 4 = 0
7x(x + 2) + 2(x + 2) = 0
(7x + 2)(x + 2) = 0
Roots of x (x के मूल): -2, -2/7 (≈ -0.28)
Equation II: 3y² + 8y + 4 = 0
3y² + 6y + 2y + 4 = 0
3y(y + 2) + 2(y + 2) = 0
(3y + 2)(y + 2) = 0
Roots of y (y के मूल): -2, -2/3 (≈ -0.67)
Comparison / तुलना:
x y Relation / संबंध -2 -2 x = y -2 -0.67 x < y -0.28 -2 x > y -0.28 -0.67 x > y Since the relationship is not consistent, it cannot be established.
Final Answer (अंतिम उत्तर): (E) relationship cannot be established
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I. x² – 19x + 88 = 0
II. y² – 12y + 35 = 0Detailed Solution / विस्तृत समाधान
Equation I: x² – 19x + 88 = 0
(x – 8)(x – 11) = 0
Roots of x (x के मूल): 8, 11
Equation II: y² – 12y + 35 = 0
(y – 5)(y – 7) = 0
Roots of y (y के मूल): 5, 7
Comparison / तुलना:
x y Relation / संबंध 8 5 x > y 8 7 x > y 11 5 x > y 11 7 x > y In all cases, x is greater than y.
Final Answer (अंतिम उत्तर): (A) x > y
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I. 2x² + x – 1 = 0
II. 6y² – 13y + 5 = 0Detailed Solution / विस्तृत समाधान
Equation I: 2x² + x – 1 = 0
2x² + 2x – x – 1 = 0
2x(x + 1) – 1(x + 1) = 0
(2x – 1)(x + 1) = 0
Roots of x (x के मूल): -1, 1/2 (0.5)
Equation II: 6y² – 13y + 5 = 0
6y² – 10y – 3y + 5 = 0
2y(3y – 5) – 1(3y – 5) = 0
(2y – 1)(3y – 5) = 0
Roots of y (y के मूल): 1/2 (0.5), 5/3 (≈ 1.67)
Comparison / तुलना:
x y Relation / संबंध -1 0.5 x < y -1 1.67 x < y 0.5 0.5 x = y 0.5 1.67 x < y Here, x is always less than or equal to y.
Final Answer (अंतिम उत्तर): (D) x ≤ y
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I. x² + 10x + 21 = 0
II. y² + 11y + 28 = 0Detailed Solution / विस्तृत समाधान
Equation I: x² + 10x + 21 = 0
(x + 3)(x + 7) = 0
Roots of x (x के मूल): -3, -7
Equation II: y² + 11y + 28 = 0
(y + 4)(y + 7) = 0
Roots of y (y के मूल): -4, -7
Comparison / तुलना:
x y Relation / संबंध -3 -4 x > y -3 -7 x > y -7 -4 x < y -7 -7 x = y Since the relationship is not consistent, it cannot be established.
Final Answer (अंतिम उत्तर): (E) relationship cannot be established
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I. 15x² – 11x + 2 = 0
II. 10y² – 9y + 2 = 0Detailed Solution / विस्तृत समाधान
Equation I: 15x² – 11x + 2 = 0
15x² – 6x – 5x + 2 = 0
3x(5x – 2) – 1(5x – 2) = 0
(3x – 1)(5x – 2) = 0
Roots of x (x के मूल): 1/3 (≈ 0.33), 2/5 (0.4)
Equation II: 10y² – 9y + 2 = 0
10y² – 5y – 4y + 2 = 0
5y(2y – 1) – 2(2y – 1) = 0
(5y – 2)(2y – 1) = 0
Roots of y (y के मूल): 2/5 (0.4), 1/2 (0.5)
Comparison / तुलना:
x y Relation / संबंध 0.33 0.4 x < y 0.33 0.5 x < y 0.4 0.4 x = y 0.4 0.5 x < y Here, x is always less than or equal to y.
Final Answer (अंतिम उत्तर): (D) x ≤ y