arun singh rawat - maths by arun sir

ARUN SINGH RAWAT

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Maths by Arun Singh Rawat sir ARUN SINGH RAWAT

❑ QUADRATIC EQUATION LIVE TEST

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a) x > yb) x ≥ yc) x < yd) x ≤ ye) x = y or relationship between x and y cannot be established.

I. 7x² – 23x + 18 = 0

II. 4y² – 9y + 5 = 0

Sol.I. 7x² – 23x + 18 = 0 7x² – 14x – 9x + 18 = 0 (7x – 9) (x – 2) = 0

x =9

7, 2

II. 4y² – 9y + 5 = 0 4y² – 4y – 5y + 5 = 0 4y (y – 1) – 5(y – 1) = 0 (4y – 5) (y – 1) = 0

y =5

4, 1

⇒ x > y

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I. 10x² - 11x – 6 = 0

II. y² + 5y + 6 = 0

a) x > yb) x ≥ yc) x < yd) x ≤ ye) x = y or relationship between x and y cannot be established.

Sol.I. 10x² - 11x – 6 = 0 5x (2x – 3) + 2 (2x – 3) = 0 (5x + 2) (2x – 3) = 0

x = –2

5,

3

2

II. y² + 5y + 6 = 0 y² + 2y + 3y + 6 = 0 y (y + 2) +3 (y + 2) = 0 (y + 3) (y + 2) = 0 y = – 2, –3

⇒ x > y

❑ QUADRATIC EQUATION LIVE TEST

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I. 13x² – 29x + 16 = 0 II. 6y² – 11y + 5 = 0

a) x > yb) x ≥ yc) x < yd) x ≤ ye) x = y or relationship between x and y cannot be established.

Sol.I. 13x² – 29x + 16 = 0 13x² – 13x – 16x + 16 = 0 13x (x – 1) –16 (x – 1) = 0(13x – 16) (x – 1) = 0

x =16

13, 1

II. 6y² – 11y + 5 = 0 6y² – 6y – 5y + 5 = 0 6y (y – 1) – 5 (y – 1) = 0(6y – 5) (y – 1) = 0

y =5

6, 1

⇒ x ≥ y

❑ QUADRATIC EQUATION LIVE TEST

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I. x = 𝟏𝟔𝟗II. 𝐲 – 𝟏 𝟐 = 144

a) x > yb) x ≥ yc) x < yd) x ≤ ye) x = y or relationship between x and y cannot be established.

Sol.

I. x = 169 = 13II. (y – 1)² = 144y – 1 = 12 or y – 1 = – 12y = 13 or y = – 11⇒ x ≥ y

❑ QUADRATIC EQUATION LIVE TEST

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I. 2x² - 9x + 9 = 0

II. 6y² - 17y + 12 = 0

a) x > yb) x ≥ yc) x < yd) x ≤ ye) x = y or relationship between x and y cannot be established.

Sol.

I. 2x² - 9x + 9 = 0

2x² - 6x – 3x + 9 = 0

(2x-3)(x-3)=0

𝑥 =3

2, 3

II. 6y² - 17y + 12 = 0

6y² - 8y – 9y + 12 = 0

(2y-3)(3y-4)=0

𝑦 =3

2,4

3∴ x ≥ y

❑ QUADRATIC EQUATION LIVE TEST

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I. 7y + 3x = 27

II. 9y + 7x = 53

a) x > yb) x ≥ yc) x < yd) x ≤ ye) x = y or relationship between x and y cannot be established.

Sol.

Multiply (i) by 7 and multiply (ii) by 3 and subtract (ii)

from (i)

y = 15

11

𝑥 =64

11∴ x > y

❑ QUADRATIC EQUATION LIVE TEST

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I. 3x² + 5x – 112 = 0

II. 2y² + 19y + 42 =0

a) x > yb) x ≥ yc) x < yd) x ≤ ye) x = y or relationship between x and y cannot be established.

Sol.

I. 3x² + 5x – 112 = 0

3x² + 21x – 16x – 112 = 0

(3x-16)(x+7)=0

𝑥 = −7,16

3II. 2y² + 19y + 42 = 0

2y² + 12y + 7y + 42 = 0

(2y+7)(y+6)=0

y = -6, −7

2

no relation

❑ QUADRATIC EQUATION LIVE TEST

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I. 28x² + 33x + 9 = 0

II. 25y² + 50y + 24 = 0

a) x > yb) x ≥ yc) x < yd) x ≤ ye) x = y or relationship between x and y cannot be established.

I. 28x² + 33x +9 =0

28x² + 21x + 12x + 9 = 0

(7x+3)(4x+3)=0

𝑥 = −3

4, −

3

7II. 25y² + 50y + 24 = 0

25y² + 30y + 20y + 24 = 0

(5y+6)(5y+4)=0

𝑦 = −4

5, −

6

5∴ x > y

❑ QUADRATIC EQUATION LIVE TEST

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(I) 16x2 - 16x + 3 = 0(II) 2y2 - y - 3 = 0

a) x > yb) x ≥ yc) x < yd) x ≤ ye) x = y or relationship between x and y cannot be established.

Sol.

(I) 16x2 - 16x + 3 = 0=> 16x2 - 12x - 4x + 3 = 0=> 4x(4x - 3) - (4x - 3) = 0=> (4x - 3)(4x - 1) = 0=> x = ¾ or ¼=> x = 0.75 or 0.25(II) 2y2 - y - 3 = 0=> 2y2 - 3y + 2y - 3 = 0=> y(2y - 3) + (2y - 3) = 0=> (2y - 3)(y + 1) = 0=> y = 3/2, or -1=> y = 1.5 or -1When x = 0.75 and y = 1.5, so, x < yAnd, when x = 0.75 and y = -1 so, x > yAnd, when x = 0.25 and y = 1.5 so, x < yAnd, when x = 0.25 and y = -1 so, x > yHence, the relation cannot be established between x and y

❑ QUADRATIC EQUATION LIVE TEST

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(I) 2x2 - 7x + 5 = 0

(II) y =𝟑

𝟖

a) x > yb) x ≥ yc) x < yd) x ≤ ye) x = y or relationship between x and y cannot be established.

Sol.

(I) 2x2 - 7x + 5 = 0=> 2x2 - 2x - 5x + 5 = 0=> 2x(x - 1) - 5(x - 1) = 0=> (2x - 5)(x - 1) = 0=> x = 5/2 or 1=> x = 2.5 or 1

And, y = 3

8=> y = 2When x = 2.5 and y = 2 so, x > yAnd, when x = 1 and y = 2 so, x < yHence, relationship cannot be established between x and y.

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(I) x2 - 11x + 10 = 0(II) y2 - 5y + 6 = 0

a) x > yb) x ≥ yc) x < yd) x ≤ ye) x = y or relationship between x and y cannot be established.

Sol.

x2 - 11x + 10 = 0=> x2 - x - 10x + 10 = 0=> (x - 10)(x - 1) = 0=> x = 10, 1(II) y2 - 5y + 6 = 0=> y2 - 2y - 3y + 6 = 0=> y(y - 2) - 3(y - 2) = 0=> (y - 2)(y - 3) = 0=> y = 2 or 3When x = 10 and y = 2, so, x > yAnd, when x = 10 and y = 3, so, x > yAnd, when x = 1 and y = 2, so, x < yAnd, when x = 1 and y = 3, so, x < yHence, relationship cannot be established between x and y.

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(I) 3x2 + 7x - 20 = 0(II) 6y2 - y - 15 = 0

a) x > yb) x ≥ yc) x < yd) x ≤ ye) x = y or relationship between x and y cannot be established.

Sol.

3x2 + 7x - 20 = 0(x + 4)(3x - 5) = 0x = -4 and 5/36y2 - y - 15 = 0(2y + 3)(3y - 5) = 0y = -3/2 and 5/3Hence, relation can't be determined.

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I. 3x² – 22x + 40 = 0

II. 2y² – 17y + 36 = 0

a) x > yb) x ≥ yc) x < yd) x ≤ ye) x = y or relationship between x and y cannot be established.

Sol.

I. 3x² – 22x + 40 = 0

3x² – 12x – 10x + 40 = 0

3x(x – 4) – 10 (x – 4) = 0

(3x – 10) (x –4) = 0

𝑥 =10

3, 4

II. 2y² – 17y + 36 = 0

2y² – 8y – 9y + 36 = 0

2y(y – 4) –9 (y – 4) = 0

(2y–9) (y – 4) = 0

y = 9

2, 4

y ≥ x

❑ QUADRATIC EQUATION LIVE TEST

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I. 5x² + 7x – 90 = 0

II. 3y² + 28y + 64 = 0

a) x > yb) x ≥ yc) x < yd) x ≤ ye) x = y or relationship between x and y cannot be established.

Sol.

I. 5x² + 7x – 90 = 0

5x² + 25x – 18x – 90 = 0

5x (x + 5) – 18 (x + 5)= 0

(x + 5) (5x – 18) = 0

𝑥 = −5,18

5II. 3y² + 28y + 64 = 0

3y² + 12y + 16y + 64 = 0

3y (y + 4) + 16 (y + 4) = 0

(3y + 16) (y +4) = 0

y = −16

3. −4

No relation can be established between x & y

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I. x² – 14x + 48 = 0

II. 2y² – 19y + 45 = 0

a) x > yb) x ≥ yc) x < yd) x ≤ ye) x = y or relationship between x and y cannot be established.

Sol.

I. x² – 14x + 48 =0

x² – 6x – 8x + 48 = 0

x(x – 6) – 8 (x – 6) = 0

(x – 8) (x – 6) = 0

x= 8, 6

II. 2y² – 19y + 45 = 0

2y² – 10y – 9y + 45 = 0

2y (y – 5) – 9(y – 5) = 0

(2y – 9) (y – 5) = 0

𝑦 =9

2, 5

x > y

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I. 2x² + 13x + 20 = 0

II. 3y² + 26y + 56 = 0

a) x > yb) x ≥ yc) x < yd) x ≤ ye) x = y or relationship between x and y cannot be established.

Sol.

I. 2x² + 13x + 20 = 0

2x² + 8x + 5x + 20 = 0

2x(x + 4) + 5 (x + 4) = 0

(2x + 5) (x + 4) = 0

𝑥 =−5

2, −4

II. 3y² + 26y +56 = 0

3y² + 12y + 14y + 56 = 0

3y (y + 4) + 14 (y + 4) = 0

(3y + 14) (y + 4) =0

𝑦 = −14

3, −4

x ≥ y

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I. 15x² + 11x + 2 = 0

II. 24y² + 11y + 1 = 0

a) x > yb) x ≥ yc) x < yd) x ≤ ye) x = y or relationship between x and y cannot be established.

Sol.

I. 15x² + 5x + 6x + 2 = 0

5x (3x + 1) + 2 (3x + 1) = 0

(5x + 2) (3x + 1) = 0

𝑥 =−2

5, −

1

3II. 24y² + 8y + 3y + 1 = 0

8y (3y +1) + 1 (3y + 1) = 0

(8y + 1) (3y + 1) = 0

𝑦 = −1

3,−1

8⇒ x ≤ y

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I. x² – 30x + 221 = 0

II. y² – 17y + 60 = 0

a) x > yb) x ≥ yc) x < yd) x ≤ ye) x = y or relationship between x and y cannot be established.

Sol.

I. x² – 13x – 17x + 221 = 0

x (x – 13) – 17 (x – 13) = 0

(x – 17) (x – 13) = 0

x = 13, 17

II. y² – 12y – 5y + 60 = 0

y (y – 12) – 5 (y – 12) = 0

(y – 5) (y – 12) = 0

y = 5, 12

⇒ x > y

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I. x² + 6x + 8 = 0

II. 8y² + 22y + 15 = 0

a) x > yb) x ≥ yc) x < yd) x ≤ ye) x = y or relationship between x and y cannot be established.

Sol.

I. x² + 6x + 8 = 0

x² + 2x + 4x + 8 = 0

x (x + 2) + 4 (x + 2) = 0

(x + 4) (x + 2) = 0

x = –2, –4

II. 8y² + 22y + 15 = 0

8y² + 10y + 12y + 15 = 0

2y (4y + 5) +3(4y + 5) = 0

(2y + 3) (4y + 5) = 0

𝑦 =−3

2, −

5

4⇒ x < y

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I. x² – 20x + 96 = 0

II. y² – 15y + 56 = 0

a) x > yb) x ≥ yc) x < yd) x ≤ ye) x = y or relationship between x and y cannot be established.

Sol.

I. x² – 20x + 96 = 0

x² – 8x – 12x + 96 = 0

x (x – 8) – 12 (x – 8) = 0

(x – 12) (x – 8) = 0

x = 12, 8

II. y² – 15y + 56 = 0

y² – 7y – 8y + 56 = 0

(y – 7) (y – 8) = 0

y = 7, 8

⇒ x ≥ y

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(i) x² = (33)² – 62 × 4

(ii) y² + 62y + 957 = 0

a) x > yb) x ≥ yc) x < yd) x ≤ ye) x = y or relationship between x and y cannot be established.

Sol.

(i) x² = 1089 – 248

x² = 841

x = ± 29

(ii) y² + 33y + 29y + 957 = 0

y (y + 33) + 29 (y + 33) = 0

(y + 29) (y + 33) = 0

y = – 29, – 33

x ≥ y

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(i) 3x² – 45x + 168= 0

(ii) 3y² – 39y + 126 = 0

a) x > yb) x ≥ yc) x < yd) x ≤ ye) x = y or relationship between x and y cannot be established.

Sol.

(i) 3x² – 24x – 21x + 168 = 0

3x (x – 8) – 21 (x – 8) = 0

(3x – 21) (x – 8) = 0

x = 7, 8

(ii) 3y² – 21y – 18y + 126= 0

3y (y – 7) – 18 (y – 7) = 0

(y – 7) (3y – 18) = 0

y = 7, 6

x ≥ y

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(i) 36x² + 12x - 8= 0

(ii) 45y² + 51y + 12 = 0

a) x > yb) x ≥ yc) x < yd) x ≤ ye) x = y or relationship between x and y cannot be established.

(i) 36x² + 24x – 12x – 8 = 0

12x (3x + 2) – 4 (3x + 2) = 0

(3x + 2) (12x – 4) = 0

𝑥 =–2

3,

1

3

(ii) 45y² + 36y + 15y + 12 = 0

9y (5y + 4) + 3 (5y + 4) = 0

(5y + 4) (9y + 3) = 0

y = –4

5,

–1

3

No relation

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(i) x² – 40x + 336= 0

(ii) y²– 60y + 896 = 0

a) x > yb) x ≥ yc) x < yd) x ≤ ye) x = y or relationship between x and y cannot be established.

Sol.

(i)x² – 28x – 12x + 336 = 0

x (x – 28) – 12 (x – 28) = 0

(x – 12) (x – 28) = 0

x = 12, 28

(ii) y² – 32y – 28y + 896 = 0

y (y – 32) – 28 (y – 32) = 0

(y – 32) (y – 28)

y = 28, 32

x ≤ y

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(i) x² – 721 = 800

(ii) y = 𝟏𝟓𝟐𝟏

a) x > yb) x ≥ yc) x < yd) x ≤ ye) x = y or relationship between x and y cannot be established.

Sol.

x² = 800 + 721

x = ±39

y = 1521

y = 39

x ≤ y

❑ QUADRATIC EQUATION LIVE TEST

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I. 16x – 15y = 22II. 2x – 5y = 4

a) x > yb) x ≥ yc) x < yd) x ≤ ye) x = y or relationship between x and y cannot be established.

Sol.

16x – 15y = 22(2x – 5y = 4)×3 (make y terms equal in both the equations and on subtracbarrelg the II eqn from the I )⇒ 16x–15y = 22

6x–15y = 12⇒10x = 10

⇒ x = 1Put x = 1 in eqn 2x–5y = 4⇒ 2×1–5y = 4⇒ 5y = 2–4 = –2

⇒ y = −2

5

𝑖. 𝑒. x > y

❑ QUADRATIC EQUATION LIVE TEST

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I. 4x² – 7x – 15 = 0II. 33y = –(9y² + 18)

a) x > yb) x ≥ yc) x < yd) x ≤ ye) x = y or relationship between x and y cannot be established.

Sol.I. 4x2–7x–15=0⇒ 4x2–12x+5x-15=0⇒ 4x (x-3)+5(x-3)=0⇒ (4x+5)(x-3)=0

x=-5

4; x=3

II. 33y =–(9y2+18)⇒9y2+18+33y=0⇒ 9y2+27y+6y+18=0⇒ (y+3)(9y+6)=0

Y=–3 ; y=−6

9=

−2

3

Relationship can’t be established

❑ QUADRATIC EQUATION LIVE TEST

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I. 𝒙 +𝟑

𝟐𝒙=

𝟓

𝟐

II. 𝒚𝟐 = 𝟗

a) x > yb) x ≥ yc) x < yd) x ≤ ye) x = y or relationship between x and y cannot be established.

Sol.

I. x+3

2𝑥=

5

2

⇒2𝑥2+3

2𝑥=

5

2

⇒ 4x2+6=10x

⇒ 4x2–10x+6=0

⇒ 4x2–4x–6x+6=0

⇒ (4x-6) (x-1)=0

⇓ ⇓

x=+3

2x=1

II. 𝑦2 = 9y = – 3, + 3

no relation

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I. 𝒙 = 𝟐𝟖𝟗II. 𝒚𝟐– 𝟑𝟒𝒚 + 𝟗𝟕 =– 𝟏𝟗𝟐

a) x > yb) x ≥ yc) x < yd) x ≤ ye) x = y or relationship between x and y cannot be established.

Sol.

I. x = 289⇒ x = 17

II. y2 – 34y + 97 = –192⇒ y2 – 34y + 289 = 0⇒ (y – 17)2 ⇒ y = 17

i.e. x = y

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I. 2𝒙𝟐 + 𝟐𝟎𝒙 =– 𝟓𝟎II. 𝒚𝟐 = 𝟐𝟓

a) x > yb) x ≥ yc) x < yd) x ≤ ye) x = y or relationship between x and y cannot be established.

Sol.I. 2x2 + 20x + 50 = 0

⇒ x2 + 10x + 25 = 0 ⇒ (x + 5)2 = 0⇒ x = -5II. y2 = 25y = + 5

i. e. x < y

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I. x2 + 25x + 156 = 0

II. y2 + 28y + 195 = 0

a) x > yb) x ≥ yc) x < yd) x ≤ ye) x = y or relationship between x and y cannot be established.

sol.From I:x2 + 25x + 156 = 0x2 + 13x + 12x + 156 = 0x(x + 13) + 12(x + 13) = 0(x + 13)(x + 12) = 0x = -13, -12From II:y2 + 28y + 195 = 0y2 + 15y + 13y + 195 = 0y(y + 15) + 13(y + 15) = 0(y + 15)(y + 13) = 0y = -15, -13So, y ≤ x.

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I. x2 + 7x – 198 = 0

II. y2 + 38y + 361 = 0

a) x > yb) x ≥ yc) x < yd) x ≤ ye) x = y or relationship between x and y cannot be established.

sol.From I:x2 + 7x – 198 = 0x2 – 11x + 18x – 198 = 0x(x – 11) + 18(x – 11) = 0(x + 18)(x – 11) = 0x = -18, 11From II:y2 + 38y + 361 = 0y2 + 19y + 19y + 361 = 0y(y + 19) + 19(y + 19) = 0(y + 19)(y + 19) = 0y = -19, -19So, x > y.

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I. x2 – 31x + 228 = 0

II. y2 – 30y + 224 = 0

a) x > yb) x ≥ yc) x < yd) x ≤ ye) x = y or relationship between x and y cannot be established.

From I:x2– 31x + 228 = 0x2 – 12x – 19x + 228 = 0x(x – 12) – 19(x – 12) = 0(x – 12)(x – 19) = 0x = 12, 19From II:y2 – 30y + 224 = 0y2 – 14y – 16y + 224 = 0y(y – 14) – 16(y – 14) = 0(y – 14)(y – 16) = 0y = 14, 16So, no relation can be established between x and y.

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I. 𝟔𝟐𝟓𝒙𝟐 – 𝟑𝟕𝟓 = 𝟎

II. 𝟐𝟏𝟏𝟔𝐲 – 𝟕𝟑𝟔 = 𝟎

a) x > yb) x ≥ yc) x < yd) x ≤ ye) x = y or relationship between x and y cannot be established.

Sol.

I. 625𝑥2– 375 = 0⇒ 625𝑥2 = 3752

⇒ x = ±15

II. 2116𝑦– 736 = 0⇒ 46y = 736 ⇒ y = 16 y > x

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I. 3x² + 5x + 2 = 0 II. 12y² + 11y + 2 = 0

a) x > yb) x ≥ yc) x < yd) x ≤ ye) x = y or relationship between x and y cannot be established.

Sol.I. 3x² + 5x + 2 = 0

⇒ 3x² + 3x + 2x + 2 = 0 ⇒ 3x ( x + 1) + 2 (x + 1) = 0⇒ (3x + 2) (x + 1) = 0

𝑥 =–2

3, –1

II. 12y² + 11y + 2 = 0 ⇒ 12y² + 8y + 3y + 2 = 0 ⇒ 4y (3y + 2) + 1 (3y + 2) = 0⇒ (4y + 1) (3y + 2) = 0

𝑦 =–1

4,

–2

3

y ≥ x

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I. 16x + 11y = 154II. 11x + 16y = 89

a) x > yb) x ≥ yc) x < yd) x ≤ ye) x = y or relationship between x and y cannot be established.

Sol.16x + 11y = 15411x + 16y = 89 On solving these two equationsx =11 and y = – 2 x > y

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I. 3x² + 14x – 5 = 0 II. 3y² – 19y + 6 = 0

a) x > yb) x ≥ yc) x < yd) x ≤ ye) x = y or relationship between x and y cannot be established.

Sol.I. 3x² + 14x – 5 = 0

⇒ 3x² + 15x – x – 5 = 0 ⇒ 3x (x + 5) – 1 (x + 5) = 0 ⇒ (3x – 1) (x + 5) = 0

x = 1

3, –5

II. 3y² – 19y + 6 = 0 ⇒ 3y² – 18y – y + 6 = 0 ⇒ 3y ( y – 6) – 1 (y – 6) = 0⇒ (3y – 1) (y – 6) = 0

𝑦 =1

3, 6

𝑦 ≥ 𝑥

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I. 7x² – 11x – 6 = 0 II. 7y² – 10y + 3 = 0

a) x > yb) x ≥ yc) x < yd) x ≤ ye) x = y or relationship between x and y cannot be established.

Sol.I. 7x² – 11x – 6 = 0

⇒ 7x² – 14x + 3x − 6 = 0 ⇒ 7x (x – 2) + 3 (x – 2) = 0 ⇒ (7y + 3) (x – 2) = 0

𝑥 =–3

7, 2

II. 7y² – 10y + 3 = 0 ⇒ 7y² – 7y – 3y + 3 = 0 ⇒ 7y (y – 1) – 3 (y – 1) = 0 ⇒ (7y – 3) (y – 1) = 0

y =3

7, 1

No relation can be established between x and y

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(i) x² – 12x + 32 = 0

(ii) y² – 20y + 96 = 0

a) x > yb) x ≥ yc) x < yd) x ≤ ye) x = y or relationship between x and y cannot be established.

Sol.

(i) x² – 12x + 32 = 0

x² – 8x – 4x + 32 = 0

x (x – 8) – 4 (x – 8) = 0

(x – 8) (x – 4) = 0

x = 8, 4

(ii) y² – 20y + 96 = 0

y² – 12 y – 8y + 96 = 0

y(y – 12) – 8 (y – 12) = 0

(y – 8) (y – 12) = 0

y = 8, 12

y ≥ x

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(i) 2x² – 3x – 20 = 0

(ii) 2y² + 11y + 15 = 0

a) x > yb) x ≥ yc) x < yd) x ≤ ye) x = y or relationship between x and y cannot be established.

Sol.

(i) 2x² – 3x – 20 = 0

2x² – 8x + 5x – 20 = 0

2x (x – 4) + 5(x – 4) = 0

(x – 4) (2x + 5) = 0

x = 4, –5

2

(ii) 2y² + 11y + 15 = 0

2y² + 6y + 5y + 15 = 0

2y (y + 3) + 5 (y + 3) =0

(2y + 5) (y + 3) =0

y = –5

2, –3

x ≥ y

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(i) x² – x – 6 = 0

(ii) y² – 6y + 8 = 0

a) x > yb) x ≥ yc) x < yd) x ≤ ye) x = y or relationship between x and y cannot be established.

Sol.

(i) x² – x – 6 = 0

x² – 3x + 2x – 6 = 0

x (x – 3) +2 (x – 3) = 0

(x – 3) (x + 2) = 0

x = 3, –2

(ii) y² – 6y + 8 = 0

y² – 2y – 4y + 8 = 0

y (y – 2) – 4 (y – 2) = 0

(y – 2) (y – 4) = 0

y = 2, 4

No relation can be established between x and y

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(i) x² + 14x - 32 = 0

(ii) y² – y – 12 = 0

a) x > yb) x ≥ yc) x < yd) x ≤ ye) x = y or relationship between x and y cannot be established.

Sol.

(i) x² + 14x - 32 = 0

x² + 16x -2x - 32 = 0

x (x +16) -2 (x +16) = 0

(x -2) (x + 16) = 0

x = –16, 2

(ii) y² – y – 12 = 0

y² – 4y + 3y – 12 = 0

y(y – 4) + 3(y – 4) = 0

(y + 3) ( y – 4) = 0

y = – 3, 4

No relation can be established between x and y

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(i) x² – 9x + 20 = 0

(ii) 2y² – 12y + 18 = 0

a) x > yb) x ≥ yc) x < yd) x ≤ ye) x = y or relationship between x and y cannot be established.

Sol.

(i) x² – 9y + 20 = 0

x² – 5y – 4y + 20 = 0

x (x – 5) –4 (y – 5) = 0

(x – 4) (x – 5) = 0

x = 4, 5

(ii) 2y² – 12y + 18 = 0

2y² – 6y – 6y + 18 = 0

2y (y – 3) –6 (y – 3) =0

(2y – 6) (y – 3) = 0

y = 3, 3

x > y

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I. 𝟏𝟒𝒙 − 𝟐𝟓 = 𝟓𝟗 − 𝟕𝒙

II. 𝒚 + 𝟐𝟐𝟐 − 𝟑𝟔 = 𝟖𝟏

a) x > yb) x ≥ yc) x < yd) x ≤ ye) x = y or relationship between x and y cannot be established.

Sol. 𝑥 = 4𝑦 = 3

∴ 𝑥 > 𝑦

S52. Ans.(e)

Sol. 𝑥 = 4, 1.8

𝑦 = −1.5, 1.8

∴ 𝑥 ≥ 𝑦

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I. 5x2 – 29x + 36 = 0

II. 10y2 – 3y – 27 =0

a) x > yb) x ≥ yc) x < yd) x ≤ ye) x = y or relationship between x and y cannot be established.

Sol. 𝑥 = 4, 1.8

𝑦 = −1.5, 1.8

∴ 𝑥 ≥ 𝑦

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I. 𝐱 −𝟔

𝐱= 𝟎

II. 𝐲𝟑 − 𝟔 𝟑⁄𝟐 = 𝟎

a) x > yb) x ≥ yc) x < yd) x ≤ ye) x = y or relationship between x and y cannot be established.

Sol. I. 𝑥 2 = 6

or, 𝑥 = 6

II. 𝑦3 = 63

2

or, 𝑦 = 6∴ x=y

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I. 3x2+5x−𝟖=0II. y2−𝟒y+3=0

a) x > yb) x ≥ yc) x < yd) x ≤ ye) x = y or relationship between x and y cannot be established.

Sol. 3𝑥2 + 8𝑥 − 3𝑥 − 8 = 0𝑥 3𝑥 + 8 − 1 3𝑥 + 8 = 0

𝑥 = 1,−8

3

𝑦2 − 𝑦 − 3𝑦 + 3 = 0𝑦 𝑦 − 1 − 3 𝑦 − 1 = 0𝑦 = 1, 3𝑥 ≤ 𝑦

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𝐈. 𝐱𝟐 + 𝐲𝟐 = 𝟑𝟑𝟒𝟏II. 𝐱𝟐−𝐲𝟐 = 𝟖𝟗𝟏

a) x > yb) x ≥ yc) x < yd) x ≤ ye) x = y or relationship between x and y cannot be established.

Sol. On adding, 2𝑥2 = 4232Or, x = 46,-46On subtracting, 2𝑦2 = 2450Or, y = 35,-35No relation

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I. 𝐱𝟐 − 𝟕𝐱 + 𝟏𝟐 = 𝟎𝐈𝐈. 𝐲𝟐 − 𝟖𝐲 + 𝟏𝟐 = 𝟎

a) x > yb) x ≥ yc) x < yd) x ≤ ye) x = y or relationship between x and y cannot be established.

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𝐈. 𝟐𝐱𝟐 + 𝐱 − 𝟐𝟖 = 𝟎𝐈𝐈. 𝟐𝐲𝟐 − 𝟐𝟑𝐲 + 𝟓𝟔 = 𝟎

a) x > yb) x ≥ yc) x < yd) x ≤ ye) x = y or relationship between x and y cannot be established.