Algebra For SBI PO : Set – 32
D.1-5) : In each of these questions, two equations (I) and (II) are given. You have to solve both the equations and give answer
(a) if x > y
(b) if x ≥ y
(c) if x < y
(d) if x ≤ y
(e) if x = y or no relation can be established between x and y.
1) I. 3x2 – 29x + 56 = 0
II. 3y2 – 5y – 8 = 0
(a) x > y
(b) x ≥ y
(c) x < y
(d) x ≤ y
(e) x = y or the relationship between ‘x’ and ‘y’ cannot be established.
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(b)
3x2 – 29x + 56 = 0
or 3x2 – 21x – 8x + 56 = 0
or 3x(x – 7) – 8(x – 7) = 0
or (3x – 8) (x – 7) = 0
x = , 7
II. 3y2 – 5y – 8 = 0
or 3y2 + 3y – 8y – 8 = 0
or 3y(y + 1) – 8(y + 1) = 0
or (3y – 8) (y + 1) = 0
or (3y – 8) (y + 1) = 0
y = -1,
x≥ y
2) I. 5x2 + 26x – 24 = 0
II. 5y2 – 34y + 24 = 0
(a) x > y
(b) x ≥ y
(c) x < y
(d) x ≤ y
(e) x = y or the relationship between ‘x’ and ‘y’ cannot be established.
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(d)
5x2 + 26x – 24 = 0
or 5x2 + 30x – 4x – 24 = 0
or 5x(x + 6) – 4(x + 6) = 0
or (5x – 4) (x + 6) = 0
x = 4/5, 6
II. 5y2 – 30y – 4y + 24 = 0
or 5y(y – 6) – 4(y – 6) = 0
or (5y – 4) (y – 6) = 0 4
y =4/5 6
x≤y
3) I. x2– 7x = 0
II. 2y2 + 5y + 3 = 0
(a) x > y
(b) x ≥ y
(c) x < y
(d) x ≤ y
(e) x = y or the relationship between ‘x’ and ‘y’ cannot be established.
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(a)
x2 – 7x = 0
or x (x – 7) = 0
x = 0, 7
II. 2y2 + 5y + 3 = 0
or 2y2 + 2y + 3y + 3 = 0
or 2y(y + 1) + 3(y + 1) = 0
or (2y + 3) (y + 1) = 0
y = -1, -3/2
x > y
4) I. 7x – 4y = 40
II. 8x + 8y = 8
(a) x > y
(b) x ≥ y
(c) x < y
(d) x ≤ y
(e) x = y or the relationship between ‘x’ and ‘y’ cannot be established.
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(a)
7x – 4y = 40 …(i)
and 8x + 8y = 8
or x+y= 1 …(ii)
Solving (i) and (ii), we have
x = 4, y = -3
x > y
5) I. 15x2 – 41x + 14 = 0
II. 2y2 – 13y + 20 = 0
(a) x > y
(b) x ≥ y
(c) x < y
(d) x ≤ y
(e) x = y or the relationship between ‘x’ and ‘y’ cannot be established.
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(c)
15x2 – 4x + 14 = 0
or 15x2 – 6x – 35x + 14 = 0
or 3x(5x – 2) – 7(5x – 2) = 0
or (3x – 7)(5x – 2) = 0
x = 7/3, 2/5
II. 2y2 – 13y + 20 = 0
or 2y2 – 8y – 5y + 20 = 0
or 2y(y – 4) – 5(y – 4) = 0
or (2y – 5) (y – 4) = 0
y = 4, 5/2
x < y
D.6-10) : In each of these questions, two equations (I) and (II) are given. You have to solve both the equations and give answer.
(a) if x > y
(b) if x ≥ y
(c) if x < y
(d) if x ≤ y
(e) if x = y or no relation can be established between x and y.
6) I. x2-8√3x+45=0
II. y2-√2y-24=0
(a) if x > y
(b) if x ≥ y
(c) if x < y
(d) if x ≤ y
(e) if x = y or no relation can be established between x and y.
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(e)
I.x2 -8√3x + 45= 0
or x2 – 5√3x + 3√3 (x – 5√3) = 0
or, (x + 3√3) (x – 5√3) = 0
X = 3√3, 5√3
II. y2 – √2y – 24 = 0
Or y2 – 4√2y + 3 √2y – 24 = 0
Or (y-4√2y) (y + 2√2)
y = -3 √2, 4√2
Hence relation cannot be established between x and y.
7) x-7√2x+24=0
II. y-5√2y+12=0
(a) if x > y
(b) if x ≥ y
(c) if x < y
(d) if x ≤ y
(e) if x = y or no relation can be established between x and y.
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x – 7 √2x + 24 = 0
Or x – 4√2x – 3 √2x + 24 = 0
Or √x (√x – 4√2) – 3√2 (√x – 4√2) = 0
Or (√x – 3√2) (√x – 4√2) = 0
Now, if √x -3√2 = 0
then √x = 3√2
x = 9 × 2 = 18
If √x – 4√2 = 0
then √x = 4√2
x = 16 × 2 = 32
II. y – 5√2y + 12 = 0
y -3√2y – 2√2y + 12 = 0
Or √y (√y – 3√2) -2√2y + 12 = 0
Or (√y – 2√2) – (√y -3√2) = 0
If (√y -2√2) = 0
Then √y = 2√2
y = 4 2 = 18
If √y -3√2 = 0
Then, √y -√2
y = 9 × 2 = 18
x ≥ y
8. I. 12x2 – 17x + 6 = 0
II. 20y2 – 31y + 12 = 0
(a) if x > y
(b) if x ≥ y
(c) if x < y
(d) if x ≤ y
(e) if x = y or no relation can be established between x and y.
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(d)
12x2 – 17x + 6 = 0
or 12x2 – 9x – 8x + 6 = 0
or 3x(4x – 3) – 2(4x – 3) = 0
or (3x – 2) (4x – 3) = 0
If 3x – 2 = 0
then 3x = 2
x= 2/3
If 4x – 3 = 0
then x = 3/4
II. 20y2 – 31y + 12 = 0
or 20y2 – 16y – 15y + 12 = 0
or 4y (5y – 4) – 3 (5y – 4) = 0
or (4y – 3) (5y – 4) = 0
y=3/4,4/5
Hence x ≤ y
9. I. 3x2 – 8x + 4 = 0
II. 4y2 – 15y + 9 = 0
(a) if x > y
(b) if x ≥ y
(c) if x < y
(d) if x ≤ y
(e) if x = y or no relation can be established between x and y.
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(e)
3x2 – 8x + 4 = 0
or 3x2 – 6x – 2x + 4 = 0
or (3x – 2) (x – 2) = 0
x=2,2/3
II. 4y2 – 15y + 9 = 0
or 4y2 – 12y – 3y + 9 = 0
or 4y(y – 3) – 3(y – 3) = 0
or (4y – 3) (y – 3) = 0
y = 3/4, 3
Relation cannot be established between x and y.
10. I. x2 -16x + 63 = 0
II. y2 – 2y – 35 = 0
(a) if x > y
(b) if x ≥ y
(c) if x < y
(d) if x ≤ y
(e) if x = y or no relation can be established between x and y.
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(a)
I. x2 – 16x + 63 = 0
or x2 – 9x – 7x + 63 = 0
or x(x – 9) – 7(x – 9) = 0
or (x – 7) (x – 9) = 0
x = 7, 9
II. y2 – 2y – 35 = 0
or y2 – 17y + 5y – 35 = 0
or y(y – 7) + 5(y – 7) = 0
or (y + 5) (y – 7) = 0
y = -5, 7
Hence, x ≥ y