Exercise Problem and Solution Part: 2 (Equation, Solving Equations, Systems of liner equations, Quadratic equations, Inequalities)

Question: 8 (a) Find the value of a if (a - 3)  = 10/a
                    (b) Find the value of X if 2X ² - 1 = 3X ² - 2X

Explore: (a )
(a-3) = 10/a
=> a(a-3) = 10
=> a ² -3a = 10
=> a ²-3a - 10 = 0
=> a ² -5a + 2a - 10 = 0
=> a(a - 5) + 2(a - 5) = 0
=> (a -5) (a + 2) = 0
:. a - 5 = 0  Or, a + 2 = 0
=> a = 5     Or, => a = - 2
Answer: a = 5, - 2

(b) 2X ²- 1 = 3X ² - 2X
=>  2X ² -1 -3X ²  + 2X = 0
=> - X ² + 2X - 1 = 0
=> X ² -2X + 1 = 0
=> X ² - 2.X.1 + 1 ²= 0
=> (X -1) <sup>2
=> X = 1
Answer: X = 1

Question: 9 If X + Y = 1 & X - Y = 5, Then what  X is equal to ?
Option: (a) 2 (b) 4 (c) 5 (d) 3 (e) None
Explore:

X+Y = 1
X -Y = 5
----------- [ Adding the equations]
:. 2X = 6
:. X =3

Answer: X = 3

Question: 10 If 2X - Y = 4 then 6X - 3y = ?
Option: (a) 4 (b) 6 (c) 8 (d) 10 (e) 12

Explore: At first it may occur to you that since we have two variables ( X,Y) we need two given equations to solve this problem. B ut not that you have not been asked to find out the values of X & Y. If you observe carefully, you will find that.
 
     3 x (2X -Y) = 6X - 3Y
Or, 3 x 4 - 6X - 3Y
Or, 6X - 3Y = 12

Answer: (e)

Question: 11 If r = 12 & r/s = 24/Y, then /s = ?
Option: (a) 0.5 (b) 2 (c) 3 (d) 4 (e) None

Explore: r/s = 24/Y=> 12/s = 24/Y
        Or, y/s = 24/12 = 2

Answer:  (b)

Question: 12 If (3 - X)/ (1 + X) = 1, then X = ?
Option: (a) 10 (b) 1 (c) 2 (d) -2 (e) None

Explore: 
     (3 - X)/(1 + X) = 1
Or, 3 - X = 1 + X 
Or, 3 - 1 = 2X
Or, 2X = 2
Or, X = 1

Answer: (b)

Question: 13 If X/Y = -1 and XY = -1, then X - Y =?
Option: (a) 2 (b) 1 (c) 0 (d) -2 (e) None

Explore: 
X/Y = -1 => X = - Y
XY = -1 => (- Y) .Y = - 1
Or,</sup> y ²  = 1
^{:. Y = 1}
 
Question: 14 If Y ² + XY + 1 = 16 and X - Y = 1 then X + Y = ?
Option: (a) -6 (b) 5 (c) 4 (d) 5 or -6 (e) None 

 Explore: X - Y = 1 Or, X = 1 + Y
The other equation says
X ² + XY + 1 = 16
Or, (1 + Y) ² + (1 + Y)Y + 1 = 16
Or 1 + 2Y + y ² + Y + y ² + 1 = 16
Or, 2y ² +3y - 14 = 0
Or, 2y ² + 7Y - 4Y - 14 = 0
Or y(2Y + 7) - 2(2Y + 7) = 0
Or, (2Y + 7)(Y - 2) = 0
:. 2Y + 7 = 0 => Y = -(7/2)
Or, y - 2 = 0 => y = 2
:. X = 1 + Y = 1 + 2 = 3 and X = 1 + Y = 1 - (7/2) = -(5/2)
:. X + Y = 3 + 2 = 5 and X + Y = -(5/2) -(7/2) = - 6
Answer: (d)

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