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Algebra: Functions Test4
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Question 1 
The following functions have been defined:
$ \displaystyle \begin{array}{l}la\,\,\left( x,y,z \right)\,=\min \,\left( x+y,y+z \right)\\le\,\,\left( x,y,z \right)\,=\max \,\left( xy,yz \right)\\ma\,\,\left( x,y,z \right)\,=\left( 1/2 \right)\,\left[ le\,\left( x,y,z \right)+la\,\left( x,y,z \right) \right]\end{array}$
For x = 15, y = 10 and z = 9, find the value of : le (x, min (y,x  z), le (9, 8, ma (x, y, z)))
For x = 15, y = 10 and z = 9, find the value of : le (x, min (y,x  z), le (9, 8, ma (x, y, z)))
5  
12  
9  
4 
Question 1 Explanation:
Segregating and simplifying,
ma (15, 10, 9)= 1/2 [{le (15,10,9)}+(la (15,10,9)}]
=1/2 [{max (5,1)}+(min (25,19)}]
=1/2 [5+19]
=12
le (x, min (y,x  z), le (9, 8, 12))
= le (15, min (10,6), le (9, 8, 12))
= le (15, 6, max(1,4))
= le (15, 6,1)
= max (9,5)
= 9
ma (15, 10, 9)= 1/2 [{le (15,10,9)}+(la (15,10,9)}]
=1/2 [{max (5,1)}+(min (25,19)}]
=1/2 [5+19]
=12
le (x, min (y,x  z), le (9, 8, 12))
= le (15, min (10,6), le (9, 8, 12))
= le (15, 6, max(1,4))
= le (15, 6,1)
= max (9,5)
= 9
Question 2 
The following operations are defined for real numbers a # b = a + b if a and b both are positive else a # b = 1 .a∇b = (ab)^{a+ b} if ab is positive else a∇b = 1.
(2 # 1)/(1∇2) =
1/8  
1  
3/8  
3 
Question 2 Explanation:
Segregating and simplifying,
(1∇2)
= (1 x 2)^{1+ 2} , since both are positive.
= (2)^{3}
=8 .
2#1= 2+1 , since both are positive.
=3
Therefore the value of the given expression is 3/8.
(1∇2)
= (1 x 2)^{1+ 2} , since both are positive.
= (2)^{3}
=8 .
2#1= 2+1 , since both are positive.
=3
Therefore the value of the given expression is 3/8.
Question 3 
The following operations are defined for real numbers a # b = a + b if a and b both are positive else a # b = 1 .a∇b = (ab)<sup>a+ b</sup> if ab is positive else a∇b = 1.
{((I # 1) # 2)  (10<sup>1.3</sup> ∇log<sub>10</sub> 0·1)}/(1∇2) =
3/8  
4log_{10}0·1/8  
(4+10^{1.3})/8  
None of these 
Question 3 Explanation:
Segregating and simplifying,
{((I # 1) # 2)  (10^{1.3} ∇log_{10} 0·1)}/(1∇2)
={((I # 1) # 2)  (10^{1.3} ∇log_{10} 0·1)}/(8)
={((I # 1) # 2)  (10^{1.3} ∇1)}/(8)
={((I # 1) # 2)  1}/(8)
={(2 # 2)  1}/(8)
={4  1}/(8)
=3/8
{((I # 1) # 2)  (10^{1.3} ∇log_{10} 0·1)}/(1∇2)
={((I # 1) # 2)  (10^{1.3} ∇log_{10} 0·1)}/(8)
={((I # 1) # 2)  (10^{1.3} ∇1)}/(8)
={((I # 1) # 2)  1}/(8)
={(2 # 2)  1}/(8)
={4  1}/(8)
=3/8
Question 4 
The following operations are defined for real numbers a # b = a + b if a and b both are positive else a # b = 1 .a∇b = (ab)<sup>a+ b</sup> if ab is positive else a∇b = 1.
. ((X #  Y)/( X∇Y)) =3/8, then which of the following must be true ?
X = 2, Y= 1  
X> 0, Y< 0  
X, Y both positive  
X, Y both negative

Question 4 Explanation:
Checking by options:
a) doesn't satisfy. Incorrect option
b) Absolutely possible since for no real value of x denominator would be able to take 8/3 as
it is expressed as (ab)^{a+b} but x+y can take a positive value 3/8 if both are
positive which can happen only when y <0. Correct option. Checking other.
c) X, Y both positive , doesn’t satisfy. Incorrect option
d) X, Y both negative. Then both numerator and denominator is 1.
a) doesn't satisfy. Incorrect option
b) Absolutely possible since for no real value of x denominator would be able to take 8/3 as
it is expressed as (ab)^{a+b} but x+y can take a positive value 3/8 if both are
positive which can happen only when y <0. Correct option. Checking other.
c) X, Y both positive , doesn’t satisfy. Incorrect option
d) X, Y both negative. Then both numerator and denominator is 1.
Question 5 
If x and yare real numbers, the functions are defined as f(x, y) = I x + Y I, F (x, y) =  f (x, y) and G (x, y) =  F (x, y). Now with the help of this information answer the following questions:
Which of the following will be necessarily true
G( f(x, y), F (x, y))> F (f(x, y), G (x, y))  
F (F (x, y), F (x, y)) = F (G (x, y), G (x, y))  
F (G (x, y), (x + y) ≠ G (F (x, y), (x  y)))  
f (f(x, y), F (x  y)) = G (F (x, y),f (x  y)) 
Question 5 Explanation:
Take some values of x and y and put in the given expression find which satisfies the answer choices.
Going by option elimination.
(a) will be invalid when$ \displaystyle x+y=0$
(b) is the correct option as both sides $ \displaystyle 2\,\left \,\,x+y\,\,\, \right$as the result.
(c) will be equal when$ \displaystyle \left( x+y \right)=0$
(d) is not necessarily equal (plug values and check)
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