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## Algebra: Functions Test-4

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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( x-y,y-z \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)

= (2)

=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

={((I # 1) # 2) - (10

={((I # 1) # 2) - (10

={((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)

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 arepositive 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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