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Algebra: Functions Test-3

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Question 1
Largest value of
$ \displaystyle \min \left( 2+{{x}^{2}},6-3x \right)when\,\,x>0$
A
1
B
2
C
3
D
4
Question 1 Explanation: 
$ \displaystyle \begin{array}{l}Equating\,\,\,\,2+{{x}^{2}}=6-3x\\\Rightarrow \,\,\,{{x}^{2}}+3x-4=0\Rightarrow {{x}^{2}}+4x-x-4=0\\or\,\,\,\left( x+4 \right)\,\left( x-1 \right)=0\\\Rightarrow \,x=-4\,\,\,or\,\,\,1\\But\,\,\,x>0\,\,\,so\,\,\,x=1,\,\,\,so\,\,\,LHS=RHS=2+1=3\end{array}$
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Therefore we can conclude that the largest value of the function when x>0 occurs at x=1.
The value of y for x=1 is 3. Therefore option (c).
Question 2
A, S, M and D are functions of x and y, and they are defined as follows:
$ \displaystyle \begin{array}{l}A\,\left( x,y \right)=x+y\\S\,\left( x,y \right)=x-y\\M\,\left( x,y \right)=xy\\D\,\left( x,y \right)=x/y\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,where\,y\ne 0.\end{array}$
What is the value of
$ \displaystyle M\,\left( M\,(A\,(M\,\left( x,y \right),S\left( y,x \right),x),A\,\left( y,x \right) \right)\,\,\,\,for\,\,\,x=2,\,y=3$
A
50
B
140
C
25
D
70
Question 2 Explanation: 
$ \displaystyle \begin{array}{l}M\,\left( M\,(A\,(M\,\left( x,y \right),S\left( y,x \right),x),A\,\left( y,x \right) \right)\\M\left( M\left( A\left( 6,1 \right),2 \right),A\left( 3,2 \right) \right)\\M\left( M\left( 7,2 \right),A\left( 3,2 \right) \right)\\M\left( 14,5 \right)=70\end{array}$
Question 3
A, S, M and D are functions of x and y, and they are defined as follows: $ \displaystyle \begin{array}{l}A\,\left( x,y \right)=x+y\\S\,\left( x,y \right)=x-y\\M\,\left( x,y \right)=xy\\D\,\left( x,y \right)=x/y\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,where\,y\ne 0.\end{array}$
What is the value of
$ \displaystyle S\,\left[ M\,\left( D\,(A\,\left( a,b \right),2 \right),\,D\,\left( A\,\left( a,b \right),2) \right),\,(M(D\,(S\left( a,b \right),2),\,D(S\,\left( a,b \right),2))\, \right]$  
A
a2+b2
B
ab
C
a2-b2
D
a/b
Question 3 Explanation: 
$ \displaystyle \begin{array}{l}S\,\left[ M\,\left( D\,(A\,\left( a,b \right),2 \right),\,D\,\left( A\,\left( a,b \right),2) \right),\,(M(D\,(S\left( a,b \right),2),\,D(S\,\left( a,b \right),2))\, \right]\\=S\,\left[ M\,\left( a+b)/2 \right),\,\left( a+b)/2) \right),\,(M((a-b)/2,\,(a-b)/2))\, \right]\\=S\,\left[ {{(a+b)}^{2}}/4,{{(a-b)}^{2}}/4\, \right]\\={{(a+b)}^{2}}/4+{{(a-b)}^{2}}/4\,\\=4ab/4\\=ab\end{array}$
Question 4
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}$
Given that $ \displaystyle x>y>z>0$
Which of the following is necessarily true?
A
$ \displaystyle la\,\left( x,y,z \right)<\,le\,\left( x,y,z \right)$
B
$ \displaystyle ma\,\left( x,y,z \right)<\,la\,\left( x,y,z \right)$
C
$ \displaystyle ma\,\left( x,y,z \right)<\,le\,\left( x,y,z \right)$
D
None of these
Question 4 Explanation: 
Taking values as x= 40, y =30, z=2.
la(x,y,z)= min (70,32) =32
le(x,y,z)=max (10,28)=28
ma(x,y,z)=1/2[32+28]=1/2[60]=30
Options (a) and (c) can be easily eliminated by taking random value of x and y (close and very much larger than z).
Taking values as x= 100, y =2, z=1.
la(x,y,z)= min (102,3) =3
le(x,y,z)=max (98,1)=98
ma(x,y,z)=1/2[3+98]=1/2[101]=55.5
Here when we take a >> b>c we find that
ma(x,y,z) > la(x,y,z)= min (102,3) =3
Question 5
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}$
What is the value of
$ \displaystyle ma\,\left( 10,\,4,\,le\,(\,la\,\left( 10,5,3 \right),5,3) \right)$
A
7.0
B
6.5
C
8.0
D
7.5
Question 5 Explanation: 
=$\displaystyle ma\,\left( 10,\,4,\,le\,(\,\min \,\left( 15,8 \right),5,3) \right)$
= ma (10,4, max (8-5,5-3))
= ma (10,4, max (3,2))
= ma (10,4,3)
=1/2 [{la (10,4,3)}+(le (10,4,3)}]
=1/2 [{min (14,7)}+(max(6,1)}]
=1/2 [7+6]
=6.5
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