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Algebra: Functions Test-5
Question 1 |
If x and y are 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:
If Y = which of the following will give x2 as the final value
f(x, y) G (x, y) 4 | |
G (f (x, y)f(x, y)) F (x, y)/8 | |
- F (x, y) G (x, y)/log216 | |
– f(x, y) G (x, y) F (x, y)/F (3x, 3y) |
Question 1 Explanation:
−F(x,y).G(x,y)=−[−|x+y|.|x+y|]−(−2x.2x)log216=4x2∵[log216=log224=4]thereforeoptioncistherightanswer
Question 2 |
Using the relation answer the questions given below
@ (A, B) = average of A and B
/(A, B) = product of A and B,
x (A, B) = the result when A is divided by B
The sum of A and B is given by
\ (@ (A, B), 2) | |
@ ( \ (A, B), 2) | |
@ (X (A, B), 2) | |
none of these |
Question 2 Explanation:
Going through the various options
∖((A,B),2)=∖(A+B2,2)=A+B2×2=A+B
∖((A,B),2)=∖(A+B2,2)=A+B2×2=A+B
Question 3 |
Using the relation answer the questions given below
@ (A, B) = average of A and B
/(A, B) = product of A and B,
x (A, B) = the result when A is divided by B
The average of A, Band C is given by
@ (x (\ (@ (A, B), 2), C), 3) | |
\ (x (\ (@ (A, B)), C 2)) | |
X (@ (\ (@ (A, B), 2), C,3)) | |
X (\ (@ (\ (@ (A, B), 2), C), 2), 3) |
Question 3 Explanation:
(a) (x (∖ ( (A, B), 2), C), 3)= (x (∖(A+B2,2),C), 3)= (x (A+B, C), 3)=(A+BC,3)(b) ∖ (x (∖ ( (A, B)), C 2))= ∖ (x(∖(A+B2,C2)))=∖(x(A+B2×C2))(c)X((∖((A,B),2),C,3))=X((∖(A+B2,2),C,3))=X((A+B,C,3))=X(A+B+C2,3)=A+B+C6(d)X(∖((∖((A,B),2),C),2),3)=X(∖((∖((A,B),2),C),2),3)=X(∖((∖(A+B2,2),C),2),3)=X(∖((A+B,C),2),3)=X(∖(A+B+C2,2),3)=X(∖(A+B+C,3))=A+B+C3
Question 4 |
x and yare non-zero real numbersf (x, y) = + (x +y)0.5,if (x +y)0.5is real otherwise = (x +y)2g (x, y) = (x +y)2 if (x + y)0.5is real, otherwise =- (x +y)For which of the following is f (x, y) necessarily greater than g (x, y)option
x and y are positive | |
x and y are negative | |
x and y are greater than - 1 | |
None of these |
Question 4 Explanation:
When both x and y are positive
f (x, y) = + (x +y)0.5g (x, y) = (x +y)2 so g(x,y)>f(x,y)
When both x and y are negative
f(x, y) = (x +y)2g (x, y) = -(x +y)
If x+y>-1
g(x,y)>f(x,y)
When x and y are greater than -1. They can be both positive as well as in the range 0 to -1
so both a and b option cases arise.
f (x, y) = + (x +y)0.5g (x, y) = (x +y)2 so g(x,y)>f(x,y)
When both x and y are negative
f(x, y) = (x +y)2g (x, y) = -(x +y)
If x+y>-1
g(x,y)>f(x,y)
When x and y are greater than -1. They can be both positive as well as in the range 0 to -1
so both a and b option cases arise.
Question 5 |
x and yare non-zero real numbers
f (x, y) = + (x +y)0.5,if (x +y)0.5is real otherwise = (x +y)2
g (x, y) = (x +y)2 if (x + y)0.5is real, otherwise =- (x +y)
Which of the following is necessarily false?
f(x,y) >g(x,y) for 0 | |
f(x,y) > g (x,y) when x,y<-1 | |
f (x, y) > g (x, y) for x, y > 1 | |
None of these |
Question 5 Explanation:
In this case ,f (x, y) = + (x +y)0.5 g (x, y) = (x +y)2
This is true cause (x+y)<1 for which (x +y)0.5>(x +y)2
In this case, f(x, y) = (x +y)2 g (x, y) = -(x +y)
This is also true cause for (x+y)<-2 (x +y)2>-(x+y)
In this case f (x, y) = + (x +y)0.5 g (x, y) = (x +y)2
This is false because for(x+y)>2 (x +y)2>(x +y)0.5
So the answer is c
This is true cause (x+y)<1 for which (x +y)0.5>(x +y)2
In this case, f(x, y) = (x +y)2 g (x, y) = -(x +y)
This is also true cause for (x+y)<-2 (x +y)2>-(x+y)
In this case f (x, y) = + (x +y)0.5 g (x, y) = (x +y)2
This is false because for(x+y)>2 (x +y)2>(x +y)0.5
So the answer is c
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