CAT 2004 | Question: 61

Question

Answer the questions on the basis of the information given below:

• $f_{1}(x) = \left\{\begin{matrix} x & 0 \leq x \leq 1 \\  1 & x \geq 1 \\ 0 & \text{otherwise} \end{matrix}\right.$  
• $f_{2}(x) =  f_{1}(-x) \;\; \text{for all} \; x $
• $f_{3}(x) =  -f_{2}(-x) \;\; \text{for all} \; x $
• $f_{4}(x) =  f_{3}(-x) \;\; \text{for all} \; x $

 How many of the following products are necessarily zero for every $x: f_1(x)f_2(x), \: f_2(x)f_3(x), \: f_2(x)f_4(x)$

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Answer 1

Q1. 

  f1(x).f2(x) = 0 for all x, because f1(x) = 0 for x < 0. and f2(x) = 0 for x >=0.

 f2(x). f3(x) = -1 for x <=-1. 

                   = -x^2 for  -1<x<= 0

                   = 0 , x >0.

f2(x).f4(x) = 0 for all x, because f2(x) = 0 for x >0 and f4(x) = 0 for x <=0.

Ans - Option 3. 2 

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f2(x)=f1(−x)
f3(x)=− f2(x) = − f1(−x)
f4(x)= f3(−x) = − f2(−x) = − f1(x)

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2. Which of the following is necessarily true?

1. f4(x)=f1(x)      for all x.   false -    { f4(x)= - f1(x) }  
2. f1(x)=f3(−x)    for all x.   false -    { f1(x)= - f3(−x) }
3. f2(−x)=f4(x)    for all x.   false.     { f2(−x)= - f4(x) }
4. f1(x)+f3(x)=0  for all x.   false .    { f1(x)= - f3(−x) }

Ans - None.

Thank you Arjun sir for correction. 

 

Comments

$f_2(x) = f_1(-x)$

$f_3(x) = -f_2(x) = -f_1(-x)$

$f_4(x) = f_3(-x) = -f_2(-x) = -f_1(x)$