GATE2014 EC-2: GA-7

Question

Lights of four colors (red, blue, green, yellow) are hung on a ladder. On every step of the ladder there are two lights. If one of the lights is red, the other light on that step will always be blue. If one of the lights on a step is green, the other light on that step will always be yellow. Which of the following statements is not necessarily correct? 

• A. The number of red lights is equal to the number of blue lights 
• B. The number of green lights is equal to the number of yellow lights 
• C. The sum of the red and green lights is equal to the sum of the yellow and blue lights 
• D. The sum of the red and blue lights is equal to the sum of the green and yellow lights

Answer 1

Question says "On every step of ladder there are $2$ bulbs". 

"If one is Green other must be Yellow". So one pair of lights possible = $\color{maroon} {(G,Y)}$

"If one is Red other must be Blue". Other pair possible = $\color{maroon} {(R,B)}$

So, from the above statements, we can conclude that there will be no pair as $(G,B)$ or $(G,R)$ or $(R,Y)$. But, the pair $\color{maroon} {(Y,B)}$ is a possible combination of lights in one of the ladder.

If we consider that there are $3$ steps in ladder as $(G,Y)$ , $(B,Y)$ and $(B,Y)$.

NONE OF THE ABOVE OPTION MATCHES (Official key was D though)

Comments

According to question,

R ⇒ B or G ⇒ Y

it means same colour of lights can also be hung on ladder, i.e (B, B) or (Y, Y) in addition to (R, B) and (G, Y). And this condition eliminates all option.

Note - (R, R) and (G, G) pair cant be hung.

Answer 2

The answer should be D.

Comments

Seems the best pick. But I don't see why even the others must be necessarily true. What if two lights are "blue" on one step.

Answer 3

(D) is not necessarily true.

here red light= blue light

and green light= yellow light

Comments

can't we have both blue colors? or blue and yellow?

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No we cannot have.

because

in 1 step a ladder has 2 lights

among that 2 light if one is red other must be blue.

So, both blue colors never possible .

it must be red = blue (in a 2 light ladder)

 

Next blue = yellow might be possible.(but not always)

Say no. of light red color = m (say)

then  "   "     "     blue color = m

Now, "    "     "   green  "   = n (say)

       "     "     "    yellow  "  = n

Now, if m=n (means all lights are of same number in the ladder)

then blue = yellow

________________________________________________________________________

But here other options can never be false

So, (D) should be the answer

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No you are wrong. I don't know how this question came in GATE :O 

If one of the lights is red, the other light on that step will always be blue

This says "if" and not "if and only". So, this is a simple implication- not a double implication in mathematical logic. i.e., if one light is red, then the other is blue. It does not say what happens if one light is blue. So, both blue lights clearly possible. As per the option, this statement should have been

If and only if one of the lights is red, other light is blue

My English grammar is not good but still I'm pretty sure this is correct and the question is wrongly framed.