Answer Very Carefully!
Which of the following statements is/are correct?
[ 1 ] An irrational number times a rational number is always an irrational number.
[ 2 ] The statement tan − 1 x + tan − 1 y = tan − 1 ( 1 − x y x + y ) is true for all real numbers x and y .
[ 3 ] If x y = y x , then x = y where x and y are positive real numbers.
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3 solutions
Nice set of statements! I nearly missed [1] till i remembered that 0 was rational.
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Yeah same, i didn't consider o as one of the values but other rational numbers times irrational numbers is always irrational.
my words, man!!!!!!!!
@Mursalin Habib your mistake was that in all the other options, you put statement [ 1 ]. I knew that statement [ 1 ] was wrong & didn't need to read the other options.
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I actually did that on purpose. I knew that most people would [incorrectly] assume that [ 1 ] is correct and immediately disregard the correct answer ["none of them are correct"]. You looked passed that, but the 65% of the people who attempted the problem didn't.
Nice thinking Ameya!, I also got this by canceling all others!
I only know that #1 is not true and the choices helped me pick the right answer.
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Did the same :D
Me too! Every answer had a #1, so of course, A.
nice one...
Good question indeed!
Sorry for raising this question here....Your previous question about "DO U KNOW ABT ROOTS"....the third statement seems to be wrong...It states that if a,b,c >0... The quadratic has both roots negative....Applying Descartes rule we get this....but consider the case where roots are irrational....Descartes rule will only tell us that the real part of the root is negative....so I think the answer should be none of the above....do clarify!!!
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I'm not sure I follow. Remember, the statement states that the quadratic equation had real solutions. So, I don't understand what you are trying to say with "the real part of the root is negative.". And the statement is also true if the roots are irrational.
Let me know if you need more help.
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Sorry...I meant to say the rational part of the irrational root.... Forgive me for not using LaTeX as I am not comfortable using it... For eg: A root is sqrt(b)+a Using descarte, we only get that the rational part i.e. a is negative, it says nothing about the sign b4 the irrational part of the root
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@Tanya Gupta – The statement still stands if the roots are irrational. If a , b and c are positive numbers, then b 2 − 4 a c is less than b 2 . That means b 2 − 4 a c is less than b [be careful, this statement's not true if b < 0 , but I digress].
The roots of a x 2 + b x + c = 0 are 2 a 1 ( − b − b 2 − 4 a c ) and 2 a 1 ( b 2 − 4 a c − b ) .
The first root is obviously negative. The second one is also negative because b > b 2 − 4 a c . If you need any more help, let me know.
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@Mursalin Habib – Thanks for clearing it up...got ahead of myself :D
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@Tanya Gupta – No problem! You could also take a look at @Christopher Boo 's solution for it.
Holy shoot.........I didn't even think that 0 is a rational number to check cases.......
As soon as I saw the first statement I knew that the were all incorrect.
That zero ! Why ! :/
I too missed [1}!
why [2] is wrong ?? i couldnt understand how did u get that xy>1 makes the statment wrong ?? actually i proved it with the rule: tan(a+b)=[tan(a)+tan(b)]/[1-tan(a)tan(b)] ....what is the mistake in my proof ?? and thanks
Mhh, I can't properly understand your consideration on the 2nd relation.. Could you elaborate more deeply?
nice solution, bhaiya
Should have included 'Only [2]" as one of the options to make it interesting.
[1] is false as multiplying any irrational number by 0 (a rational number) gives 0 (a rational number). The only answer which doesn't state [1] is true is "none".
Exactly. That's enough to know.
In statement [3] (x,y)=(-1,-1) also violates the statement.
Positive real*
Let's consider the statements one by one.
Is [ 1 ] true? Not necessarily. 0 is a rational number and any number regardless of its rationality-status [is that even a word?] multiplied by 0 is 0 , a rational number.
[ 2 ] is not true when x y > 1 . The inverse tangent function is something that takes a real number as input and returns an angle in the range ( 2 − π , 2 π ) as output. Try putting x and y equal to 3 . What happens when you use the formula? What is the real answer?
[ 3 ] is not true because ( x , y ) = ( 2 , 4 ) satisfies the given equation.