If f(x) = 100 – x2, is f(a) > f(b) ?
Answer:
f(x) = 100 - x2
Therefore, f(a) = 100 - a2 and f(b) = 100 - b2
Is f(a) > f(b)?
Or, is 100 - a2 > 100 - b2 ?
Or, is - a2 > - b2 ?
Or, is a2 < b2 ?
1) a2> b2 , clearly its sufficient to tell that a2 is not less than b2 ;SUFFICIENT.
2) a/b > 1
=> (a/b) -1 > 0
=> (a-b)/b > 0, which means either both a-b and b are positive, or both a-b and b are negative.
=> If b>0, a-b>0 means a>b. So a2> b2.
And if b<0, a-b<0 means a2 > b, which means a is bigger negative number than b.(For eg. a=-4, b=-2). So, a2> b2.
So, either way we can say that a2> b2 ; SUFFICIENT.
Alternatively, the statement 2 could also have been analyzed in the following manner.
a/b > 1 → that the magnitude of numerator is more than denominator which would mean a2> b2.
The correct answer is D; each statement alone is sufficient.
a’ =10/a for all positive prime numbers. And a’ = -a2for all positive non-primes, which of the following is the largest?
(A) 5’ + 6’ + 2’
(B) 7’ – 4’
(C) 4’ – 7’
(D) 4’ + 8’ – 5’
(E) 4’ + 8’ – 2’
Ans:
Since, for all positive prime numbers;
a' = 10/a
And, for all positive non-prime numbers;
a' = -a2
Thus, using these functions to find out the values given in the question, we get;
(A) 5' + 6' + 2' = (10/5 - 62 + 10/2) = (5 - 36 +2) = -29
(B) 7' - 4' = (10/7 - (-42)) = (10/7 + 16) = 122/7
(C) 4' - 7' = ( - 42 - 10/7) = ( -16 - 10/7) = -122/7
(D) 4' + 8' - 5' = ( -42 - 82 - 10/5) = ( -16 -64 - 2) = -82
(E) 4' + 8' - 2' = (-42 - 82 - 10/2) = (-16 - 64 -5) = -85
Thus, among all the options we can see the largest value is obtained in option B
Alternatively, you could have also analyzed that all the options will generate a negative value except option B. hence that is the largest.
The correct answer is B.
If [a] represents least integer greater than or equal to a, what is the value of [–3.3] + [–2.9] + [0.1] + [1] + [1.9] ?
(A) –1
(B) –2
(C) –3
(D) –4
(E) –5
Since [a] represents the least integer greater than or equal to a, this implies that if a is an integer than [a] = a only. But if a is a fraction than, [a] = the least integer greater than a.
Thus, [-3.3]=-3
And,[-2.9]=-2
And,[0.1]=1
And,[1]=1
And,[1.9]=2
Hence, [-3.3]+[-2.9]+[0.1]+[1]+[1.9]
=-3-2+1+1+2
=-1
The correct answer is A.
In which of the following functions, is f(x) = f(1/x), for all values of x greater than 1?
(A) f(x) = 1/x + 1/ (x + 1)
(B) f(x) = |x|
(C) f(x) = 1-x
(D) f(x) = (1 - x2 )2/x2
(E) f(x) = (1 - x2)/x2
Answer:
(A) f(x) = 1/x + 1/(x + 1) and f(1/x) = 1/(1/x) + 1/(1/x + 1) = x + x/(1 + x)
Thus, f(x) ≠ f(1/x) for all values of x > 1
(B) f(x) = |x| and f(1/x) = |1/x| = 1/(|x|)
Thus, f(x) ≠ f(1/x) for all values of x >1
(C) f(x) = 1 - x and f(1/x) = 1 - 1/x = (x - 1)/x
Thus, f(x)≠f(1/x) for all values of x>1
(D) f(x)=(1 - x2 )2/x2 and f(1/x)=[1 - (1/x)2 ]2/(1/x)2 =[1 - 1/x2 ]2/(1/x2 )=[(x2 - 1)/x2 ]2/(1/x2 )=(x2 - 1)2/(x4 × 1/x2 )=(x2 - 1)2/x2 =(1 - x2 )2/x2
Thus, f(x) = f(1/x) for all values of x > 1
(E) f(x) = (1 - x2)/x2 and f(1/x) = (1 - (1/x)2)/(1/x)2 = (1 - 1/x2 )/(1/x2 ) = ((x2 - 1)/x2 )/(1/x2 ) = (x2- 1)/(x2×1/x2 ) = x2 - 1
Thus, f(x) ≠ f(1/x) for all values of x>1
Thus we can see only for (D); f(x) = f(1/x)
The correct answer is D.
In a sequence, an + 2 = an + 1 – an + (n + 2) for all n ≥1, where an + 2 represents the (n + 2)th term. If the second and the fourth terms of the sequence are 2 and 5 respectively then what is the value of the 6th term of the sequence?
(A) 5
(B) 7
(C) 8
(D) 9
(E) None of these
For n≥1,
a(n + 2) = a(n + 1) - an + (n + 2) ……Equation(1)
Putting value of n = 2 in the above equation, we get:
a4 = a3 - a2 + (2 + 2)
In the question, it’s also mentioned that a2 = 2 , a4 = 5. Putting in the above equation, we get the value of a3 = 3
Now by putting value of n = 3 in equation(1), we get:
a5 = a4 - a3 + (3 + 2)
a5 = 5 - 3 + (3 + 2)
a5 = 7
Now by putting value of n = 4 in equation(1),we get:
a6 = a5 - a4 + (4 + 2)
a6 = 7 - 5 + (4 + 2)
a6 = 8
The correct answer is C.
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