Sets, Relations & Functions · JEE Main previous year questions

2005 6 questions

1. Q72005 · 28 Apr 2005 · Shift 1

   If the coefficient of $x^7$ in ${\left[a{x}^2+\left(\frac{1}{bx}\right)\right]}^{11}$ equals the coefficient of $x^{-7}$ in ${\left[ax-\left(\frac{1}{b{x}^2}\right)\right]}^{11}$, then $a$ and $b$ satisfy the relation

   1. A.$a-b=1$
   2. B.$a+b=1$
   3. C.$\frac{a}{b}=1$
   4. D.$ab=1$
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   Answer: D

2. Q442005 · 28 Apr 2005 · Shift 1

   Let $f:(-1,1)\to B$, be a function defined by $$f(x)={\tan }^{-1}\frac{2x}{1-x^2},$$ then $f$ is both one-one and onto when B is the interval

   1. A.$\left(0,\frac{\pi }{2}\right)$
   2. B.$\left[0,\frac{\pi }{2}\right)$
   3. C.$\left[-\frac{\pi }{2},\frac{\pi }{2}\right]$
   4. D.$\left(-\frac{\pi }{2},\frac{\pi }{2}\right)$
   Show answer

   Answer: D

3. Q452005 · 28 Apr 2005 · Shift 1

   A function is matched below against an interval where it is supposed to be increasing. Which of the following pairs is incorrectly matched?

   1. A.Interval: $(-\infty ,\infty )$, Function: $x^3-3x^2+3x+3$
   2. B.Interval: $[2,\infty )$, Function: $2x^3-3x^2-12x+6$
   3. C.Interval: $\left(-\infty ,\frac{1}{3}\right]$, Function: $3x^2-2x+1$
   4. D.Interval: $(-\infty ,-4)$, Function: $x^3+6x^2+6$
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   Answer: C

4. Q462005 · 28 Apr 2005 · Shift 1

   A real valued function f(x) satisfies the functional equation f(x - y) = f(x)f(y) - f(a - x)f(a + y) where a is a given constant and f(0) = 1, f(2a - x) is equal to

   1. A.- f(x)
   2. B.f(x)
   3. C.f(a) + f(a - x)
   4. D.f(- x)
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   Answer: A

5. Q492005 · 28 Apr 2005 · Shift 1

   If $f$ is a real valued differentiable function satisfying $$|f(x)-f(y)|\le (x-y{)}^2,x,y\in R$$ and $f(0)=0$, then $f(1)$ equals

   1. A.$-1$
   2. B.$0$
   3. C.$2$
   4. D.$1$
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   Answer: B

6. Q522005 · 28 Apr 2005 · Shift 1

   Let $R=\{(3,3),(6,6),(9,9),(12,12),(6,12),(3,9),(3,12),(3,6)\}$ be a relation on the set $A=\{3,6,9,12\}$. The relation is :

   1. A.reflexive and symmetric only
   2. B.an equivalence relation
   3. C.reflexive only
   4. D.reflexive and transitive only
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   Answer: D