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1

IIT-JEE 2008

MCQ (Single Correct Answer)
Consider the function $$f:\left( { - \infty ,\infty } \right) \to \left( { - \infty ,\infty } \right)$$ defined by
$$f\left( x \right) = {{{x^2} - ax + 1} \over {{x^2} + ax + 1}},0 < a < 2.$$

Which of the following is true?

A
$$f(x)$$ is decreasing on $$(-1,1)$$ and has a local minimum at $$x=1$$
B
$$f(x)$$ is increasing on $$(-1,1)$$ and has a local minimum at $$x=1$$
C
$$f(x)$$ is increasing on $$(-1,1)$$ but has neither a local maximum nor a local minimum at $$x=1$$
D
$$f(x)$$ is decreasing on $$(-1,1)$$ but has neither a local maximum nor a local minimum at $$x=1$$
2

IIT-JEE 2008

MCQ (Single Correct Answer)
Consider the function $$f:\left( { - \infty ,\infty } \right) \to \left( { - \infty ,\infty } \right)$$ defined by
$$f\left( x \right) = {{{x^2} - ax + 1} \over {{x^2} + ax + 1}},0 < a < 2.$$

Which of the following is true?

A
$${\left( {2 + a} \right)^2}f''\left( 1 \right) + {\left( {2 - a} \right)^2}f''\left( { - 1} \right) = 0$$
B
$${\left( {2 - a} \right)^2}f''\left( 1 \right) - {\left( {2 + a} \right)^2}f''\left( { - 1} \right) = 0$$
C
$$f'\left( 1 \right)f'\left( { - 1} \right) = {\left( {2 - a} \right)^2}$$
D
$$f'\left( 1 \right)f'\left( { - 1} \right) = -{\left( {2 + a} \right)^2}$$
3

IIT-JEE 2008

MCQ (Single Correct Answer)
Consider the functions defined implicitly by the equation
$${y^3} - 3y + x = 0$$ on various intervals in the real line. If
$$x \in \left( { - \infty , - 2} \right) \cup \left( {2,\infty } \right),$$ the equation implicitly defines a unique
real valued differentiable function $$y = f\left( x \right).$$ If $$x \in \left( { - 2,2} \right),$$ the
equation implicitly defines a unique real valued differentiable function
$$y=g(x)$$ satisfying $$g(0)=0.$$

$$\int\limits_{ - 1}^1 {g'\left( x \right)dx = } $$

A
$$2g(-1)$$
B
$$0$$
C
$$-2g(1)$$
D
$$2g(1)$$
4

IIT-JEE 2008

MCQ (Single Correct Answer)
Consider the functions defined implicitly by the equation
$${y^3} - 3y + x = 0$$ on various intervals in the real line. If
$$x \in \left( { - \infty , - 2} \right) \cup \left( {2,\infty } \right),$$ the equation implicitly defines a unique
real valued differentiable function $$y = f\left( x \right).$$ If $$x \in \left( { - 2,2} \right),$$ the
equation implicitly defines a unique real valued differentiable function
$$y=g(x)$$ satisfying $$g(0)=0.$$

The area of the region bounded by the curve $$y=f(x),$$ the
$$x$$-axis, and the lines $$x=a$$ and $$x=b$$, where $$ - \infty < a < b < - 2,$$ is

A
$$\int\limits_a^b {{x \over {3\left( {{{(f(x))}^2} - 1} \right)}}} dx + bf\left( b \right) - af\left( a \right)$$
B
$$ - \int\limits_a^b {{x \over {3\left( {{{(f(x))}^2} - 1} \right)}}} dx + bf\left( b \right) - af\left( a \right)$$
C
$$\int\limits_a^b {{x \over {3\left( {{{(f(x))}^2} - 1} \right)}}} dx - bf\left( b \right) + af\left( a \right)$$
D
$$ - \int\limits_a^b {{x \over {3\left( {{{(f(x))}^2} - 1} \right)}}} dx - bf\left( b \right) + af\left( a \right)$$

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