IIT-JEE 2008 Paper 2 Offline

The distance of the point \((1, 1, 1)\) from the plane passing through the point \((-1, -2, -1)\) and whose normal is perpendicular to both the lines \({L_1}\) and \({L_2}\) is :

The area of the region between the curves \(y = \sqrt {{{1 + \sin x} \over {\cos x}}}\)
and \(y = \sqrt {{{1 - \sin x} \over {\cos x}}}\) bounded by the lines \(x=0\) and \(x = {\pi \over 4}\) is

Consider
\(\,{L_1}:\,\,2x\,\, + \,\,3y\, + \,p\,\, - \,\,3 = 0\)
\(\,{L_2}:\,\,2x\,\, + \,\,3y\, + \,p\,\, + \,\,3 = 0\)
where p is a real number, and \(\,C:\,{x^2}\, + \,{y^2}\, + \,6x\, - 10y\, + \,30 = 0\)
STATEMENT-1 : If line $...

A particle P stats from the point \({z_0}\) = 1 +2i, where \(i = \sqrt { - 1}\). It moves horizontally away from origin by 5 unit and then vertically away from origin by 3 units to reach a point \({z_1}\). From \({z_1}\) the particle moves...

Let \(g\left( x \right) = \int\limits_0^{{e^x}} {{{f'\left( t \right)} \over {1 + {t^2}}}} \,dt.\) Which of the following is true?

Let a solution \(y=y(x)\) of the differential equation,
\(x\sqrt {{x^2} - 1} \,\,dy - y\sqrt {{y^2} - 1} \,dx = 0\) satify \(y\left( 2 \right) = {2 \over {\sqrt 3 }}.\)
STATEMENT-1 : $$y\left( x \right) = \sec \left( {{{\sec }^{ - 1}}x - ...

Let \(g(x) = \log f(x)\), where \(f(x)\) is a twice differentiable positive function on (0, \(\infty\)) such that \(f(x + 1) = xf(x)\). Then for N = 1, 2, 3, ..., \(g''\left( {N + {1 \over 2}} \right) - g''\left( {{1 \over 2}} \right) =\)

Which of the following is true?

Consider a branch of the hyperbola
\(${x^2} - 2{y^2} - 2\sqrt 2 x - 4\sqrt 2 y - 6 = 0\)$
with vertex at the point \(A\). Let \(B\) be one of the end points of its latus rectum. If \(C\) is the focus of the hyperbola nearest to the point ...

Let \(I = \int {{{{e^x}} \over {{e^{4x}} + {e^{2x}} + 1}}dx,\,\,J = \int {{{{e^{ - x}}} \over {{e^{ - 4x}} + {e^{ - 2x}} + 1}}dx.} }\) Then
for an arbitrary constant \(C\), the value of \(J -I\) equals :

Let the function \(g:\left( { - \infty ,\infty } \right) \to \left( { - {\pi \over 2},{\pi \over 2}} \right)\) be given by
\(g\left( u \right) = 2{\tan ^{ - 1}}\left( {{e^u}} \right) - {\pi \over 2}.\) Then, \(g\) is

Which of the following is true?

Consider all possible permutations of the letters of the word ENDEANOEL. Match the Statements/Expressions in Column I with the Statements/Expressions in Column II.

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An experiment has 10 equally likely outcomes. Let A and B be two non-empty events of the experiment. If A consists of 4 outcomes, the number of outcomes that B must have so that A and B are independent is :

Let \(a,\,b,c\), \(p,q\) be real numbers. Suppose \(\alpha ,\,\beta\) are the roots of the equation \({x^2} + 2px + q = 0\) and \(\alpha ,{1 \over \beta }\) are the roots of the equation \(a{x^2} + 2bx + c = 0,\) where $${\beta ^2} \in \le...

Suppose four distinct positive numbers \({a_1},\,{a_{2\,}},\,{a_3},\,{a_4}\,\) are in G.P. Let \({b_1} = {a_1},{b_2} = {b_1} + {a_2},\,{b_3} = {b_2} + {a_{3\,\,}}\,\,\,and\,\,\,{b_4} = {b_3} + {a_4}\).
STATEMENT-1: The numbers $${b_1},\,...

Consider three points \(P = ( - \sin (\beta - \alpha ), - cos\beta ),Q = (cos(\beta - \alpha ),\sin \beta )\) and \(R = (\cos (\beta - \alpha + \theta ),\sin (\beta - \theta ))\) where \(0 < \alpha ,\beta ,\theta < {\pi \over 4}\). T...

Consider the lines given by:
\({L_1}:x + 3y - 5 = 0\)
\({L_2}:3x - ky - 1 = 0\)
\({L_3}:5x + 2y - 12 = 0\)
Match the Statement/Expressions in Column I with the Statements/Expressions in Column II.

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Match the Statements/Expressions in Column I with the Statements/Expressions in Column II.

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Let two non-collinear unit vectors \(\widehat a\) and \(\widehat b\) form an acute angle. A point \(P\) moves so that at any time \(t\) the position vector \(\overrightarrow {OP}\) (where \(O\) is the origin) is given by $$\widehat a\cos t...

The unit vector perpendicular to both \({L_1}\) and \({L_2}\) is :

The shortest distance between \({L_1}\) and \({L_2}\) is :