IIT-JEE 2008 Paper 1 Offline

Consider three planes
\(${P_1}:x - y + z = 1\)$
\(${P_2}:x + y - z = 1\)$
\(${P_3}:x - 3y + 3z = 2\)$
Let \({L_1},\) \({L_2},\) \({L_3}\) be the lines of intersection of the planes \({P_2}\) and \({P_3},\) \({P_3}\) and \({P_1},\) $${P...

The total number of local maxima and local minima of the function \(f(x) = \left\{ {\matrix{ {{{(2 + x)}^3},} & { - 3 < x \le - 1} \cr {{x^{2/3}},} & { - 1 < x < 2} \cr } } \right.\) is

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 :

The equation of circle C is

Points E and F are given by

Equations of the sides QR, RP are

The number of elements in the set \(A \cap B \cap C\) is

Let z be any point in \(A \cap B \cap C\)
Then, \({\left| {z + 1 - i} \right|^2} + {\left| {z - 5 - i} \right|^2}\) lies between :

Let z be any point \(A \cap B \cap C\) and let w be any point satisfying \(\left| {w - 2 - i} \right| < 3\,\). Then, \(\left| z \right| - \left| w \right| + 3\) lies between :

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

If \(f\left( { - 10\sqrt 2 } \right) = 2\sqrt 2 ,\) then \(f''\left( { - 10\sqrt 2 } \right) =\)

Let \(f\) and \(g\) be real valued functions defined on interval \((-1, 1)\) such that \(g''(x)\) is continuous, \(g\left( 0 \right) \ne 0.\) \(g'\left( 0 \right) = 0\), \(g''\left( 0 \right) \ne 0\), and $$f\left( x \right) = g\left( x \ri...

Let \(P\left( {{x_1},{y_1}} \right)\) and \(Q\left( {{x_2},{y_2}} \right),{y_1} < 0,{y_2} < 0,\) be the end points of the latus rectum of the ellipse \({x^2} + 4{y^2} = 4.\) The equations of parabolas with latus rectum \(PQ\) are :

Consider the two curves \({C_1}:{y^2} = 4x,\,{C_2}:{x^2} + {y^2} - 6x + 1 = 0\). Then,

If \(0 < x < 1\), then
\(\sqrt {1 + {x^2}} {\left[ {{{\left\{ {x\cos \left( {{{\cot }^{ - 1}}x} \right) + \sin \left( {{{\cot }^{ - 1}}x} \right)} \right\}}^2} - 1} \right]^{1/2}} =\)

Let \(g(x) = {{{{(x - 1)}^n}} \over {\log {{\cos }^m}(x - 1)}};0 < x < 2,m\) and \(n\) are integers, \(m \ne 0,n > 0\), and let \(p\) be the left hand derivative of \(|x - 1|\) at \(x = 1\). If $$\mathop {\lim }\limits_{x \to {1^ + }} g(x) ...

Let \(f(x)\) be a non-constant twice differentiable function defined on \(\left( { - \infty ,\infty } \right)\)
such that \(f\left( x \right) = f\left( {1 - x} \right)\) and \(f'\left( {{1 \over 4}} \right) = 0.\) Then,

Consider the system of equations:
\(x-2y+3z=-1\)
\(-x+y-2z=k\)
\(x-3y+4z=1\)
Statement - 1 : The system of equations has no solution for \(k\ne3\).
and
Statement - 2 : The determinant $$\left| {\matrix{
1 & 3 & { - 1} \cr
{ - 1} & {...

Consider the system of equations \(ax+by=0; cx+dy=0,\)
where \(a,b,c,d\) \(\in \left\{ {0,1} \right\}\)
STATEMENT - 1 : The probability that the system of equations has a unique solution is \({3 \over 8}.\) and
STATEMENT - 2 : The proba...

Let \({S_n} = \sum\limits_{k = 1}^n {{n \over {{n^2} + kn + {k^2}}}}\) and \({T_n} = \sum\limits_{k = 0}^{n - 1} {{n \over {{n^2} + kn + {k^2}}}}\) for \(n\) \(=1, 2, 3, ............\) Then,

Let a and b be non-zero real numbers. Then, the equation \((a{x^2} + b{y^2} + c)({x^2} - 5xy + 6{y^2}) = 0\) represents :

A straight line through the vertex p of a triangle PQR intersects the side QR at the point S and the circumcircle of the triangle PQR at the point T. If S is not the centre of the circumcircle, then :

The edges of a parallelopiped are of unit length and are parallel to non-coplanar unit vectors \(\overrightarrow a \,,\,\overrightarrow b ,\overrightarrow c\) such that $$\widehat a\,.\,\widehat b = \widehat b\,.\,\widehat c = \widehat c\,...