IIT-JEE 2003 Screening

The value of \(k\) such that \({{x - 4} \over 1} = {{y - 2} \over 1} = {{z - k} \over 2}\) lies in the plane \(2x -4y +z = 7,\) is

Tangent is drawn to ellipse
\({{{x^2}} \over {27}} + {y^2} = 1\,\,\,at\,\left( {3\sqrt 3 \cos \theta ,\sin \theta } \right)\left( {where\,\,\theta \in \left( {0,\pi /2} \right)} \right)\).
Then the value of \(\theta\) such that sum of in...

In \(\left[ {0,1} \right]\) Languages Mean Value theorem is NOT applicable to

The area bounded by the curves \(y = \sqrt x ,2y + 3 = x\) and
\(x\)-axis in the 1st quadrant is

The centre of circle inscibed in square formed by the lines \({x^2} - 8x + 12 = 0\,\,and\,{y^2} - 14y + 45 = 0\), is

If \(\,\left| z \right| = 1\) and \(\omega = {{z - 1} \over {z + 1}}\) (where \(z \ne - 1\)), then \({\mathop{\rm Re}\nolimits} \left( \omega \right)\) is

If \(f\left( x \right) = \int\limits_{{x^2}}^{{x^2} + 1} {{e^{ - {t^2}}}} dt,\) then \(f(x)\) increases in

If \(l\left( {m,n} \right) = \int\limits_0^1 {{t^m}{{\left( {1 + t} \right)}^n}dt,}\) then the expression for \(l(m, n)\) in terms of \(l(m+n, n-1)\) is

If \(y(t)\) is a solution of \(\left( {1 + t} \right){{dy} \over {dt}} - ty = 1\) and \(y\left( 0 \right) = - 1,\) then \(y(1)\) is equal to

The area of the quadrilateral formed by the tangents at the end points of latus rectum to the ellipse \({{{x^2}} \over 9} + {{{y^2}} \over 5} = 1,\) is

For hyperbola \({{{x^2}} \over {{{\cos }^2}\alpha }} - {{{y^2}} \over {{{\sin }^2}\alpha }} = 1\) which of the following remains constant with change in \('\alpha '\)

Coefficient of \({t^{24}}\) in \({\left( {1 + {t^2}} \right)^{12}}\left( {1 + {t^{12}}} \right)\left( {1 + {t^{24}}} \right)\) is

The focal chord to \({y^2} = 16x\) is tangent to \({\left( {x - 6} \right)^2} + {y^2} = 2,\) then the possible values of the slope of the chord, are

If \(P\left( B \right) = {3 \over 4},P\left( {A \cap B \cap \overline C } \right) = {1 \over 3}\) and
\(P\left( {\overline A \cap B \cap \overline C } \right) = {1 \over 3},\,\,\) then \(P\left( {B \cap C} \right)\) is

Two numbers are selected randomly from the set \(S = \left\{ {1,2,3,4,5,6} \right\}\) without replacement one by one. The probability that minimum of the two numbers is less than \(4\) is

If the angles of a triangle are in the ratio \(4:1:1\), then the ratio of the longest side to the perimeter is

If \(\,\alpha \in \left( {0,{\pi \over 2}} \right)\,\,then\,\,\sqrt {{x^2} + x} + {{{{\tan }^2}\alpha } \over {\sqrt {{x^2} + x} }}\) is always greater than or equal to

Orthocentre of triangle with vertices \(\left( {0,0} \right),\left( {3,4} \right)\) and \(\left( {4,0} \right)\) is

The number of integral points (integral point means both the coordinates should be integer) exactly in the interior of the triangle with vertices \(\left( {0,0} \right),\left( {0,21} \right)\) and \(\left( {21,0} \right)\), is

The value of \('a'\) so that the volume of parallelopiped formed by \(\widehat i + a\widehat j + \widehat k,\widehat j + a\widehat k\) and \(a\widehat i + \widehat k\) becomes minimum is