IIT-JEE 2002 Screening

The point(s) in the curve \({y^3} + 3{x^2} = 12y\) where the tangent is vertical, is (are)

The length of a longest interval in which the function \(3\,\sin x - 4{\sin ^3}x\) is increasing, is

Let \(f\left( x \right) = \int\limits_1^x {\sqrt {2 - {t^2}} \,dt.}\) Then the real roots of the equation
\({x^2} - f'\left( x \right) = 0\) are

The area bounded by the curves \(y = \left| x \right| - 1\) and \(y = - \left| x \right| + 1\) is

If the tangent at the point P on the circle \({x^2} + {y^2} + 6x + 6y = 2\) meets a straight line 5x - 2y + 6 = 0 at a point Q on the y-axis, then the lenght of PQ is

If \(a > 2b > 0\) then the positive value of \(m\) for which \(y = mx - b\sqrt {1 + {m^2}}\) is a common tangent to \({x^2} + {y^2} = {b^2}\) and \({\left( {x - a} \right)^2} + {y^2} = {b^2}\) is

For all complex numbers \({z_1},\,{z_2}\) satisfying \(\left| {{z_1}} \right| = 12\) and \(\left| {{z_2} - 3 - 4i} \right| = 5,\)
the minimum value of \(\left| {{z_1} - {z_2}} \right|\) is

Let \(T>0\) be a fixed real number . Suppose \(f\) is a continuous
function such that for all \(x \in R\), \(f\left( {x + T} \right) = f\left( x \right)\).
If \(I = \int\limits_0^T {f\left( x \right)dx}\) then the value of $$\int\limits...

Let \(T>0\) be a fixed real number . Suppose \(f\) is a continuous
function such that for all \(x \in R\), \(f\left( {x + T} \right) = f\left( x \right)\).
If \(I = \int\limits_0^T {f\left( x \right)dx}\) then the value of $$\int\limits...

The integral \(\int\limits_{ - 1/2}^{1/2} {\left( {\left[ x \right] + \ell n\left( {{{1 + x} \over {1 - x}}} \right)} \right)dx}\) equal to

The sum $$\sum\limits_{i = 0}^m {\left( {\matrix{
{10} \cr
i \cr

} } \right)\left( {\matrix{
{20} \cr
{m - i} \cr

} } \right),\,\left( {where\left( {\matrix{
p \cr
q \cr

} } \right) = 0\,\,if\,\,p < q} \r...

The equation of the common tangent to the curves \({y^2} = 8x\) and \(xy = - 1\) is

The locus of the mid-point of the line segment joining the focus to a moving point on the parabola \({y^2} = 4ax\) is another parabola with directrix

The number of arrangements of the letters of the word BANANA in which the two N's do not appear adjacently is

Which of the following pieces of data does NOT uniquely determine an acute-angled triangle \(ABC\) (\(R\) being the radius of the circumcircle)?

If \({a_1},{a_2}.......,{a_n}\) are positive real numbers whose product is a fixed number c, then the minimum value of \({a_1} + {a_2} + ..... + {a_{n - 1}} + 2{a_n}\) is

The set of all real numbers x for which \({x^2} - \left| {x + 2} \right| + x > 0\), is

Suppose \(a, b, c\) are in A.P. and \({a^2},{b^2},{c^2}\) are in G.P. If \(a < b < c\) and \(a + b + c = {3 \over 2},\) then the value of \(a\) is

Let \(P = \left( { - 1,\,0} \right),\,Q = \left( {0,\,0} \right)\) and \(R = \left( {3,\,3\sqrt 3 } \right)\) be three points.
Then the equation of the bisector of the angle \(PQR\) is

A straight line through the origin \(O\) meets the parallel lines \(4x+2y=9\) and \(2x+y+6=0\) at points \(P\) and \(Q\) respectively. Then the point \(O\) divides the segemnt \(PQ\) in the ratio

Let \(0 < \alpha < {\pi \over 2}\) be fixed angle. If \(P = \left( {\cos \theta ,\,\sin \theta } \right)\) and \(Q = \left( {\cos \left( {\alpha - \theta } \right),\,\sin \left( {\alpha - \theta } \right)} \right),\) then \(Q\) is obtai...

The number of integral values of \(k\) for which the equation \(7\cos x + 5\sin x = 2k + 1\) has a solution is

If \({\overrightarrow a }\) and \({\overrightarrow b }\) are two unit vectors such that \({\overrightarrow a + 2\overrightarrow b }\) and \({5\overrightarrow a - 4\overrightarrow b }\) are perpendicular to each other then the angle betwee...

Let \(\overrightarrow V = 2\overrightarrow i + \overrightarrow j - \overrightarrow k\) and \(\overrightarrow W = \overrightarrow i + 3\overrightarrow k .\) If \(\overrightarrow U\) is a unit vector, then the maximum value of the sca...