IIT-JEE 2000 Screening

Let \(f\left( x \right) = \int {{e^x}\left( {x - 1} \right)\left( {x - 2} \right)dx.}\) Then \(f\) decreases in the interval

If the normal to the curve \(y = f\left( x \right)\) and the point \((3, 4)\) makes an angle \({{{3\pi } \over 4}}\) with the positive \(x\)-axis, then \(f'\left( 3 \right) =\)

Let \(f\left( x \right) = \left\{ {\matrix{ {\left| x \right|,} & {for} & {0 < \left| x \right| \le 2} \cr {1,} & {for} & {x = 0} \cr } } \right.\) then at \(x=0\), \(f\) has

For all \(x \in \left( {0,1} \right)\)

Consider the following statements in \(S\) and \(R\)
\(S:\) \(\,\,\,\)$ Both \(\sin \,\,x\) and \(\cos \,\,x\) are decreasing functions in the interval \(\left( {{\pi \over 2},\pi } \right)\)
\(R:\)\(\,\,\,\) If a differentiable function ...

The triangle PQR is inscribed in the circle \({x^2}\, + \,\,{y^2} = \,25\). If Q and R have co-ordinates (3, 4) and ( - 4, 3) respectively, then \(\angle \,Q\,P\,R\) is equal to

If the circles \({x^2}\, + \,{y^2}\, + \,\,2x\, + \,2\,k\,y\,\, + \,6\,\, = \,\,0,\,\,{x^2}\, + \,\,{y^2}\, + \,2ky\, + \,k\, = \,0\) intersect orthogonally, then k is

If \(\arg \left( z \right) < 0,\) then \(\arg \left( { - z} \right) - \arg \left( z \right) =\)

If \({z_1},\,{z_2}\) and \({z_3}\) are complex numbers such that \(\left| {{z_1}} \right| = \left| {{z_2}} \right| = \left| {{z_3}} \right| = \left| {{1 \over {{z_1}}} + {1 \over {{z_2}}} + {1 \over {{z_3}}}} \right| = 1,\) then $$\left| {{...

If \(f\left( x \right) = \left\{ {\matrix{ {{e^{\cos x}}\sin x,} & {for\,\,\left| x \right| \le 2} \cr {2,} & {otherwise,} \cr } } \right.\) then \(\int\limits_{ - 2}^3 {f\left( x \right)dx = }\)

Let \(g\left( x \right) = \int\limits_0^x {f\left( t \right)dt,}\) where f is such that
\({1 \over 2} \le f\left( t \right) \le 1,\) for \(t \in \left[ {0,1} \right]\) and \(\,0 \le f\left( t \right) \le {1 \over 2},\) for $$t \in \left[ ...

The value of the integral \(\int\limits_{{e^{ - 1}}}^{{e^2}} {\left| {{{{{\log }_e}x} \over x}} \right|dx}\) is :

If \({x^2} + {y^2} = 1,\) then

For \(2 \le r \le n,\,\,\,\,\left( {\matrix{ n \cr r \cr } } \right) + 2\left( {\matrix{ n \cr {r - 1} \cr } } \right) + \left( {\matrix{ n \cr {r - 2} \cr } } \right) =\)

If \(x + y = k\) is normal to \({y^2} = 12x,\) then \(k\) is

If the line \(x - 1 = 0\) is the directrix of the parabola \({y^2} - kx + 8 = 0,\) then one of the values of \(k\) is

How many different nine digit numbers can be formed from the number 223355888 by rearranging its digits so that the odd digits occupy even positions?

In a triangle \(ABC\), \(2ac\,\sin {1 \over 2}\left( {A - B + C} \right) =\)

A pole stands vertically inside a triangular park \(\Delta ABC\). If the angle of elevation of the top of the pole from each corner of the park is same, then in \(\Delta ABC\) the foot of the pole is at the

In a triangle \(ABC\), let \(\angle C = {\pi \over 2}\). If \(r\) is the inradius and \(R\) is the circumradius of the triangle, then \(2(r+R)\) is equal to

If \(\alpha \,\text{and}\,\beta\) \((\alpha \, < \,\beta )\) are the roots of the equation \({x^2} + bx + c = 0\,\), where \(c < 0 < b\), then

For the equation \(3{x^2} + px + 3 = 0\). p > 0, if one of the root is square of the other, then p is equal to

If a, b, c, d are positive real numbers such that a + b + c + d = 2, then M = (a + b) (c + d) satisfies the relation

If b > a, then the equation (x - a) (x - b) - 1 = 0 has

Consider an infinite geometric series with first term a and common ratio \(r\). If its sum is 4 and the second term is 3/4, then

The incentre of the triangle with vertices \(\left( {1,\,\sqrt 3 } \right),\left( {0,\,0} \right)\) and \(\left( {2,\,0} \right)\) is

Let \(PS\) be the median of the triangle with vertices \(P(2, 2),\) \(Q(6, -1)\) and \(R(7, 3).\) The equation of the line passing through \((1, -1)\) and parallel to \(PS\) is

Let \(f\left( \theta \right) = \sin \theta \left( {\sin \theta + \sin 3\theta } \right)\). Then \(f\left( \theta \right)\) is

If the vectors \(\overrightarrow a ,\overrightarrow b\) and \(\overrightarrow c\) form the sides \(BC,\) \(CA\) and \(AB\) respectively of a triangle \(ABC,\) then

Let the vectors \(\overrightarrow a ,\overrightarrow b ,\overrightarrow c\) and \(\overrightarrow d\) be such that
$$\left( {\overrightarrow a \times \overrightarrow b } \right) \times \left( {\overrightarrow c \times \overrightarrow d...

If \(\overrightarrow a \,,\,\overrightarrow b\) and \(\overrightarrow c\) are unit coplanar vectors, then the scalar triple product $$\left[ {2\overrightarrow a - \overrightarrow b ,2\overrightarrow b - \overrightarrow c ,2\overrightarr...