AIEEE 2003

If the sum of the roots of the quadratic equation \(a{x^2} + bx + c = 0\) is equal to the sum of the squares of their reciprocals, then \({a \over c},\,{b \over a}\) and \({c \over b}\) are in

The sum of the serier \({1 \over {1.2}} - {1 \over {2.3}} + {1 \over {3.4}}..............\) up to \(\infty\) is equal to

The median of a set of 9 distinct observations is 20.5. If each of the largest 4 observations of
the set is increased by 2, then the median of the new set :

In an experiment with 15 observations on \(x\), then following results were available:
\(\sum {{x^2}} = 2830\), \(\sum x = 170\)
One observation that was 20 was found to be wrong and was replaced by the correct value
30. Then the correcte...

If \({x_1},{x_2},{x_3}\) and \({y_1},{y_2},{y_3}\) are both in G.P. with the same common ratio, then the points \(\left( {{x_1},{y_1}} \right),\left( {{x_2},{y_2}} \right)\) and \(\left( {{x_3},{y_3}} \right)\) :

Locus of centroid of the triangle whose vertices are \(\left( {a\cos t,a\sin t} \right),\left( {b\sin t, - b\cos t} \right)\) and \(\left( {1,0} \right),\) where \(t\) is a parameter, is :

A square of side a lies above the \(x\)-axis and has one vertex at the origin. The side passing through the origin makes an angle \(\alpha \left( {0 < \alpha < {\pi \over 4}} \right)\) with the positive direction of x-axis. The equation o...

If the equation of the locus of a point equidistant from the point \(\left( {{a_{1,}}{b_1}} \right)\) and \(\left( {{a_{2,}}{b_2}} \right)\) is
\(\left( {{a_1} - {a_2}} \right)x + \left( {{b_1} - {b_2}} \right)y + c = 0\) , then the valu...

If the pair of straight lines \({x^2} - 2pxy - {y^2} = 0\) and \({x^2} - 2qxy - {y^2} = 0\) be such that each pair bisects the angle between the other pair, then :

If \(\overrightarrow u \,,\overrightarrow v\) and \(\overrightarrow w\) are three non-coplanar vectors, then $$\,\left( {\overrightarrow u + \overrightarrow v - \overrightarrow w } \right).\left( {\overrightarrow u - \overrightarrow v ...

If \(\overrightarrow a \times \overrightarrow b = \overrightarrow b \times \overrightarrow c = \overrightarrow c \times \overrightarrow a\) then \(\overrightarrow a + \overrightarrow b + \overrightarrow c =\)

\(\overrightarrow a \,,\overrightarrow b \,,\overrightarrow c\) are \(3\) vectors, such that \(\overrightarrow a + \overrightarrow b + \overrightarrow c = 0\) , $$\left| {\overrightarrow a } \right| = 1\,\,\,\left| {\overrightarrow b }...

A tetrahedron has vertices at \(O(0,0,0), A(1,2,1) B(2,1,3)\) and \(C(-1,1,2).\) Then the angle between the faces \(OAB\) and \(ABC\) will be :

Consider points \(A, B, C\) and \(D\) with position
vectors \(7\widehat i - 4\widehat j + 7\widehat k,\widehat i - 6\widehat j + 10\widehat k, - \widehat i - 3\widehat j + 4\widehat k\) and \(5\widehat i - \widehat j + 5\widehat k\) re...

If \(\left| {\matrix{ a & {{a^2}} & {1 + {a^3}} \cr b & {{b^2}} & {1 + {b^3}} \cr c & {{c^2}} & {1 + {c^3}} \cr } } \right| = 0\) and vectors \(\left( {1,a,{a^2}} \right),\,\,\)
\(\left( {1,b,{b^2}} \right)\) and $$\left( {...

Let \(\overrightarrow u = \widehat i + \widehat j,\,\overrightarrow v = \widehat i - \widehat j\) and \(\overrightarrow w = \widehat i + 2\widehat j + 3\widehat k\,\,.\) If \(\widehat n\) is a unit vector such that $$\overrightarrow u .\...

The vectors \(\overrightarrow {AB} = 3\widehat i + 4\widehat k\,\,\& \,\,\overrightarrow {AC} = 5\widehat i - 2\widehat j + 4\widehat k\) are the sides of triangle \(ABC.\) The length of the median through \(A\) is :