AIEEE 2005

Let \(A\) and \(B\) two events such that \(P\left( {\overline {A \cup B} } \right) = {1 \over 6},\) \(P\left( {A \cap B} \right) = {1 \over 4}\) and \(P\left( {\overline A } \right) = {1 \over 4},\) where \({\overline A }\) stands for comp...

A random variable \(X\) has Poisson distribution with mean \(2\).
Then \(P\left( {X > 1.5} \right)\) equals :

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

If in a \(\Delta ABC\), the altitudes from the vertices \(A, B, C\) on opposite sides are in H.P, then \(\sin A,\sin B,\sin C\) are in :

If the roots of the equation \({x^2} - bx + c = 0\) be two consecutive integers, then \({b^2} - 4c\) equals

If both the roots of the quadratic equation \({x^2} - 2kx + {k^2} + k - 5 = 0\) are less than 5, then \(k\) lies in the interval

The value of \(a\) for which the sum of the squares of the roots of the equation \({x^2} - \left( {a - 2} \right)x - a - 1 = 0\) assume the least value is

The value of \(a\) for which the sum of the squares of the roots of the equation \({x^2} - \left( {a - 2} \right)x - a - 1 = 0\)
assume the least value is :

If the roots of the equation \({x^2} - bx + c = 0\) be two consecutive integers, then \({b^2} - 4c\) equals

In a triangle \(PQR,\;\;\angle R = {\pi \over 2}.\,\,If\,\,\tan \,\left( {{P \over 2}} \right)\) and \(\tan \left( {{Q \over 2}} \right)\) are the roots of \(a{x^2} + bx + c = 0,\,\,a \ne 0\) then

The sum of the series \(1 + {1 \over {4.2!}} + {1 \over {16.4!}} + {1 \over {64.6!}} + .......\) ad inf. is

If \(x = \sum\limits_{n = 0}^\infty {{a^n},\,\,y = \sum\limits_{n = 0}^\infty {{b^n},\,\,z = \sum\limits_{n = 0}^\infty {{c^n},} } } \,\,\) where a, b, c are in A.P and $$\,\left| a \right| < 1,\,\left| b \right| < 1,\,\left| c \right| <...

Let $R=\{(3,3),(6,6),(9,9),(12,12),(6,12)$, $(3,9),(3,12),(3,6)\}$ be a relation on the set $A=\{3,6,9,12\}$. The relation is :

If in a frequency distribution, the mean and median are 21 and 22 respectively, then
its mode is approximately :

Let x1, x2,...........,xn be n observations such that
\(\sum {x_i^2} = 400\) and \(\sum {{x_i}} = 80\). Then a
possible value of n among the following is

If a vertex of a triangle is \((1, 1)\) and the mid points of two sides through this vertex are \((-1, 2)\) and \((3, 2)\) then the centroid of the triangle is :

If non zero numbers \(a, b, c\) are in \(H.P.,\) then the straight line \({x \over a} + {y \over b} + {1 \over c} = 0\) always passes through a fixed point. That point is :

The line parallel to the \(x\) - axis and passing through the intersection of the lines \(ax + 2by + 3b = 0\) and \(bx - 2ay - 3a = 0,\) where \((a, b)\) \(\ne\) \((0, 0)\) is :

Let \(\overrightarrow a \,\, = \,\,\widehat i - \widehat k,\,\,\,\,\,\overrightarrow b \,\,\, = \,\,\,x\widehat i + \widehat j\,\,\, + \,\,\,\left( {1 - x} \right)\widehat k\) and $$\overrightarrow c \,\, = \,\,y\widehat i + x\widehat j + \...

Let \(a, b\) and \(c\) be distinct non-negative numbers. If the vectors \(a\widehat i + a\widehat j + c\widehat k,\,\,\widehat i + \widehat k\) and \(c\widehat i + c\widehat j + b\widehat k\) lie in a plane, then \(c\) is :

For any vector \({\overrightarrow a }\) , the value of \({\left( {\overrightarrow a \times \widehat i} \right)^2} + {\left( {\overrightarrow a \times \widehat j} \right)^2} + {\left( {\overrightarrow a \times \widehat k} \right)^2}\) is...

If \(C\) is the mid point of \(AB\) and \(P\) is any point outside \(AB,\) then :

If \(\overrightarrow a ,\overrightarrow b ,\overrightarrow c\) are non coplanar vectors and \(\lambda\) is a real number then $$\left[ {\lambda \left( {\overrightarrow a + \overrightarrow b } \right)\,\,\,\,\,\,\,\,{\lambda ^2}\overright...