AIEEE 2004

Let two numbers have arithmetic mean 9 and geometric mean 4. Then these numbers are the roots of the quadratic equation

The sum of the first n terms of the series \({1^2} + {2.2^2} + {3^2} + {2.4^2} + {5^2} + {2.6^2} + ....\,is\,{{n{{(n + 1)}^2}} \over 2}\) when n is even. When n is odd the sum is

The sum of series \({1 \over {2\,!}} + {1 \over {4\,!}} + {1 \over {6\,!}} + ........\) is

Let \({{T_r}}\) be the rth term of an A.P. whose first term is a and common difference is d. If for some positive integers m, n, \(m \ne n,\,\,{T_m} = {1 \over n}\,\,and\,{T_n} = {1 \over m},\,\) then a - d equals

Let $R=\{(1,3),(4,2),(2,4),(2,3),(3,1)\}$ be a relation on the set $A=\{1,2,3,4\}$. The relation $R$ is :

Consider the following statements:
(a) Mode can be computed from histogram
(b) Median is not independent of change of scale
(c) Variance is independent of change of origin and scale.
Which of these is/are correct?

In a series of 2n observations, half of them equal \(a\) and remaining half equal \(–a\). If the
standard deviation of the observations is 2, then \(|a|\) equals

If one of the lines given by \(6{x^2} - xy + 4c{y^2} = 0\) is \(3x + 4y = 0,\) then \(c\) equals :

If the sum of the slopes of the lines given by \({x^2} - 2cxy - 7{y^2} = 0\) is four times their product \(c\) has the value :

The equation of the straight line passing through the point \((4, 3)\) and making intercepts on the co-ordinate axes whose sum is \(-1\) is :

Let \(A\left( {2, - 3} \right)\) and \(B\left( {-2, 1} \right)\) be vertices of a triangle \(ABC\). If the centroid of this triangle moves on the line \(2x + 3y = 1\), then the locus of the vertex \(C\) is the line :

Let \(\alpha ,\,\beta\) be such that \(\pi < \alpha - \beta < 3\pi\).
If \(sin{\mkern 1mu} \alpha + \sin \beta = - {{21} \over {65}}\) and \(\cos \alpha + \cos \beta = - {{27} \over {65}}\) then the value of $$\cos {{\alpha - ...

If \(u = \sqrt {{a^2}{{\cos }^2}\theta + {b^2}{{\sin }^2}\theta } + \sqrt {{a^2}{{\sin }^2}\theta + {b^2}{{\cos }^2}\theta }\)
then the difference between the maximum and minimum values of \({u^2}\) is given by :

Let \(\overrightarrow u ,\overrightarrow v ,\overrightarrow w\) be such that \(\left| {\overrightarrow u } \right| = 1,\,\,\,\left| {\overrightarrow v } \right|2,\,\,\,\left| {\overrightarrow w } \right|3.\) If the projection $${\overright...

Let \(\overrightarrow a ,\overrightarrow b\) and \(\overrightarrow c\) be non-zero vectors such that $$\left( {\overrightarrow a \times \overrightarrow b } \right) \times \overrightarrow c = {1 \over 3}\left| {\overrightarrow b } \right...

A particle acted on by constant forces \(4\widehat i + \widehat j - 3\widehat k\) and \(3\widehat i + \widehat j - \widehat k\) is displaced from the point \(\widehat i + 2\widehat j + 3\widehat k\) to the point $$\,5\widehat i + 4\widehat ...

If \({\overrightarrow a ,\overrightarrow b ,\overrightarrow c }\) are non-coplanar vectors and \(\lambda\) is a real number, then the vectors $${\overrightarrow a + 2\overrightarrow b + 3\overrightarrow c ,\,\,\lambda \overrightarrow b ...

Let \(\overrightarrow a ,\overrightarrow b\) and \(\overrightarrow c\) be three non-zero vectors such that no two of these are collinear. If the vector \(\overrightarrow a + 2\overrightarrow b\) is collinear with \(\overrightarrow c\) ...