AIEEE 2006

The image of the point \((-1, 3,4)\) in the plane \(x-2y=0\) is :

The two lines \(x=ay+b, z=cy+d;\) and \(x=a'y+b' ,\) \(z=c'y+d'\) are perpendicular to each other if :

A triangular park is enclosed on two sides by a fence and on the third side by a straight river bank. The two sides having fence are of same length \(x\). The maximum area enclosed by the park is

The function \(f\left( x \right) = {x \over 2} + {2 \over x}\) has a local minimum at

Angle between the tangents to the curve \(y = {x^2} - 5x + 6\) at the points \((2,0)\) and \((3,0)\) is

For natural numbers \(m\) , \(n\), if \({\left( {1 - y} \right)^m}{\left( {1 + y} \right)^n}\,\, = 1 + {a_1}y + {a_2}{y^2} + ..........\) and \({a_1} = {a_2} = 10,\) then \(\left( {m,\,n} \right)\) is

If the expansion in powers of \(x\) of the function \({1 \over {\left( {1 - ax} \right)\left( {1 - bx} \right)}}\) is \({a_0} + {a_1}x + {a_2}{x^2} + {a_3}{x^3}.....\) then \({a_n}\) is

If the lines \(3x - 4y - 7 = 0\) and \(2x - 3y - 5 = 0\) are two diameters of a circle of area \(49\pi\) square units, the equation of the circle is :

Let \(C\) be the circle with centre \((0, 0)\) and radius \(3\) units. The equation of the locus of the mid points of the chords of the circle \(C\) that subtend an angle of \({{2\pi } \over 3}\) at its center is :

The value of \(\sum\limits_{k = 1}^{10} {\left( {\sin {{2k\pi } \over {11}} + i\,\,\cos {{2k\pi } \over {11}}} \right)}\) is :

If \({z^2} + z + 1 = 0\), where z is complex number, then value of $${\left( {z + {1 \over z}} \right)^2} + {\left( {{z^2} + {1 \over {{z^2}}}} \right)^2} + {\left( {{z^3} + {1 \over {{z^3}}}} \right)^2} + .......... + {\left( {{z^6} + {1 \...

The value of \(\int\limits_1^a {\left[ x \right]} f'\left( x \right)dx,a > 1\) where \({\left[ x \right]}\) denotes the greatest integer not exceeding \(x\) is

\(\int\limits_{ - {{3\pi } \over 2}}^{ - {\pi \over 2}} {\left[ {{{\left( {x + \pi } \right)}^3} + {{\cos }^2}\left( {x + 3\pi } \right)} \right]} dx\) is equal to

\(\int\limits_0^\pi {xf\left( {\sin x} \right)dx}\) is equal to

The differential equation whose solution is \(A{x^2} + B{y^2} = 1\)
where \(A\) and \(B\) are arbitrary constants is of

If \({x^m}.{y^n} = {\left( {x + y} \right)^{m + n}},\) then \({{{dy} \over {dx}}}\) is

In the ellipse, the distance between its foci is \(6\) and minor axis is \(8\). Then its eccentricity is :

The set of points where \(f\left( x \right) = {x \over {1 + \left| x \right|}}\) is differentiable is

If \(A\) and \(B\) are square matrices of size \(n\, \times \,n\) such that
\({A^2} - {B^2} = \left( {A - B} \right)\left( {A + B} \right),\) then which of the following will be always true?

Let \(A = \left( {\matrix{ 1 & 2 \cr 3 & 4 \cr } } \right)\) and \(B = \left( {\matrix{ a & 0 \cr 0 & b \cr } } \right),a,b \in N.\) Then

The locus of the vertices of the family of parabolas
\(y = {{{a^3}{x^2}} \over 3} + {{{a^2}x} \over 2} - 2a\) is :

At an election, a voter may vote for any number of candidates, not greater than the number to be elected. There are 10 candidates and 4 are of be selected, if a voter votes for at least one candidate, then the number of ways in which he can...

At a telephone enquiry system the number of phone cells regarding relevant enquiry follow Poisson distribution with an average of \(5\) phone calls during \(10\) minute time intervals. The probability that there is at the most one phone cal...

If the roots of the quadratic equation \({x^2} + px + q = 0\) are \(\tan {30^ \circ }\) and \(\tan {15^ \circ }\), respectively, then the value of \(2 + q - p\) is

If \(x\) is real, the maximum value of \({{3{x^2} + 9x + 17} \over {3{x^2} + 9x + 7}}\) is

All the values of \(m\) for which both roots of the equation \({x^2} - 2mx + {m^2} - 1 = 0\) are greater than \(- 2\) but less then 4, lie in the interval

If \({{a_1},{a_2},....{a_n}}\) are in H.P., then the expression \({{a_1}\,{a_2} + \,{a_2}\,{a_3}\, + .... + {a_{n - 1}}\,{a_n}}\) is equal to

Let \({a_1}\), \({a_2}\), \({a_3}\).....be terms on A.P. If \({{{a_1} + {a_2} + .....{a_p}} \over {{a_1} + {a_2} + .....{a_q}}} = {{{p^2}} \over {{q^2}}},\,p \ne q,\,then\,{{{a_6}} \over {{a_{21}}}}\,\) equals

Let $W$ denote the words in the English dictionary. Define the relation $R$ by
$R=\{(x, y) \in W \times W \mid$ the words $x$ and $y$ have at least one letter in common}. Then, $R$ is

Suppose a population A has 100 observations 101, 102,........, 200, and another
population B has 100 observations 151, 152,......., 250. If VA and VB represent the
variances of the two populations, respectively, then $${{{V_A}} \over {{V_B}...

If \(\left( {a,{a^2}} \right)\) falls inside the angle made by the lines \(y = {x \over 2},\) \(x > 0\) and \(y = 3x,\) \(x > 0,\) then a belong to :

A straight line through the point \(A (3, 4)\) is such that its intercept between the axes is bisected at \(A\). Its equation is :

The number of values of \(x\) in the interval \(\left[ {0,3\pi } \right]\,\) satisfying the equation \(2{\sin ^2}x + 5\sin x - 3 = 0\) is

If \(0 < x < \pi\) and \(\cos x + \sin x = {1 \over 2},\) then \(\tan x\) is :

The values of a, for which the points \(A, B, C\) with position vectors \(2\widehat i - \widehat j + \widehat k,\,\,\widehat i - 3\widehat j - 5\widehat k\) and \(a\widehat i - 3\widehat j + \widehat k\) respectively are the vertices of a ...

If \(\left( {\overrightarrow a \times \overrightarrow b } \right) \times \overrightarrow c = \overrightarrow a \times \left( {\overrightarrow b \times \overrightarrow c } \right)\) where \({\overrightarrow a ,\overrightarrow b }\) and $...