AIEEE 2009

Let the line \(\,\,\,\,\,\) \({{x - 2} \over 3} = {{y - 1} \over { - 5}} = {{z + 2} \over 2}\) lie in the plane \(\,\,\,\,\,\) \(x + 3y - \alpha z + \beta = 0.\) Then \(\left( {\alpha ,\beta } \right)\) equals

The projections of a vector on the three coordinate axis are \(6,-3,2\) respectively. The direction cosines of the vector are :

Given \(P\left( x \right) = {x^4} + a{x^3} + b{x^2} + cx + d\) such that \(x=0\) is the only
real root of \(P'\,\left( x \right) = 0.\) If \(P\left( { - 1} \right) < P\left( 1 \right),\) then in the interval \(\left[ { - 1,1} \right]:\)

The area of the region bounded by the parabola \({\left( {y - 2} \right)^2} = x - 1,\) the tangent of the parabola at the point \((2, 3)\) and the \(x\)-axis is :

The remainder left out when \({8^{2n}} - {\left( {62} \right)^{2n + 1}}\) is divided by 9 is :

If \(P\) and \(Q\) are the points of intersection of the circles
\({x^2} + {y^2} + 3x + 7y + 2p - 5 = 0\) and \({x^2} + {y^2} + 2x + 2y - {p^2} = 0\) then there is a circle passing through \(P,Q\) and \((1, 1)\) for :

Three distinct points A, B and C are given in the 2 -dimensional coordinates plane such that the ratio of the distance of any one of them from the point \((1, 0)\) to the distance from the point \((-1, 0)\) is equal to \({1 \over 3}\). Then...

If \(\,\left| {z - {4 \over z}} \right| = 2,\) then the maximum value of \(\,\left| z \right|\) is equal to :

\(\int\limits_0^\pi {\left[ {\cot x} \right]dx,}\) where \(\left[ . \right]\) denotes the greatest integer function, is equal to:

The differential equation which represents the family of curves \(y = {c_1}{e^{{c_2}x}},\) where \({c_1}\) , and \({c_2}\) are arbitrary constants, is

Let \(y\) be an implicit function of \(x\) defined by \({x^{2x}} - 2{x^x}\cot \,y - 1 = 0\). Then \(y'(1)\) equals

The ellipse \({x^2} + 4{y^2} = 4\) is inscribed in a rectangle aligned with the coordinate axex, which in turn is inscribed in another ellipse that passes through the point \((4,0)\). Then the equation of the ellipse is :

For real x, let f(x) = x3 + 5x + 1, then

Let \(f\left( x \right) = {\left( {x + 1} \right)^2} - 1,x \ge - 1\)
Statement - 1 : The set \(\left\{ {x:f\left( x \right) = {f^{ - 1}}\left( x \right)} \right\} = \left\{ {0, - 1} \right\}\).
Statement - 2 : \(f\) is a bijection.

Let \(f\left( x \right) = x\left| x \right|\) and \(g\left( x \right) = \sin x.\)
Statement-1: gof is differentiable at \(x=0\) and its derivative is continuous at that point.
Statement-2: gof is twice differentiable at \(x=0\).

Statement-1 : \(\sim \left( {p \leftrightarrow \sim q} \right)\) is equivalent to \({p \leftrightarrow q}\).
Statement-2 : \(\sim \left( {p \leftrightarrow \sim q} \right)\) is a tautology.

Let \(a, b, c\) be such that \(b\left( {a + c} \right) \ne 0\) if
$$\left| {\matrix{
a & {a + 1} & {a - 1} \cr
{ - b} & {b + 1} & {b - 1} \cr
c & {c - 1} & {c + 1} \cr

} } \right| + \left| {\matrix{
{a + 1} & {b + 1} & {...

Let \(A\) be a \(\,2 \times 2\) matrix
Statement - 1 : \(adj\left( {adj\,A} \right) = A\)
Statement - 2 :\(\left| {adj\,A} \right| = \left| A \right|\)

From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are to be selected and arranged in a row on a shelf so that the dictionary is always in the middle. Then the number of such arrangement is :

One ticket is selected at random from \(50\) tickets numbered \(00, 01, 02, ...., 49.\) Then the probability that the sum of the digits on the selected ticket is \(8\), given that the product of these digits is zer, equals :

In a binomial distribution \(B\left( {n,p = {1 \over 4}} \right),\) if the probability of at least one success is greater than or equal to \({9 \over {10}},\) then \(n\) is greater than :

If the roots of the equation \(b{x^2} + cx + a = 0\) imaginary, then for all real values of \(x\), the expression \(3{b^2}{x^2} + 6bcx + 2{c^2}\) is :

The sum to infinite term of the series \(1 + {2 \over 3} + {6 \over {{3^2}}} + {{10} \over {{3^3}}} + {{14} \over {{3^4}}} + .....\) is

If $A, B$ and $C$ are three sets such that $A \cap B=A \cap C$ and $A \cup B=A \cup C$, then :

Statement - 1 : The variance of first n even natural numbers is \({{{n^2} - 1} \over 4}\)
Statement - 2 : The sum of first n natural numbers is \({{n\left( {n + 1} \right)} \over 2}\) and the sum of squares of first n natural numbers is $${...

If the mean deviation of number 1, 1 + d, 1 + 2d,........, 1 + 100d from their mean is 255, then the d is
equal to

The lines \(p\left( {{p^2} + 1} \right)x - y + q = 0\) and \(\left( {{p^2} + 1} \right){}^2x + \left( {{p^2} + 1} \right)y + 2q\) \(=0\) are perpendicular to a common line for :

The shortest distance between the line \(y - x = 1\) and the curve \(x = {y^2}\) is :

Let A and B denote the statements
A: \(\cos \alpha + \cos \beta + \cos \gamma = 0\)
B: \(\sin \alpha + \sin \beta + \sin \gamma = 0\)
If $$\cos \left( {\beta - \gamma } \right) + \cos \left( {\gamma - \alpha } \right) + \cos \left(...

If \(\overrightarrow u ,\overrightarrow v ,\overrightarrow w\) are non-coplanar vectors and \(p,q\) are real numbers, then the equality $$\left[ {3\overrightarrow u \,\,p\overrightarrow v \,\,p\overrightarrow w } \right] - \left[ {p\overri...