AIEEE 2011

If the angle between the line \(x = {{y - 1} \over 2} = {{z - 3} \over \lambda }\) and the plane
\(x+2y+3z=4\) is \({\cos ^{ - 1}}\left( {\sqrt {{5 \over {14}}} } \right),\) then \(\lambda\) equals :

Statement - 1 : The point \(A(1,0,7)\) is the mirror image of the point
\(B(1,6,3)\) in the line : \({x \over 1} = {{y - 1} \over 2} = {{z - 2} \over 3}\)
Statement - 2 : The line \({x \over 1} = {{y - 1} \over 2} = {{z - 2} \over 3}\) bi...

For \(x \in \left( {0,{{5\pi } \over 2}} \right),\) define \(f\left( x \right) = \int\limits_0^x {\sqrt t \sin t\,dt.}\) Then \(f\) has

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

The area of the region enclosed by the curves \(y = x,x = e,y = {1 \over x}\) and the positive \(x\)-axis is :

The coefficient of \({x^7}\) in the expansion of \({\left( {1 - x - {x^2} + {x^3}} \right)^6}\) is

The two circles x2 + y2 = ax, and x2 + y2 = c2 (c > 0) touch each other if :

Let \(\alpha \,,\beta\) be real and z be a complex number. If \({z^2} + \alpha z + \beta = 0\) has two distinct roots on the line Re z = 1, then it is necessary that :

If \(\omega ( \ne 1)\) is a cube root of unity, and \({(1 + \omega )^7} = A + B\omega \,\). Then \((A,B)\) equals :

The value of \(\int\limits_0^1 {{{8\log \left( {1 + x} \right)} \over {1 + {x^2}}}} dx\) is

Let \(I\) be the purchase value of an equipment and \(V(t)\) be the value after it has been used for \(t\) years. The value \(V(t)\) depreciates at a rate given by differential equation $${{dv\left( t \right)} \over {dt}} = - k\left( {T - ...

If \({{dy} \over {dx}} = y + 3 > 0\,\,\) and \(y(0)=2,\) then \(y\left( {\ln 2} \right)\) is equal to :

\({{{d^2}x} \over {d{y^2}}}\) equals:

Equation of the ellipse whose axes of coordinates and which passes through the point \((-3,1)\) and has eccentricity \(\sqrt {{2 \over 5}}\) is :

The domain of the function f(x) = \({1 \over {\sqrt {\left| x \right| - x} }}\) is

The value of \(p\) and \(q\) for which the function
$$f\left( x \right) = \left\{ {\matrix{
{{{\sin (p + 1)x + \sin x} \over x}} & {,x < 0} \cr
q & {,x = 0} \cr
{{{\sqrt {x + {x^2}} - \sqrt x } \over {{x^{3/2}}}}} & {,x > 0} ...

\(\mathop {\lim }\limits_{x \to 2} \left( {{{\sqrt {1 - \cos \left\{ {2(x - 2)} \right\}} } \over {x - 2}}} \right)\)

Consider the following statements
P : Suman is brilliant
Q : Suman is rich
R : Suman is honest
The negation of the statement,
“Suman is brilliant and dishonest if and only if Suman is rich” can be
expressed as :

Let \(A\) and \(B\) be two symmetric matrices of order \(3\).
Statement - 1 : \(A(BA)\) and \((AB)\)\(A\) are symmetric matrices.
Statement - 2 : \(AB\) is symmetric matrix if matrix multiplication of \(A\) with \(B\) is commutative.

The number of values of \(k\) for which the linear equations
\(4x + ky + 2z = 0,kx + 4y + z = 0\) and \(2x+2y+z=0\) possess a non-zero solution is :

These are 10 points in a plane, out of these 6 are collinear, if N is the number of triangles formed by joining these points. then:

Statement - 1: The number of ways of distributing 10 identical balls in 4 distinct boxes such that no box is emply is \({}^9{C_3}\).
Statement - 2: The number of ways of choosing any 3 places from 9 different places is \({}^9{C_3}\).

If \(C\) and \(D\) are two events such that \(C \subset D\) and \(P\left( D \right) \ne 0,\) then the correct statement among the following is :

Consider \(5\) independent Bernoulli's trials each with probability of success \(p.\) If the probability of at least one failure is greater than or equal to \({{31} \over 32},\) then \(p\) lies in the interval :

A man saves ₹ 200 in each of the first three months of his service. In each of the subsequent months his saving increases by ₹ 40 more than the saving of immediately previous month. His total saving from the start of service will be ₹ 110...

Let $R$ be the set of real numbers.
Statement I : $A=\{(x, y) \in R \times R: y-x$ is an integer $\}$ is an equivalence relation on $R$.
Statement II : $ B=\{(x, y) \in R \times R: x=\alpha y$ for some rational number $\alpha\}$ is an equiv...

If the mean deviation about the median of the numbers a, 2a,........., 50a is 50, then |a| equals

The lines \({L_1}:y - x = 0\) and \({L_2}:2x + y = 0\) intersect the line \({L_3}:y + 2 = 0\) at \(P\) and \(Q\) respectively. The bisector of the acute angle between \({L_1}\) and \({L_2}\) intersects \({L_3}\) at \(R\).
Statement-1: The ...

If \(A = {\sin ^2}x + {\cos ^4}x,\) then for all real \(x\):

The vectors \(\overrightarrow a\) and \(\overrightarrow b\) are not perpendicular and \(\overrightarrow c\) and \(\overrightarrow d\) are two vectors satisfying $$\overrightarrow b \times \overrightarrow c = \overrightarrow b \times ...

Let \(\overrightarrow a\), \(\overrightarrow b\), \(\overrightarrow c\) be three non-zero vectors which are pairwise non-collinear. If $\overrightarrow a+3 \overrightarrow b$ is collinear with $\overrightarrow c$ and $\overrightarrow b+2...

If \(\overrightarrow a = {1 \over {\sqrt {10} }}\left( {3\widehat i + \widehat k} \right)\) and \(\overrightarrow b = {1 \over 7}\left( {2\widehat i + 3\widehat j - 6\widehat k} \right),\) then the value
of $$\left( {2\overrightarrow a ...