AIEEE 2010

A line \(AB\) in three-dimensional space makes angles \({45^ \circ }\) and \({120^ \circ }\) with the positive \(x\)-axis and the positive \(y\)-axis respectively. If \(AB\) makes an acute angle \(\theta\) with the positive \(z\)-axis, th...

Statement-1 : The point \(A(3, 1, 6)\) is the mirror image of the point \(B(1, 3, 4)\) in the plane \(x-y+z=5.\)
Statement-2 : The plane \(x-y+z=5\) bisects the line segment joining \(A(3, 1, 6)\) and \(B(1, 3, 4).\)

Let \(f:R \to R\) be defined by
\($f\left( x \right) = \left\{ {\matrix{ {k - 2x,\,\,if} & {x \le - 1} \cr {2x + 3,\,\,if} & {x > - 1} \cr } } \right.\)$
If \(f\)has a local minimum at \(x=-1\), then a possible value of \(k\)...

The equation of the tangent to the curve \(y = x + {4 \over {{x^2}}}\), that
is parallel to the \(x\)-axis, is

Let \(f:R \to R\) be a continuous function defined by
\($f\left( x \right) = {1 \over {{e^x} + 2{e^{ - x}}}}\)$
Statement - 1 : \(f\left( c \right) = {1 \over 3},\) for some \(c \in R\).
Statement - 2 : $$0 < f\left( x \right) \le {1 \over...

The area bounded by the curves \(y = \cos x\) and \(y = \sin x\) between the ordinates \(x=0\) and \(x = {{3\pi } \over 2}\) is

Let \({s_1} = \sum\limits_{j = 1}^{10} {j\left( {j - 1} \right){}^{10}} {C_j}\),
\({{s_2} = \sum\limits_{j = 1}^{10} {} } j.{}^{10}{C_j}\) and \({{s_3} = \sum\limits_{j = 1}^{10} {{j^2}.{}^{10}{C_j}.} }\)
Statement-1 : $${{S_3} = 55 \times...

The circle \({x^2} + {y^2} = 4x + 8y + 5\) intersects the line \(3x - 4y = m\) at two distinct points if :

The number of complex numbers z such that \(\left| {z - 1} \right| = \left| {z + 1} \right| = \left| {z - i} \right|\) equals :

Let \(p(x)\) be a function defined on \(R\) such that \(p'(x)=p'(1-x),\) for all \(x \in \left[ {0,1} \right],p\left( 0 \right) = 1\) and \(p(1)=41.\) Then \(\int\limits_0^1 {p\left( x \right)dx}\) equals :

Solution of the differential equation
\(\cos x\,dy = y\left( {\sin x - y} \right)dx,\,\,0 < x <{\pi \over 2}\) is :

Let \(f:\left( { - 1,1} \right) \to R\) be a differentiable function with \(f\left( 0 \right) = - 1\) and \(f'\left( 0 \right) = 1\). Let \(g\left( x \right) = {\left[ {f\left( {2f\left( x \right) + 2} \right)} \right]^2}\). Then $$g'\lef...

Let \(f:R \to R\) be a positive increasing function with
\(\mathop {\lim }\limits_{x \to \infty } {{f(3x)} \over {f(x)}} = 1\). Then \(\mathop {\lim }\limits_{x \to \infty } {{f(2x)} \over {f(x)}} =\)

Let S be a non-empty subset of R. Consider the following statement:
P : There is a rational number x ∈ S such that x > 0.
Which of the following statements is the negation of the statement P?

Let \(A\) be a \(\,2 \times 2\) matrix with non-zero entries and let \({A^2} = I,\)
where \(I\) is \(2 \times 2\) identity matrix. Define
\(Tr\)\((A)=\) sum of diagonal elements of \(A\) and \(\left| A \right| =\) determinant of matrix $...

The number of \(3 \times 3\) non-singular matrices, with four entries as \(1\) and all other entries as \(0\), is :

Consider the system of linear equations;
\($\matrix{ {{x_1} + 2{x_2} + {x_3} = 3} \cr {2{x_1} + 3{x_2} + {x_3} = 3} \cr {3{x_1} + 5{x_2} + 2{x_3} = 1} \cr }\)$
The system has :

If two tangents drawn from a point \(P\) to the parabola \({y^2} = 4x\) are at right angles, then the locus of \(P\) is

There are two urns. Urn A has 3 distinct red balls and urn B has 9 distinct blue balls. From each urn two balls are taken out at random and then transferred to the other. The number of ways in which this can be done is

An urn contains nine balls of which three are red, four are blue and two are green. Three balls are drawn at random without replacement from the urn. The probability that the three balls have different colours is :

Four numbers are chosen at random (without replacement) from the set \(\left\{ {1,2,3,....20} \right\}.\)
Statement - 1: The probability that the chosen numbers when arranged in some order will form an AP is \({1 \over {85}}.\)
Statement -...

For a regular polygon, let \(r\) and \(R\) be the radii of the inscribed and the circumscribed circles. A \(false\) statement among the following is :

If \(\alpha\) and \(\beta\) are the roots of the equation \({x^2} - x + 1 = 0,\) then \({\alpha ^{2009}} + {\beta ^{2009}} =\)

A person is to count 4500 currency notes. Let \({a_n}\) denote the number of notes he counts in the \({n^{th}}\) minute. If \({a_1}\) = \({a_2}\) = ....= \({a_{10}}\)= 150 and \({a_{10}}\), \({a_{11}}\),.... are in an AP with common differe...

Consider the following relations
$R=\{(x, y) \mid x, y$ are real numbers and $x=w y$ for some rational number $w\}$;
$S=\left\{\left(\frac{m}{n}, \frac{p}{q}\right) \mid m, n, p\right.$ and $q$ are integers such that $n, q \neq 0$ and $q ...

For two data sets, each of size 5, the variances are given to be 4 and 5 and the corresponding
means are given to be 2 and 4, respectively. The variance of the combined data set is

The line \(L\) given by \({x \over 5} + {y \over b} = 1\) passes through the point \(\left( {13,32} \right)\). The line K is parrallel to \(L\) and has the equation \({x \over c} + {y \over 3} = 1.\) Then the distance between \(L\) and $$K$...

Let \(\cos \left( {\alpha + \beta } \right) = {4 \over 5}\) and \(\sin \,\,\,\left( {\alpha - \beta } \right) = {5 \over {13}},\) where \(0 \le \alpha ,\,\beta \le {\pi \over 4}.\)
Then \(tan\,2\alpha\) =

If the vectors \(\overrightarrow a = \widehat i - \widehat j + 2\widehat k,\,\,\,\,\,\overrightarrow b = 2\widehat i + 4\widehat j + \widehat k\,\,\,\) and \(\,\overrightarrow c = \lambda \widehat i + \widehat j + \mu \widehat k\) are m...

Let \(\overrightarrow a = \widehat j - \widehat k\) and \(\overrightarrow c = \widehat i - \widehat j - \widehat k.\) Then the vector \(\overrightarrow b\) satisfying $$\overrightarrow a \times \overrightarrow b + \overrightarrow c = ...