AIEEE 2012

If the line \({{x - 1} \over 2} = {{y + 1} \over 3} = {{z - 1} \over 4}\) and \({{x - 3} \over 1} = {{y - k} \over 2} = {z \over 1}\) intersect, then \(k\) is equal to :

A equation of a plane parallel to the plane \(x-2y+2z-5=0\) and at a unit distance from the origin is :

A spherical balloon is filled with \(4500\pi\) cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of \(72\pi\) cubic meters per minute, then the rate (in meters per minute) at which the radius of the...

Let \(a,b \in R\) be such that the function \(f\) given by \(f\left( x \right) = In\left| x \right| + b{x^2} + ax,\,x \ne 0\) has extreme values at \(x=-1\) and \(x=2\)
Statement-1 : \(f\) has local maximum at \(x=-1\) and at \(x=2\).
Stat...

A line is drawn through the point \((1, 2)\) to meet the coordinate axes at \(P\) and \(Q\) such that it forms a triangle \(OPQ,\) where \(O\) is the origin. If the area of the triangle \(OPQ\) is least, then the slope of the line \(PQ\) i...

The area between the parabolas \({x^2} = {y \over 4}\) and \({x^2} = 9y\) and the straight line \(y=2\) is :

If \(n\) is a positive integer, then \({\left( {\sqrt 3 + 1} \right)^{2n}} - {\left( {\sqrt 3 - 1} \right)^{2n}}\) is :

The length of the diameter of the circle which touches the \(x\)-axis at the point \((1, 0)\) and passes through the point \((2, 3)\) is :

If \(z \ne 1\) and \(\,{{{z^2}} \over {z - 1}}\,\) is real, then the point represented by the complex number z lies :

If \(g\left( x \right) = \int\limits_0^x {\cos 4t\,dt,}\) then \(g\left( {x + \pi } \right)\) equals

The population \(p\) \((t)\) at time \(t\) of a certain mouse species satisfies the differential equation \({{dp\left( t \right)} \over {dt}} = 0.5\,p\left( t \right) - 450.\,\,\) If \(p(0)=850,\) then the time at which the population becom...

STATEMENT-1 : An equation of a common tangent to the parabola \({y^2} = 16\sqrt 3 x\) and the ellipse \(2{x^2} + {y^2} = 4\) is \(y = 2x + 2\sqrt 3\)
STATEMENT-2 :If line \(y = mx + {{4\sqrt 3 } \over m},\left( {m \ne 0} \right)\) is a com...

An ellipse is drawn by taking a diameter of thec circle \({\left( {x - 1} \right)^2} + {y^2} = 1\) as its semi-minor axis and a diameter of the circle \({x^2} + {\left( {y - 2} \right)^2} = 4\) is semi-major axis. If the centre of the ellip...

If the \(\int {{{5\tan x} \over {\tan x - 2}}dx = x + a\,\ln \,\left| {\sin x - 2\cos x} \right| + k,}\) then \(a\) is
equal to :

If \(f:R \to R\) is a function defined by
\(f\left( x \right) = \left[ x \right]\cos \left( {{{2x - 1} \over 2}} \right)\pi\),
where [x] denotes the greatest integer function, then \(f\) is

Consider the function, \(f\left( x \right) = \left| {x - 2} \right| + \left| {x - 5} \right|,x \in R\)
Statement - 1 : \(f'\left( 4 \right) = 0\)
Statement - 2 : \(f\) is continuous in [2, 5], differentiable in (2, 5) and \(f\)(2) = \(f\)(5...

The negation of the statement “If I become a teacher, then I will open a school” is :

Let \(P\) and \(Q\) be \(3 \times 3\) matrices \(P \ne Q.\) If \({P^3} = {Q^3}\) and
\({P^2}Q = {Q^2}P\) then determinant of \(\left( {{P^2} + {Q^2}} \right)\) is equal to :

Let \(A = \left( {\matrix{ 1 & 0 & 0 \cr 2 & 1 & 0 \cr 3 & 2 & 1 \cr } } \right)\). If \({u_1}\) and \({u_2}\) are column matrices such
that \(A{u_1} = \left( {\matrix{ 1 \cr 0 \cr 0 \cr } } \right)\) and...

Assuming the balls to be identical except for difference in colours, the number of ways in which one or more balls can be selected from 10 white, 9 green and 7 black balls is:

Three numbers are chosen at random without replacement from \(\left\{ {1,2,3,..8} \right\}.\) The probability that their minimum is \(3,\) given that their maximum is \(6,\) is :

In a \(\Delta PQR,{\mkern 1mu} {\mkern 1mu} {\mkern 1mu}\) If \(3{\mkern 1mu} \sin {\mkern 1mu} P + 4{\mkern 1mu} \cos {\mkern 1mu} Q = 6\) and \(4\sin Q + 3\cos P = 1,\) then the angle R is equal to :

The equation \({e^{\sin x}} - {e^{ - \sin x}} - 4 = 0\) has:

Statement-1: The sum of the series 1 + (1 + 2 + 4) + (4 + 6 + 9) + (9 + 12 + 16) +.....+ (361 + 380 + 400) is 8000.
Statement-2: \(\sum\limits_{k = 1}^n {\left( {{k^3} - {{(k - 1)}^3}} \right)} = {n^3}\), for any natural number n.

Let X = {1, 2, 3, 4, 5}. The number of different ordered pairs (Y, Z) that can be formed such that Y \(\subseteq\) X, Z \(\subseteq\) X and Y \(\cap\) Z is empty, is :

Let x1, x2,........., xn be n observations, and let \(\overline x\) be their arithematic mean and \({\sigma ^2}\) be their variance.
Statement 1 : Variance of 2x1, 2x2,......., 2xn is 4\({\sigma ^2}\).
Statement 2 : : Arithmetic mean of 2x...

If the line \(2x + y = k\) passes through the point which divides the line segment joining the points \((1, 1)\) and \((2, 4)\) in the ratio \(3 : 2\), then \(k\) equals :

Let \(ABCD\) be a parallelogram such that \(\overrightarrow {AB} = \overrightarrow q ,\overrightarrow {AD} = \overrightarrow p\) and \(\angle BAD\) be an acute angle. If \(\overrightarrow r\) is the vector that coincide with the altitud...

Let \(\overrightarrow a\) and \(\overrightarrow b\) be two unit vectors. If the vectors \(\,\overrightarrow c = \widehat a + 2\widehat b\) and \(\overrightarrow d = 5\widehat a - 4\widehat b\) are perpendicular to each other, then the a...