AIEEE 2005

If the angel \(\theta\) between the line \({{x + 1} \over 1} = {{y - 1} \over 2} = {{z - 2} \over 2}\) and
the plane \(2x - y + \sqrt \lambda \,\,z + 4 = 0\) is such that \(\sin \,\,\theta = {1 \over 3}\) then value of \(\lambda\) is :

The distance between the line
\(\overrightarrow r = 2\widehat i - 2\widehat j + 3\widehat k + \lambda \left( {i - j + 4k} \right),\) and the plane
\(\overrightarrow r .\left( {\widehat i + 5\widehat j + \widehat k} \right) = 5\) is

The plane \(x+2y-z=4\) cuts the sphere \({x^2} + {y^2} + {z^2} - x + z - 2 = 0\) in a circle of radius

If the plane \(2ax-3ay+4az+6=0\) passes through the midpoint of the line joining the centres of the spheres
\({x^2} + {y^2} + {z^2} + 6x - 8y - 2z = 13\) and
\({x^2} + {y^2} + {z^2} - 10x + 4y - 2z = 8\) then a equals :

The angle between the lines \(2x=3y=-z\) and \(6x=-y=-4z\) is :

A function is matched below against an interval where it is supposed to be
increasing. Which of the following pairs is incorrectly matched?

Area of the greatest rectangle that can be inscribed in the
ellipse \({{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1\)

If the equation \({a_n}{x^n} + {a_{n - 1}}{x^{n - 1}} + ........... + {a_1}x = 0\)
\({a_1} \ne 0,n \ge 2,\) has a positive root \(x = \alpha\), then the equation
$$n{a_n}{x^{n - 1}} + \left( {n - 1} \right){a_{n - 1}}{x^{n - 2}} + ..........

Let f be differentiable for all x. If f(1) = -2 and f'(x) \(\ge\) 2 for
x \(\in \left[ {1,6} \right]\), then

A spherical iron ball \(10\) cm in radius is coated with a layer of ice of uniform thickness that melts at a rate of \(50\) cm\(^3\) /min. When the thickness of ice is \(5\) cm, then the rate at which the thickness of ice decreases is

A lizard, at an initial distance of 21 cm behind an insect moves from rest with an acceleration of $2 \mathrm{~cm} / \mathrm{s}^2$ and pursues the insect which is crawling uniformly along a straight line at a speed of $20 \mathrm{~cm} / \ma...

The normal to the curve
\(x = a\left( {\cos \theta + \theta \sin \theta } \right),y = a\left( {\sin \theta - \theta \cos \theta } \right)\) at any point
\(\theta\, '\) is such that

The parabolas \({y^2} = 4x\) and \({x^2} = 4y\) divide the square region bounded by the lines \(x=4,\) \(y=4\) and the coordinate axes. If \({S_1},{S_2},{S_3}\) are respectively the areas of these parts numbered from top to bottom ; then $$...

Let \(f(x)\) be a non - negative continuous function such that the area bounded by the curve \(y=f(x),\) \(x\)-axis and the ordinates \(x = {\pi \over 4}\) and \(x = \beta > {\pi \over 4}\) is $$\left( {\beta \sin \beta + {\pi \over 4...

The area enclosed between the curve \(y = {\log _e}\left( {x + e} \right)\) and the coordinate axes is :

The value of \(\,{}^{50}{C_4} + \sum\limits_{r = 1}^6 {^{56 - r}} {C_3}\) is

If \(x\) is so small that \({x^3}\) and higher powers of \(x\) may be neglected, then \({{{{\left( {1 + x} \right)}^{{3 \over 2}}} - {{\left( {1 + {1 \over 2}x} \right)}^3}} \over {{{\left( {1 - x} \right)}^{{1 \over 2}}}}}\) may be approxi...

If the coefficients of rth, (r+1)th, and (r + 2)th terms in the binomial expansion of \({{\rm{(1 + y )}}^m}\) are in A.P., then m and r satisfy the equation

If the coefficient of \({x^7}\) in \({\left[ {a{x^2} + \left( {{1 \over {bx}}} \right)} \right]^{11}}\) equals the coefficient of \({x^{ - 7}}\) in \({\left[ {ax - \left( {{1 \over {b{x^2}}}} \right)} \right]^{11}}\), then \(a\) and \(b\)...

If a circle passes through the point (a, b) and cuts the circle \({x^2}\, + \,{y^2} = {p^2}\) orthogonally, then the equation of the locus of its centre is :

A circle touches the x-axis and also touches the circle with centre at (0, 3) and radius 2. The locus of the centre of the circle is :

If the pair of lines \(a{x^2} + 2\left( {a + b} \right)xy + b{y^2} = 0\) lie along diameters of a circle and divide the circle into four sectors such that the area of one of the sectors is thrice the area of another sector then :

If the circles \({x^2}\, + \,{y^2} + \,2ax\, + \,cy\, + a\,\, = 0\) and \({x^2}\, + \,{y^2} - \,3ax\, + \,dy\, - 1\,\, = 0\) intersect in two ditinct points P and Q then the line 5x + by - a = 0 passes through P and Q for :

If \({z_1}\) and \({z_2}\) are two non-zero complex numbers such that \(\,\left| {{z_1} + {z_2}} \right| = \left| {{z_1}} \right| + \left| {{z_2}} \right|\), then arg \({z_1}\) - arg \({z_2}\) is equal to :

If the cube roots of unity are 1, \(\omega \,,\,{\omega ^2}\) then the roots of the equation \({(x - 1)^3}\) + 8 = 0, are :

If \(\,\omega = {z \over {z - {1 \over 3}i}}\,\) and \(\left| \omega \right| = 1\), then \(z\) lies on :

If \({I_1} = \int\limits_0^1 {{2^{{x^2}}}dx,{I_2} = \int\limits_0^1 {{2^{{x^3}}}dx,\,{I_3} = \int\limits_1^2 {{2^{{x^2}}}dx} } }\) and \({I_4} = \int\limits_1^2 {{2^{{x^3}}}dx}\) then

The value of integral, \(\int\limits_3^6 {{{\sqrt x } \over {\sqrt {9 - x} + \sqrt x }}} dx\) is

Let \(f:R \to R\) be a differentiable function having \(f\left( 2 \right) = 6\),
\(f'\left( 2 \right) = \left( {{1 \over {48}}} \right)\). Then $$\mathop {\lim }\limits_{x \to 2} \int\limits_6^{f\left( x \right)} {{{4{t^3}} \over {x - 2...

The value of \(\int\limits_{ - \pi }^\pi {{{{{\cos }^2}} \over {1 + {a^x}}}dx,\,\,a > 0,}\) is

\(\mathop {\lim }\limits_{n \to \infty } \left[ {{1 \over {{n^2}}}{{\sec }^2}{1 \over {{n^2}}} + {2 \over {{n^2}}}{{\sec }^2}{4 \over {{n^2}}}.... + {1 \over n}{{\sec }^2}1} \right]\)
equals

The differential equation representing the family of curves \({y^2} = 2c\left( {x + \sqrt c } \right),\) where \(c>0,\) is a parameter, is of order and degree as follows:

If \(x{{dy} \over {dx}} = y\left( {\log y - \log x + 1} \right),\) then the solution of the equation is :

An ellipse has \(OB\) as semi minor axis, \(F\) and \(F\)' its focii and theangle \(FBF\)' is a right angle. Then the eccentricity of the ellipse is :

A real valued function f(x) satisfies the functional equation
f(x - y) = f(x)f(y) - f(a - x)f(a + y)
where a is given constant and f(0) = 1, f(2a - x) is equal to

Let \(f:( - 1,1) \to B\), be a function defined by
\(f\left( x \right) = {\tan ^{ - 1}}{{2x} \over {1 - {x^2}}}\),
then \(f\) is both one-one and onto when B is the interval

The locus of a point \(P\left( {\alpha ,\beta } \right)\) moving under the condition that the line \(y = \alpha x + \beta\) is tangent to the hyperbola \({{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1\) is :

\(\int {{{\left\{ {{{\left( {\log x - 1} \right)} \over {1 + {{\left( {\log x} \right)}^2}}}} \right\}}^2}\,\,dx}\) is equal to

If \({\cos ^{ - 1}}x - {\cos ^{ - 1}}{y \over 2} = \alpha ,\) then \(4{x^2} - 4xy\cos \alpha + {y^2}\) is equal to :

Let \(\alpha\) and \(\beta\) be the distinct roots of \(a{x^2} + bx + c = 0\), then
\(\mathop {\lim }\limits_{x \to \alpha } {{1 - \cos \left( {a{x^2} + bx + c} \right)} \over {{{\left( {x - \alpha } \right)}^2}}}\) is equal to

Suppose \(f(x)\) is differentiable at x = 1 and
\(\mathop {\lim }\limits_{h \to 0} {1 \over h}f\left( {1 + h} \right) = 5\), then \(f'\left( 1 \right)\) equals

If \(f\) is a real valued differentiable function satisfying
\(\left| {f\left( x \right) - f\left( y \right)} \right|\) \(\le {\left( {x - y} \right)^2}\), \(x, y\) \(\in R\)
and \(f(0)\) = 0, then \(f(1)\) equals

If \(A = \left[ {\matrix{ 1 & 0 \cr 1 & 1 \cr } } \right]\) and \(I = \left[ {\matrix{ 1 & 0 \cr 0 & 1 \cr } } \right],\) then which one of the following holds for all \(n \ge 1,\) by the principle of mathematical in...

The system of equations
\(\matrix{ {\alpha \,x + y + z = \alpha - 1} \cr {x + \alpha y + z = \alpha - 1} \cr {x + y + \alpha \,z = \alpha - 1} \cr }\)
has no solutions, if \(\alpha\) is :

If \({a_1},{a_2},{a_3},........,{a_n},.....\) are in G.P., then the determinant
$$$\Delta = \left| {\matrix{
{\log {a_n}} & {\log {a_{n + 1}}} & {\log {a_{n + 2}}} \cr
{\log {a_{n + 3}}} & {\log {a_{n + 4}}} & {\log {a_{n + 5}}} \...

If \({A^2} - A + 1 = 0\), then the inverse of \(A\) is :

If \({a^2} + {b^2} + {c^2} = - 2\) and
f$$\left( x \right) = \left| {\matrix{
{1 + {a^2}x} & {\left( {1 + {b^2}} \right)x} & {\left( {1 + {c^2}} \right)x} \cr
{\left( {1 + {a^2}} \right)x} & {1 + {b^2}x} & {\left( {1 + {c^2}} \rig...

Let \(P\) be the point \((1, 0)\) and \(Q\) a point on the parabola \({y^2} = 8x\). The locus of mid point of \(PQ\) is :

If the letter of the word SACHIN are arranged in all possible ways and these words are written out as in dictionary, then the word SACHIN appears at serial number

Three houses are available in a locality. Three persons apply for the houses. Each applies for one house without consulting others. The probability that all the three apply for the same house is :