AIEEE 2004

If the straight lines
\(x=1+s,y=-3\)\(- \lambda s,\) \(z = 1 + \lambda s\) and \(x = {t \over 2},y = 1 + t,z = 2 - t,\) with parameters \(s\) and \(t\) respectively, are co-planar, then \(\lambda\) equals :

A line with direction cosines proportional to \(2,1,2\) meets each of the lines \(x=y+a=z\) and \(x+a=2y=2z\) . The co-ordinates of each of the points of intersection are given by :

A line makes the same angle \(\theta\), with each of the \(x\) and \(z\) axis.
If the angle \(\beta \,\), which it makes with y-axis, is such that \(\,{\sin ^2}\beta = 3{\sin ^2}\theta ,\) then \({\cos ^2}\theta\) equals :

Distance between two parallel planes
\(\,2x + y + 2z = 8\) and \(4x + 2y + 4z + 5 = 0\) is :

The intersection of the spheres
\({x^2} + {y^2} + {z^2} + 7x - 2y - z = 13\) and
\({x^2} + {y^2} + {z^2} - 3x + 3y + 4z = 8\)
is the same as the intersection of one of the sphere and the plane

A point on the parabola \({y^2} = 18x\) at which the ordinate increases at twice the rate of the abscissa is

The normal to the curve x = a(1 + cos \(\theta\)), \(y = a\sin \theta\) at \('\theta '\) always passes through the fixed point

A function \(y=f(x)\) has a second order derivative \(f''\left( x \right) = 6\left( {x - 1} \right).\) If its graph passes through the point \((2, 1)\) and at that point the tangent to the graph is \(y = 3x - 5\), then the function is :

If \(2a+3b+6c=0\), then at least one root of the equation
\(a{x^2} + bx + c = 0\) lies in the interval

The area of the region bounded by the curves
\(y = \left| {x - 2} \right|,x = 1,x = 3\) and the \(x\)-axis is :

The coefficient of \({x^n}\) in expansion of \(\left( {1 + x} \right){\left( {1 - x} \right)^n}\) is

If \({S_n} = \sum\limits_{r = 0}^n {{1 \over {{}^n{C_r}}}} \,\,and\,\,{t_n} = \sum\limits_{r = 0}^n {{r \over {{}^n{C_r}}},\,}\)then \({{{t_{ n}}} \over {{S_n}}}\) is equal to

The coefficient of the middle term in the binomial expansion in powers of \(x\) of \({\left( {1 + \alpha x} \right)^4}\) and \({\left( {1 - \alpha x} \right)^6}\) is the same if \(\alpha\) equals

Intercept on the line y = x by the circle \({x^2}\, + \,{y^2} - 2x = 0\) is AB. Equation of the circle on AB as a diameter is :

If the lines 2x + 3y + 1 + 0 and 3x - y - 4 = 0 lie along diameter of a circle of circumference \(10\,\pi\), then the equation of the circle is :

If a circle passes through the point (a, b) and cuts the circle \({x^2}\, + \,{y^2} = 4\) orthogonally, then the locus of its centre is :

A variable circle passes through the fixed point A (p, q) and touches x-axis. The locus of the other end of the diameter through A is :

Let z and w be complex numbers such that \(\overline z + i\overline w = 0\) and arg zw = \(\pi\). Then arg z equals :

If \(\,\left| {{z^2} - 1} \right| = {\left| z \right|^2} + 1\), then z lies on :

If \(z = x - iy\) and \({z^{{1 \over 3}}} = p + iq\), then
\({{\left( {{x \over p} + {y \over q}} \right)} \over {\left( {{p^2} + {q^2}} \right)}}\) is equal to :

The value of \(I = \int\limits_0^{\pi /2} {{{{{\left( {\sin x + \cos x} \right)}^2}} \over {\sqrt {1 + \sin 2x} }}dx}\) is

If \(\int\limits_0^\pi {xf\left( {\sin x} \right)dx = A\int\limits_0^{\pi /2} {f\left( {\sin x} \right)dx,} }\) then \(A\) is

If \(f\left( x \right) = {{{e^x}} \over {1 + {e^x}}},{I_1} = \int\limits_{f\left( { - a} \right)}^{f\left( a \right)} {xg\left\{ {x\left( {1 - x} \right)} \right\}dx}\)
and $${I_2} = \int\limits_{f\left( { - a} \right)}^{f\left( a \right)...

The value of \(\int\limits_{ - 2}^3 {\left| {1 - {x^2}} \right|dx}\) is

\(\mathop {Lim}\limits_{n \to \infty } \sum\limits_{r = 1}^n {{1 \over n}{e^{{r \over n}}}}\) is

Solution of the differential equation \(ydx + \left( {x + {x^2}y} \right)dy = 0\) is

The differential equation for the family of circle \({x^2} + {y^2} - 2ay = 0,\) where a is an arbitrary constant is :

If \(x = {e^{y + {e^y} + {e^{y + .....\infty }}}}\) , \(x > 0,\) then \({{{dy} \over {dx}}}\) is

The eccentricity of an ellipse, with its centre at the origin, is \({1 \over 2}\). If one of the directrices is \(x=4\), then the equation of the ellipse is :

The graph of the function y = f(x) is symmetrical about the line x = 2, then

The domain of the function
\(f\left( x \right) = {{{{\sin }^{ - 1}}\left( {x - 3} \right)} \over {\sqrt {9 - {x^2}} }}\)

The range of the function f(x) = \({}^{7 - x}{P_{x - 3}}\) is

If \(f:R \to S\), defined by
\(f\left( x \right) = \sin x - \sqrt 3 \cos x + 1\),
is onto, then the interval of \(S\) is

A person standing on the bank of a river observes that the angle of elevation of the top of a tree on the opposite bank of the river is \({60^ \circ }\) and when he retires \(40\) meters away from the tree the angle of elevation becomes $${...

If \(\int {{{\sin x} \over {\sin \left( {x - \alpha } \right)}}dx = Ax + B\log \sin \left( {x - \alpha } \right), + C,}\) then value of
\((A, B)\) is

\(\int {{{dx} \over {\cos x - \sin x}}}\) is equal to

Let \(f(x) = {{1 - \tan x} \over {4x - \pi }}\), \(x \ne {\pi \over 4}\), \(x \in \left[ {0,{\pi \over 2}} \right]\).
If \(f(x)\) is continuous in \(\left[ {0,{\pi \over 2}} \right]\), then \(f\left( {{\pi \over 4}} \right)\) is

If \(\mathop {\lim }\limits_{x \to \infty } {\left( {1 + {a \over x} + {b \over {{x^2}}}} \right)^{2x}} = {e^2}\), then the value of \(a\) and \(b\), are

Let \(S(K)\) \(= 1 + 3 + 5... + \left( {2K - 1} \right) = 3 + {K^2}.\) Then which of the following is true

Let \(A = \left( {\matrix{ 0 & 0 & { - 1} \cr 0 & { - 1} & 0 \cr { - 1} & 0 & 0 \cr } } \right)\). The only correct
statement about the matrix \(A\) is

Let \(A = \left( {\matrix{ 1 & { - 1} & 1 \cr 2 & 1 & { - 3} \cr 1 & 1 & 1 \cr } } \right).\) and \(10\) \(B = \left( {\matrix{ 4 & 2 & 2 \cr { - 5} & 0 & \alpha \cr 1 & { - 2} & 3 \cr } } \right)\). if $...

If \({a_1},{a_2},{a_3},.........,{a_n},......\) are in G.P., then the value of the determinant
$$\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 \(a \ne 0\) and the line \(2bx+3cy+4d=0\) passes through the points of intersection of the parabolas \({y^2} = 4ax\) and \({x^2} = 4ay\), then :

The number of ways of distributing 8 identical balls in 3 distinct boxes so that none of the boxes is empty is

How many ways are there to arrange the letters in the word GARDEN with vowels in alphabetical order

The probability that \(A\) speaks truth is \({4 \over 5},\) while the probability for \(B\) is \({3 \over 4}.\) The probability that they contradict each other when asked to speak on a fact is :

The mean and the variance of a binomial distribution are \(4\) and \(2\) respectively. Then the probability of \(2\) successes is :

The sides of a triangle are \(\sin \alpha ,\,\cos \alpha\) and \(\sqrt {1 + \sin \alpha \cos \alpha }\) for some \(0 < \alpha < {\pi \over 2}\). Then the greatest angle of the triangle is :

If \(\left( {1 - p} \right)\) is a root of quadratic equation \({x^2} + px + \left( {1 - p} \right) = 0\) then its root are

If one root of the equation \({x^2} + px + 12 = 0\) is 4, while the equation \({x^2} + px + q = 0\) has equal roots,
then the value of \('q'\) is