AIEEE 2004

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

Let two numbers have arithmetic mean 9 and geometric mean 4. Then these numbers are the roots of the quadratic equation

The sum of the first n terms of the series \({1^2} + {2.2^2} + {3^2} + {2.4^2} + {5^2} + {2.6^2} + ....\,is\,{{n{{(n + 1)}^2}} \over 2}\) when n is even. When n is odd the sum is

The sum of series \({1 \over {2\,!}} + {1 \over {4\,!}} + {1 \over {6\,!}} + ........\) is

Let \({{T_r}}\) be the rth term of an A.P. whose first term is a and common difference is d. If for some positive integers m, n, \(m \ne n,\,\,{T_m} = {1 \over n}\,\,and\,{T_n} = {1 \over m},\,\) then a - d equals

Let $R=\{(1,3),(4,2),(2,4),(2,3),(3,1)\}$ be a relation on the set $A=\{1,2,3,4\}$. The relation $R$ is :

Consider the following statements:
(a) Mode can be computed from histogram
(b) Median is not independent of change of scale
(c) Variance is independent of change of origin and scale.
Which of these is/are correct?

In a series of 2n observations, half of them equal \(a\) and remaining half equal \(–a\). If the
standard deviation of the observations is 2, then \(|a|\) equals

If one of the lines given by \(6{x^2} - xy + 4c{y^2} = 0\) is \(3x + 4y = 0,\) then \(c\) equals :

If the sum of the slopes of the lines given by \({x^2} - 2cxy - 7{y^2} = 0\) is four times their product \(c\) has the value :

The equation of the straight line passing through the point \((4, 3)\) and making intercepts on the co-ordinate axes whose sum is \(-1\) is :

Let \(A\left( {2, - 3} \right)\) and \(B\left( {-2, 1} \right)\) be vertices of a triangle \(ABC\). If the centroid of this triangle moves on the line \(2x + 3y = 1\), then the locus of the vertex \(C\) is the line :

Let \(\alpha ,\,\beta\) be such that \(\pi < \alpha - \beta < 3\pi\).
If \(sin{\mkern 1mu} \alpha + \sin \beta = - {{21} \over {65}}\) and \(\cos \alpha + \cos \beta = - {{27} \over {65}}\) then the value of $$\cos {{\alpha - ...

If \(u = \sqrt {{a^2}{{\cos }^2}\theta + {b^2}{{\sin }^2}\theta } + \sqrt {{a^2}{{\sin }^2}\theta + {b^2}{{\cos }^2}\theta }\)
then the difference between the maximum and minimum values of \({u^2}\) is given by :

Let \(\overrightarrow u ,\overrightarrow v ,\overrightarrow w\) be such that \(\left| {\overrightarrow u } \right| = 1,\,\,\,\left| {\overrightarrow v } \right|2,\,\,\,\left| {\overrightarrow w } \right|3.\) If the projection $${\overright...

Let \(\overrightarrow a ,\overrightarrow b\) and \(\overrightarrow c\) be non-zero vectors such that $$\left( {\overrightarrow a \times \overrightarrow b } \right) \times \overrightarrow c = {1 \over 3}\left| {\overrightarrow b } \right...

A particle acted on by constant forces \(4\widehat i + \widehat j - 3\widehat k\) and \(3\widehat i + \widehat j - \widehat k\) is displaced from the point \(\widehat i + 2\widehat j + 3\widehat k\) to the point $$\,5\widehat i + 4\widehat ...

If \({\overrightarrow a ,\overrightarrow b ,\overrightarrow c }\) are non-coplanar vectors and \(\lambda\) is a real number, then the vectors $${\overrightarrow a + 2\overrightarrow b + 3\overrightarrow c ,\,\,\lambda \overrightarrow b ...

Let \(\overrightarrow a ,\overrightarrow b\) and \(\overrightarrow c\) be three non-zero vectors such that no two of these are collinear. If the vector \(\overrightarrow a + 2\overrightarrow b\) is collinear with \(\overrightarrow c\) ...

In a \(LCR\) circuit capacitance is changed from \(C\) to \(2\) \(C\). For the resonant frequency to remain unchaged, the inductance should be changed from \(L\) to

In an \(LCR\) series \(a.c.\) circuit, the voltage across each of the components, \(L,C\) and \(R\) is \(50V\). The voltage across the \(L.C\) combination will be :

Alternating current can not be measured by \(D.C.\) ammeter because

The binding energy per nucleon of deuteron \(\left( {{}_1^2\,H} \right)\) and helium nucleus \(\left( {{}_2^4\,He} \right)\) is \(1.1\) \(MeV\) and \(7\) \(MeV\) respectively. If two deuteron nuclei react to form a single helium nucleus, th...

An \(\alpha\)-particle of energy \(5\) \(MeV\) is scattered through \({180^ \circ }\) by a fixed uranium nucleus. The distance of closest approach is of the order of

A nucleus disintegrated into two nuclear parts which have their velocities in the ratio of \(2:1.\) The ratio of their nuclear sizes will be

A machine gun fires a bullet of mass \(40\) \(g\) with a velocity \(1200m{s^{ - 1}}.\) The man holding it can exert a maximum force of \(144\) \(N\) on the gun. How many bullets can he fire per second at the most?