AIEEE 2003

If \(f\left( x \right) = {x^n},\) then the value of
$$f\left( 1 \right) - {{f'\left( 1 \right)} \over {1!}} + {{f''\left( 1 \right)} \over {2!}} - {{f'''\left( 1 \right)} \over {3!}} + ..........{{{{\left( { - 1} \right)}^n}{f^n}\left( 1 \...

If \(f:R \to R\) satisfies \(f\)(x + y) = \(f\)(x) + \(f\)(y), for all x, y \(\in\) R and \(f\)(1) = 7, then \(\sum\limits_{r = 1}^n {f\left( r \right)}\) is

A function \(f\) from the set of natural numbers to integers defined by
\($f\left( n \right) = \left\{ {\matrix{ {{{n - 1} \over 2},\,when\,n\,is\,odd} \cr { - {n \over 2},\,when\,n\,is\,even} \cr } } \right.\)$
is

The function \(f\left( x \right)\) \(= \log \left( {x + \sqrt {{x^2} + 1} } \right)\), is

Domain of definition of the function f(x) = \({3 \over {4 - {x^2}}}\) + \({\log _{10}}\left( {{x^3} - x} \right)\), is

The foci of the ellipse \({{{x^2}} \over {16}} + {{{y^2}} \over {{b^2}}} = 1\) and the hyperbola \({{{x^2}} \over {144}} - {{{y^2}} \over {81}} = {1 \over {25}}\) coincide. Then the value of \({b^2}\) is :

The trigonometric equation \({\sin ^{ - 1}}x = 2{\sin ^{ - 1}}a\) has a solution for :

Let \(f(a) = g(a) = k\) and their nth derivatives
\({f^n}(a)\), \({g^n}(a)\) exist and are not equal for some n. Further if
\(\mathop {\lim }\limits_{x \to a} {{f(a)g(x) - f(a) - g(a)f(x) + f(a)} \over {g(x) - f(x)}} = 4\)
then the value o...

If \(f(x) = \left\{ {\matrix{ {x{e^{ - \left( {{1 \over {\left| x \right|}} + {1 \over x}} \right)}}} & {,x \ne 0} \cr 0 & {,x = 0} \cr } } \right.\)
then \(f(x)\) is

If \(\mathop {\lim }\limits_{x \to 0} {{\log \left( {3 + x} \right) - \log \left( {3 - x} \right)} \over x}\) = k, the value of k is

\(\mathop {\lim }\limits_{x \to {\pi \over 2}} {{\left[ {1 - \tan \left( {{x \over 2}} \right)} \right]\left[ {1 - \sin x} \right]} \over {\left[ {1 + \tan \left( {{x \over 2}} \right)} \right]{{\left[ {\pi - 2x} \right]}^3}}}\) is

If \(A = \left[ {\matrix{ a & b \cr b & a \cr } } \right]\) and \({A^2} = \left[ {\matrix{ \alpha & \beta \cr \beta & \alpha \cr } } \right]\), then

If the system of linear equations
\(x + 2ay + az = 0;\) \(x + 3by + bz = 0;\,\,x + 4cy + cz = 0;\)
has a non - zero solution, then \(a, b, c\).

If \(1,\) \(\omega ,{\omega ^2}\) are the cube roots of unity, then
$$\Delta = \left| {\matrix{
1 & {{\omega ^n}} & {{\omega ^{2n}}} \cr
{{\omega ^n}} & {{\omega ^{2n}}} & 1 \cr
{{\omega ^{2n}}} & 1 & {{\omega ^n}} \cr

}...

The normal at the point\(\left( {bt_1^2,2b{t_1}} \right)\) on a parabola meets the parabola again in the point \(\left( {bt_2^2,2b{t_2}} \right)\), then :

The number of ways in which 6 men and 5 women can dine at a round table if no two women are to sit together is given by

A student is to answer 10 out of 13 questions in an examination such that he must choose at least 4 from the first five questions. The number of choices available to him is

If \({}^n{C_r}\) denotes the number of combination of n things taken r at a time, then the expression \(\,{}^n{C_{r + 1}} + {}^n{C_{r - 1}} + 2\, \times \,{}^n{C_r}\) equals

The mean and variance of a random variable \(X\) having binomial distribution are \(4\) and \(2\) respectively, then \(P(X=1)\) is :

Five horses are in a race. Mr. A selects two of the horses at random and bets on them. The probability that Mr. A selected the winning horse is :

Events \(A, B, C\) are mutually exclusive events such that \(P\left( A \right) = {{3x + 1} \over 3},\) \(P\left( B \right) = {{1 - x} \over 4}\) and \(P\left( C \right) = {{1 - 2x} \over 2}\) The set of possible values of \(x\) are in the i...

In a triangle \(ABC\), medians \(AD\) and \(BE\) are drawn. If \(AD=4\),
\(\angle DAB = {\pi \over 6}\) and \(\angle ABE = {\pi \over 3}\), then the area of the \(\angle \Delta ABC\) is :

If in a \(\Delta ABC\) \(a\,{\cos ^2}\left( {{C \over 2}} \right) + c\,{\cos ^2}\left( {{A \over 2}} \right) = {{3b} \over 2},\) then the sides \(a, b\) and \(c\) :

The sum of the radii of inscribed and circumscribed circles for an \(n\) sided regular polygon of side \(a,\) is :

The value of '\(a\)' for which one root of the quadratic equation
\($\left( {{a^2} - 5a + 3} \right){x^2} + \left( {3a - 1} \right)x + 2 = 0\)$
is twice as large as the other is

The number of real solutions of the equation \({x^2} - 3\left| x \right| + 2 = 0\) is

If the sum of the roots of the quadratic equation \(a{x^2} + bx + c = 0\) is equal to the sum of the squares of their reciprocals, then \({a \over c},\,{b \over a}\) and \({c \over b}\) are in

The sum of the serier \({1 \over {1.2}} - {1 \over {2.3}} + {1 \over {3.4}}..............\) up to \(\infty\) is equal to

The median of a set of 9 distinct observations is 20.5. If each of the largest 4 observations of
the set is increased by 2, then the median of the new set :

In an experiment with 15 observations on \(x\), then following results were available:
\(\sum {{x^2}} = 2830\), \(\sum x = 170\)
One observation that was 20 was found to be wrong and was replaced by the correct value
30. Then the correcte...

If \({x_1},{x_2},{x_3}\) and \({y_1},{y_2},{y_3}\) are both in G.P. with the same common ratio, then the points \(\left( {{x_1},{y_1}} \right),\left( {{x_2},{y_2}} \right)\) and \(\left( {{x_3},{y_3}} \right)\) :

Locus of centroid of the triangle whose vertices are \(\left( {a\cos t,a\sin t} \right),\left( {b\sin t, - b\cos t} \right)\) and \(\left( {1,0} \right),\) where \(t\) is a parameter, is :

A square of side a lies above the \(x\)-axis and has one vertex at the origin. The side passing through the origin makes an angle \(\alpha \left( {0 < \alpha < {\pi \over 4}} \right)\) with the positive direction of x-axis. The equation o...

If the equation of the locus of a point equidistant from the point \(\left( {{a_{1,}}{b_1}} \right)\) and \(\left( {{a_{2,}}{b_2}} \right)\) is
\(\left( {{a_1} - {a_2}} \right)x + \left( {{b_1} - {b_2}} \right)y + c = 0\) , then the valu...

If the pair of straight lines \({x^2} - 2pxy - {y^2} = 0\) and \({x^2} - 2qxy - {y^2} = 0\) be such that each pair bisects the angle between the other pair, then :

If \(\overrightarrow u \,,\overrightarrow v\) and \(\overrightarrow w\) are three non-coplanar vectors, then $$\,\left( {\overrightarrow u + \overrightarrow v - \overrightarrow w } \right).\left( {\overrightarrow u - \overrightarrow v ...

If \(\overrightarrow a \times \overrightarrow b = \overrightarrow b \times \overrightarrow c = \overrightarrow c \times \overrightarrow a\) then \(\overrightarrow a + \overrightarrow b + \overrightarrow c =\)

\(\overrightarrow a \,,\overrightarrow b \,,\overrightarrow c\) are \(3\) vectors, such that \(\overrightarrow a + \overrightarrow b + \overrightarrow c = 0\) , $$\left| {\overrightarrow a } \right| = 1\,\,\,\left| {\overrightarrow b }...

A tetrahedron has vertices at \(O(0,0,0), A(1,2,1) B(2,1,3)\) and \(C(-1,1,2).\) Then the angle between the faces \(OAB\) and \(ABC\) will be :

Consider points \(A, B, C\) and \(D\) with position
vectors \(7\widehat i - 4\widehat j + 7\widehat k,\widehat i - 6\widehat j + 10\widehat k, - \widehat i - 3\widehat j + 4\widehat k\) and \(5\widehat i - \widehat j + 5\widehat k\) re...

If \(\left| {\matrix{ a & {{a^2}} & {1 + {a^3}} \cr b & {{b^2}} & {1 + {b^3}} \cr c & {{c^2}} & {1 + {c^3}} \cr } } \right| = 0\) and vectors \(\left( {1,a,{a^2}} \right),\,\,\)
\(\left( {1,b,{b^2}} \right)\) and $$\left( {...

Let \(\overrightarrow u = \widehat i + \widehat j,\,\overrightarrow v = \widehat i - \widehat j\) and \(\overrightarrow w = \widehat i + 2\widehat j + 3\widehat k\,\,.\) If \(\widehat n\) is a unit vector such that $$\overrightarrow u .\...

The vectors \(\overrightarrow {AB} = 3\widehat i + 4\widehat k\,\,\& \,\,\overrightarrow {AC} = 5\widehat i - 2\widehat j + 4\widehat k\) are the sides of triangle \(ABC.\) The length of the median through \(A\) is :

The core of any transformer is laminated so as to

In an oscillating \(LC\) circuit the maximum charge on the capacitor is \(Q\). The charge on the capacitor when the energy is stored equally between the electric and magnetic field is

When a \({U^{238}}\) nucleus originally at rest, decays by emitting an alpha particle having a speed \('u',\) the recoil speed of the residual nucleus is

A radioactive sample at any instant has its disintegration rate \(5000\) disintegrations per minute. After \(5\) minutes, the rate is \(1250\) disintegrations per minute. Then, the decay constant (per minute) is

Which of the following atoms has the lowest ionization potential ?

The wavelengths involved in the spectrum of deuterium \(\left( {{}_1^2\,D} \right)\) are slightly different from that of hydrogen spectrum, because

Which of the following radiations has the least wavelength ?