AIEEE 2003

Two systems of rectangular axes have the same origin. If a plane cuts then at distances \(a,b,c\) and \(a', b', c'\) from the origin then

The radius of the circle in which the sphere
\({x^2} + {y^2} + {z^2} + 2x - 2y - 4z - 19 = 0\) is cut by the plane
\(x+2y+2z+7=0\) is

The two lines \(x=ay+b,z=cy+d\) and \(x = a'y + b',z = c'y + d'\) will be perpendicular, if and only if :

The lines \({{x - 2} \over 1} = {{y - 3} \over 1} = {{z - 4} \over { - k}}\) and \({{x - 1} \over k} = {{y - 4} \over 2} = {{z - 5} \over 1}\) are coplanar if :

The shortest distance from the plane \(12x+4y+3z=327\) to the sphere \({x^2} + {y^2} + {z^2} + 4x - 2y - 6z = 155\) is

The real number \(x\) when added to its inverse gives the minimum sum at \(x\) equal :

If the function \(f\left( x \right) = 2{x^3} - 9a{x^2} + 12{a^2}x + 1,\) where \(a>0,\) attains its maximum and minimum at \(p\) and \(q\) respectively such that \({p^2} = q\) , then \(a\) equals

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

If \(x\) is positive, the first negative term in the expansion of \({\left( {1 + x} \right)^{27/5}}\) is

The number of integral terms in the expansion of \({\left( {\sqrt 3 + \root 8 \of 5 } \right)^{256}}\) is

If the two circles \({(x - 1)^2}\, + \,{(y - 3)^2} = \,{r^2}\) and \(\,{x^2}\, + \,{y^2} - \,8x\, + \,2y\, + \,\,8\,\, = 0\) intersect in two distinct point, then :

The lines 2x - 3y = 5 and 3x - 4y = 7 are diameters of a circle having area as 154 sq. units. Then the equation of the circle is :

Let \({Z_1}\) and \({Z_2}\) be two roots of the equation \({Z^2} + aZ + b = 0\), Z being complex. Further , assume that the origin, \({Z_1}\) and \({Z_2}\) form an equilateral triangle. Then :

If \(z\) and \(\omega\) are two non-zero complex numbers such that \(\left| {z\omega } \right| = 1\) and \(Arg(z) - Arg(\omega ) = {\pi \over 2},\) then \(\,\overline {z\,} \omega\) is equal to

If \({\left( {{{1 + i} \over {1 - i}}} \right)^x} = 1\) then :

Let \(f(x)\) be a function satisfying \(f'(x)=f(x)\) with \(f(0)=1\) and \(g(x)\) be a function that satisfies \(f\left( x \right) + g\left( x \right) = {x^2}\). Then the value of the integral $$\int\limits_0^1 {f\left( x \right)g\left( x \...

The value of the integral \(I = \int\limits_0^1 {x{{\left( {1 - x} \right)}^n}dx}\) is

The value of \(\mathop {\lim }\limits_{x \to 0} {{\int\limits_0^{{x^2}} {{{\sec }^2}tdt} } \over xsinx}\) is

If \(f\left( {a + b - x} \right) = f\left( x \right)\) then \(\int\limits_a^b {xf\left( x \right)dx}\) is equal to

If \(f\left( y \right) = {e^y},\) \(g\left( y \right) = y;y > 0\) and
\(F\left( t \right) = \int\limits_0^t {f\left( {t - y} \right)g\left( y \right)dy,}\) then :

\(\mathop {\lim }\limits_{n \to \infty } {{1 + {2^4} + {3^4} + .... + {n^4}} \over {{n^5}}}\) - \(\mathop {\lim }\limits_{n \to \infty } {{1 + {2^3} + {3^3} + .... + {n^3}} \over {{n^5}}}\)

The solution of the differential equation
\(\left( {1 + {y^2}} \right) + \left( {x - {e^{{{\tan }^{ - 1}}y}}} \right){{dy} \over {dx}} = 0,\) is :

The degree and order of the differential equation of the family of all parabolas whose axis is \(x\)-axis, are respectively.

Let \(f\left( x \right)\) be a polynomial function of second degree. If \(f\left( 1 \right) = f\left( { - 1} \right)\) and \(a,b,c\) are in \(A.P,\) then \(f'\left( a \right),f'\left( b \right),f'\left( c \right)\) are in

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