AIEEE 2005

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 :

Let \(A\) and \(B\) two events such that \(P\left( {\overline {A \cup B} } \right) = {1 \over 6},\) \(P\left( {A \cap B} \right) = {1 \over 4}\) and \(P\left( {\overline A } \right) = {1 \over 4},\) where \({\overline A }\) stands for comp...

A random variable \(X\) has Poisson distribution with mean \(2\).
Then \(P\left( {X > 1.5} \right)\) equals :

In a triangle \(ABC\), let \(\angle C = {\pi \over 2}\). If \(r\) is the inradius and \(R\) is the circumradius of the triangle \(ABC\), then \(2(r+R)\) equals :

If in a \(\Delta ABC\), the altitudes from the vertices \(A, B, C\) on opposite sides are in H.P, then \(\sin A,\sin B,\sin C\) are in :

If the roots of the equation \({x^2} - bx + c = 0\) be two consecutive integers, then \({b^2} - 4c\) equals

If both the roots of the quadratic equation \({x^2} - 2kx + {k^2} + k - 5 = 0\) are less than 5, then \(k\) lies in the interval

The value of \(a\) for which the sum of the squares of the roots of the equation \({x^2} - \left( {a - 2} \right)x - a - 1 = 0\) assume the least value is

The value of \(a\) for which the sum of the squares of the roots of the equation \({x^2} - \left( {a - 2} \right)x - a - 1 = 0\)
assume the least value is :

If the roots of the equation \({x^2} - bx + c = 0\) be two consecutive integers, then \({b^2} - 4c\) equals

In a triangle \(PQR,\;\;\angle R = {\pi \over 2}.\,\,If\,\,\tan \,\left( {{P \over 2}} \right)\) and \(\tan \left( {{Q \over 2}} \right)\) are the roots of \(a{x^2} + bx + c = 0,\,\,a \ne 0\) then

The sum of the series \(1 + {1 \over {4.2!}} + {1 \over {16.4!}} + {1 \over {64.6!}} + .......\) ad inf. is

If \(x = \sum\limits_{n = 0}^\infty {{a^n},\,\,y = \sum\limits_{n = 0}^\infty {{b^n},\,\,z = \sum\limits_{n = 0}^\infty {{c^n},} } } \,\,\) where a, b, c are in A.P and $$\,\left| a \right| < 1,\,\left| b \right| < 1,\,\left| c \right| <...

Let $R=\{(3,3),(6,6),(9,9),(12,12),(6,12)$, $(3,9),(3,12),(3,6)\}$ be a relation on the set $A=\{3,6,9,12\}$. The relation is :

If in a frequency distribution, the mean and median are 21 and 22 respectively, then
its mode is approximately :

Let x1, x2,...........,xn be n observations such that
\(\sum {x_i^2} = 400\) and \(\sum {{x_i}} = 80\). Then a
possible value of n among the following is

If a vertex of a triangle is \((1, 1)\) and the mid points of two sides through this vertex are \((-1, 2)\) and \((3, 2)\) then the centroid of the triangle is :

If non zero numbers \(a, b, c\) are in \(H.P.,\) then the straight line \({x \over a} + {y \over b} + {1 \over c} = 0\) always passes through a fixed point. That point is :

The line parallel to the \(x\) - axis and passing through the intersection of the lines \(ax + 2by + 3b = 0\) and \(bx - 2ay - 3a = 0,\) where \((a, b)\) \(\ne\) \((0, 0)\) is :

Let \(\overrightarrow a \,\, = \,\,\widehat i - \widehat k,\,\,\,\,\,\overrightarrow b \,\,\, = \,\,\,x\widehat i + \widehat j\,\,\, + \,\,\,\left( {1 - x} \right)\widehat k\) and $$\overrightarrow c \,\, = \,\,y\widehat i + x\widehat j + \...

Let \(a, b\) and \(c\) be distinct non-negative numbers. If the vectors \(a\widehat i + a\widehat j + c\widehat k,\,\,\widehat i + \widehat k\) and \(c\widehat i + c\widehat j + b\widehat k\) lie in a plane, then \(c\) is :

For any vector \({\overrightarrow a }\) , the value of \({\left( {\overrightarrow a \times \widehat i} \right)^2} + {\left( {\overrightarrow a \times \widehat j} \right)^2} + {\left( {\overrightarrow a \times \widehat k} \right)^2}\) is...

If \(C\) is the mid point of \(AB\) and \(P\) is any point outside \(AB,\) then :

If \(\overrightarrow a ,\overrightarrow b ,\overrightarrow c\) are non coplanar vectors and \(\lambda\) is a real number then $$\left[ {\lambda \left( {\overrightarrow a + \overrightarrow b } \right)\,\,\,\,\,\,\,\,{\lambda ^2}\overright...

The self inductance of the motor of an electric fan is \(10\) \(H\). In order to impart maximum power at \(50\) \(Hz\), it should be connected to a capacitance of

A circuit has a resistance of \(12\) \(ohm\) and an impedance of \(15\) \(ohm\). The power factor of the circuit will be

The phase difference between the alternating current and \(emf\) is \({\pi \over 2}.\) Which of the following cannot be the constituent of the circuit?