AIEEE 2002

The \(d.r.\) of normal to the plane through \((1, 0, 0), (0, 1, 0)\) which makes an angle \(\pi /4\) with plane \(x+y=3\) are :

A plane which passes through the point \((3,2,0)\) and the line
\({{x - 4} \over 1} = {{y - 7} \over 5} = {{z - 4} \over 4}\) is :

If \(2a+3b+6c=0,\) \(\left( {a,b,c \in R} \right)\) then the quadratic equation \(a{x^2} + bx + c = 0\) has

The maximum distance from origin of a point on the curve
\(x = a\sin t - b\sin \left( {{{at} \over b}} \right)\)
\(y = a\cos t - b\cos \left( {{{at} \over b}} \right),\) both \(a,b > 0\) is

The area bounded by the curves \(y = \ln x,y = \ln \left| x \right|,y = \left| {\ln {\mkern 1mu} x} \right|\) and \(y = \left| {\ln \left| x \right|} \right|\) is :

The positive integer just greater than \({\left( {1 + 0.0001} \right)^{10000}}\) is

\(r\) and \(n\) are positive integers \(\,r > 1,\,n > 2\) and coefficient of \(\,{\left( {r + 2} \right)^{th}}\) term and \(3{r^{th}}\) term in the expansion of \({\left( {1 + x} \right)^{2n}}\) are equal, then \(n\) equals

If the sum of the coefficients in the expansion of \(\,{\left( {a + b} \right)^n}\) is 4096, then the greatest coefficient in the expansion is

The coefficients of \({x^p}\) and \({x^q}\) in the expansion of \({\left( {1 + x} \right)^{p + q}}\) are

The centres of a set of circles, each of radius 3, lie on the circle \({x^2}\, + \,{y^2} = 25\). The locus of any point in the set is :

If the chord y = mx + 1 of the circle \({x^2}\, + \,{y^2} = 1\) subtends an angle of measure \({45^ \circ }\) at the major segment of the circle then value of m is :

The centre of the circle passing through (0, 0) and (1, 0) and touching the circle \({x^2}\, + \,{y^2} = 9\) is :

The equation of a circle with origin as a center and passing through an equilateral triangle whose median is of length \(3\)\(a\) is :

z and w are two nonzero complex numbers such that \(\,\left| z \right| = \left| w \right|\) and Arg z + Arg w =\(\pi\) then z equals

If \(\left| {z - 4} \right| < \left| {z - 2} \right|\), its solution is given by :

The locus of the centre of a circle which touches the circle \(\left| {z - {z_1}} \right| = a\) and\(\left| {z - {z_2}} \right| = b\,\) externally
(\(z,\,{z_1}\,\& \,{z_2}\,\) are complex numbers) will be :

\(\mathop {\lim }\limits_{n \to \infty } {{{1^p} + {2^p} + {3^p} + ..... + {n^p}} \over {{n^{p + 1}}}}\) is

\(\int\limits_0^2 {\left[ {{x^2}} \right]dx}\) is

If \(y=f(x)\) makes +\(ve\) intercept of \(2\) and \(0\) unit on \(x\) and \(y\) axes and encloses an area of \(3/4\) square unit with the axes then \(\int\limits_0^2 {xf'\left( x \right)dx}\) is

\(\int_{ - \pi }^\pi {{{2x\left( {1 + \sin x} \right)} \over {1 + {{\cos }^2}x}}} dx\) is

\(\int\limits_0^{10\pi } {\left| {\sin x} \right|dx}\) is

\({I_n} = \int\limits_0^{\pi /4} {{{\tan }^n}x\,dx}\) then \(\,\mathop {\lim }\limits_{n \to \infty } \,n\left[ {{I_n} + {I_{n + 2}}} \right]\) equals

The solution of the equation \(\,{{{d^2}y} \over {d{x^2}}} = {e^{ - 2x}}\)

The order and degree of the differential equation
\(\,{\left( {1 + 3{{dy} \over {dx}}} \right)^{2/3}} = 4{{{d^3}y} \over {d{x^3}}}\) are

If \(y = {\left( {x + \sqrt {1 + {x^2}} } \right)^n},\) then \(\left( {1 + {x^2}} \right){{{d^2}y} \over {d{x^2}}} + x{{dy} \over {dx}}\) is

The period of \({\sin ^2}\theta\) is

Which one is not periodic?

The domain of \({\sin ^{ - 1}}\left[ {{{\log }_3}\left( {{x \over 3}} \right)} \right]\) is

\({\cot ^{ - 1}}\left( {\sqrt {\cos \alpha } } \right) - {\tan ^{ - 1}}\left( {\sqrt {\cos \alpha } } \right) = x,\) then sin x is equal to :

If f(x + y) = f(x).f(y) \(\forall\) x, y and f(5) = 2, f'(0) = 3, then
f'(5) is

If \(f\left( 1 \right) = 1,{f'}\left( 1 \right) = 2,\) then
\(\mathop {\lim }\limits_{x \to 1} {{\sqrt {f\left( x \right)} - 1} \over {\sqrt x - 1}}\) is

\(\mathop {\lim }\limits_{x \to 0} {{\sqrt {1 - \cos 2x} } \over {\sqrt 2 x}}\) is

\(\mathop {\lim }\limits_{x \to 0} {{\log {x^n} - \left[ x \right]} \over {\left[ x \right]}}\), \(n \in N\), ( [x] denotes the greatest integer less than or equal to x )

\(\mathop {\lim }\limits_{x \to \infty } {\left( {{{{x^2} + 5x + 3} \over {{x^2} + x + 2}}} \right)^x}\)

f(x) and g(x) are two differentiable functions on [0, 2] such that
f''(x) - g''(x) = 0, f'(1) = 2, g'(1) = 4, f(2) = 3, g(2) = 9
then f(x) - g(x) at x = \({3 \over 2}\) is

Let \(f(2) = 4\) and \(f'(x) = 4.\)
Then \(\mathop {\lim }\limits_{x \to 2} {{xf\left( 2 \right) - 2f\left( x \right)} \over {x - 2}}\) is given by

\(f\) is defined in \(\left[ { - 5,5} \right]\) as
\(f\left( x \right) = x\) if \(x\) is rational
\(\,\,\,\,\,\,\,\,\,\,\,\,\,\) \(= - x\) if \(x\) is irrational. Then

If \({a_n} = \sqrt {7 + \sqrt {7 + \sqrt {7 + .......} } }\) having \(n\) radical signs then by methods of mathematical induction which is true

If \(a>0\) and discriminant of \(\,a{x^2} + 2bx + c\) is \(-ve\), then
\(\left| {\matrix{ a & b & {ax + b} \cr b & c & {bx + c} \cr {ax + b} & {bx + c} & 0 \cr } } \right|\) is equal to

Two common tangents to the circle \({x^2} + {y^2} = 2{a^2}\) and parabola \({y^2} = 8ax\) are :

The sum of integers from 1 to 100 that are divisible by 2 or 5 is :

Number greater than 1000 but less than 4000 is formed using the digits 0, 1, 2, 3, 4 (repetition allowed). Their number is :

Total number of four digit odd numbers that can be formed using 0, 1, 2, 3, 5, 7 (using repetition allowed) are :

Five digit number divisible by 3 is formed using 0, 1, 2, 3, 4 and 5 without repetition. Total number of such numbers are :

\(A\) and \(B\) are events such that \(P\left( {A \cup B} \right) = 3/4\),\(P\left( {A \cap B} \right) = 1/4,\)
\(P\left( {\overline A } \right) = 2/3\) then \(P\left( {\overline A \cap B} \right)\) is :

A dice is tossed \(5\) times. Getting an odd number is considered a success. Then the variance of distribution of success is :

A problem in mathematics is given to three students \(A,B,C\) and their respective probability of solving the problem is \({1 \over 2},{1 \over 3}\) and \({1 \over 4}.\) Probability that the problem is solved is :

In a triangle with sides \(a, b, c,\) \({r_1} > {r_2} > {r_3}\) (which are the ex-radii) then :

The sides of a triangle are \(3x + 4y,\) \(4x + 3y\) and \(5x + 5y\) where \(x\), \(y>0\) then the triangle is :

Difference between the corresponding roots of \({x^2} + ax + b = 0\) and \({x^2} + bx + a = 0\) is same and \(a \ne b,\) then