AIEEE 2002

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

If \(\alpha \ne \beta\) but \({\alpha ^2} = 5\alpha - 3\) and \({\beta ^2} = 5\beta - 3\) then the equation having \(\alpha /\beta\) and \(\beta /\alpha \,\,\) as its roots is

If \(a,\,b,\,c\) are distinct \(+ ve\) real numbers and \({a^2} + {b^2} + {c^2} = 1\) then \(ab + bc + ca\) is

Product of real roots of equation \({t^2}{x^2} + \left| x \right| + 9 = 0\)

If \(p\) and \(q\) are the roots of the equation \({x^2} + px + q = 0,\) then

\({1^3} - \,\,{2^3} + {3^3} - {4^3} + ... + {9^3} =\)

Sum of infinite number of terms of GP is 20 and sum of their square is 100. The common ratio of GP is

Fifth term of a GP is 2, then the product of its 9 terms is

The value of \(\,{2^{1/4}}.\,\,{4^{1/8}}.\,{8^{1/16}}...\infty\) is

If 1, \({\log _9}\,\,({3^{1 - x}} + 2),\,\,{\log _3}\,\,({4.3^x} - 1)\) are in A.P. then x equals

l, m, n are the \({p^{th}}\), \({q^{th}}\) and \({r^{th}}\) term of a G.P all positive, \(then\,\left| {\matrix{ {\log \,l} & p & 1 \cr {\log \,m} & q & 1 \cr {\log \,n} & r & 1 \cr } } \right|\,equals\)

In a class of 100 students there are 70 boys whose average marks in a subject are 75. If the average marks of the complete class is 72, then what is the average marks of the girls?

Locus of mid point of the portion between the axes of
\(x\) \(cos\) \(\alpha + y\,\sin \alpha = p\) where \(p\) is constant is :

If the pair of lines
\(a{x^2} + 2hxy + b{y^2} + 2gx + 2fy + c = 0\)
intersect on the \(y\)-axis then :

The pair of lines represented by
\($3a{x^2} + 5xy + \left( {{a^2} - 2} \right){y^2} = 0\)$
are perpendicular to each other for :

A triangle with vertices \(\left( {4,0} \right),\left( { - 1, - 1} \right),\left( {3,5} \right)\) is :

The number of solution of \(\tan \,x + \sec \,x = 2\cos \,x\) in \(\left[ {0,\,2\,\pi } \right]\) is

If \(\left| {\overrightarrow a } \right| = 4,\left| {\overrightarrow b } \right| = 2\) and the angle between \({\overrightarrow a }\) and \({\overrightarrow b }\) is \(\pi /6\) then $${\left( {\overrightarrow a \times \overrightarrow b } \...

If the vectors \(\overrightarrow c ,\overrightarrow a = x\widehat i + y\widehat j + z\widehat k\) and \(\widehat b = \widehat j\) are such that \(\overrightarrow a ,\overrightarrow c\) and \(\overrightarrow b\) form a right handed system...

If the vectors $\overrightarrow{\mathbf{a}}, \overrightarrow{\mathbf{b}}$ and $\overrightarrow{\mathbf{c}}$ from the sides $B C, C A$ and $A B$ respectively of a triangle $A B C$, then :

\(\overrightarrow a = 3\widehat i - 5\widehat j\) and \(\overrightarrow b = 6\widehat i + 3\widehat j\) are two vectors and \(\overrightarrow c\) is a vector such that \(\overrightarrow c = \overrightarrow a \times \overrightarrow b\)...

If \(\left| {\overrightarrow a } \right| = 5,\left| {\overrightarrow b } \right| = 4,\left| {\overrightarrow c } \right| = 3\) thus what will be the value of $$\left| {\overrightarrow a .\overrightarrow b + \overrightarrow b .\overrightarr...

If \(\overrightarrow a \,\,,\,\,\overrightarrow b \,\,,\,\,\overrightarrow c\) are vectors such that \(\left[ {\overrightarrow a \,\overrightarrow b \,\overrightarrow c } \right] = 4\) then $$\left[ {\overrightarrow a \, \times \overrighta...