IIT-JEE 1989

Find all maxima and minima of the function
\($y = x{\left( {x - 1} \right)^2},0 \le x \le 2\)$
Also determine the area bounded by the curve \(y = x{\left( {x - 1} \right)^2}\),
the \(y\)-axis and the line \(y-2\).

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 points, then

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

If \(\left( {{m_i},{1 \over {{m_i}}}} \right),\,{m_i}\, > \,0,\,i\, = 1,\,2,\,3,\,4\) are four distinct points on a circle, then show that \({m_1}\,{m_2}\,{m_3}\,{m_4}\, = 1\)

The area of the triangle formed by the positive x-axis and the normal and the tangent to the circle \({x^2} + {y^2} = 4\,\,at\,\,\left( {1,\sqrt 3 } \right)\) is,..................

The line x + 3y = 0 is a diameter of the circle \({x^2} + {y^2} - 6x + 2y = 0\,\).

If \(a,\,b,\,c,\) are the numbers between 0 and 1 such that the ponts \({z_1} = a + i,{z_2} = 1 + bi\) and \({z_3} = 0\) form an equilateral triangle,
then a= .......and b=..........

The value of \(\int\limits_{ - 2}^2 {\left| {1 - {x^2}} \right|dx}\) is ...............

If \(f\) and \(g\) are continuous function on \(\left[ {0,a} \right]\) satisfying
\(f\left( x \right) = f\left( {a - x} \right)\) and \(g\left( x \right) + g\left( {a - x} \right) = 2,\)
then show that $$\int\limits_0^a {f\left( x \right)...

If \(x = \sec \theta - \cos \theta\) and \(y = {\sec ^n}\theta - {\cos ^n}\theta\), then show
that \(\left( {{x^2} + 4} \right){\left( {{{dy} \over {dx}}} \right)^2} = {n^2}\left( {{y^2} + 4} \right)\)

Evaluate \(\int {\left( {\sqrt {\tan x} + \sqrt {\cot x} } \right)dx}\)

The greater of the two angles \(A = 2{\tan ^{ - 1}}\left( {2\sqrt 2 - 1} \right)\) and \(B = 3{\sin ^{ - 1}}\left( {1/3} \right) + {\sin ^{ - 1}}\left( {3/5} \right)\) is ________ .

Prove that
\({C_0} - {2^2}{C_1} + {3^2}{C_2}\,\, - \,..... + {\left( { - 1} \right)^n}{\left( {n + 1} \right)^2}{C_n} = 0,\,\,\,\,n > 2,\,\,\) where \({C_r} = {}^n{C_r}.\)

Using mathematical induction, prove that \({}^m{C_0}{}^n{C_k} + {}^m{C_1}{}^n{C_{k - 1}}\,\,\, + .....{}^m{C_k}{}^n{C_0} = {}^{\left( {m + n} \right)}{C_k},\)
where \(m,\,n,\,k\) are positive integers, and \({}^p{C_q} = 0\) for \(p < q.\)

A five-digit numbers divisible by 3 is to be formed using the numerals 0, 1, 2, 3, 4 and 5, without repetition. The total number of ways this can be done is

Suppose the probability for A to win a game against B is \(0.4.\) If \(A\) has an option of playing either a "best of \(3\) games" or a "best of \(5\) games" match against \(B\), which option should be choose so that the probability of his ...

A pair of fair dice is rolled together till a sum of either \(5\) or \(7\) is obtained. Then the probability that \(5\) comes before \(7\) is ...............

If \(E\) and \(F\) are independent events such that \(0 < P\left( E \right) < 1\) and \(0 < P\left( F \right) < 1,\) then

If the probability for \(A\) to fail in an examination is \(0.2\) and that for \(B\) is \(0.3\), then the probability that either \(A\) or \(B\) fails is \(0.5\)

\(ABC\) is a triangular park with \(AB=AC=100\) \(m\). A television tower stands at the midpoint of \(BC\). The angles of elevetion of the top of the tower at \(A, B, C\) are 45\(^ \circ\), 60\(^ \circ\), 60\(^ \circ\), respectively. Fin...

The equation \({x^{3/4{{\left( {{{\log }_2}\,\,x} \right)}^2} + {{\log }_2}\,\,x - 5/4}} = \sqrt 2\) has

Let a, b, c be real numbers, \(a \ne 0\). If \(\alpha \,\) is a root of \({a^2}{x^2} + bx + c = 0\). \(\beta \,\) is the root of \({a^2}{x^2} - bx - c = 0\) and \(0 < \alpha \, < \,\beta\), then the equation \({a^2}{x^2} + 2bx + 2c = 0\) h...

If \(\alpha\) and \(\beta\) are the roots of \({x^2}\)+ px + q = 0 and \({\alpha ^4},{\beta ^4}\) are the roots of \(\,{x^2} - rx + s = 0\), then the equation \({x^2} - 4qx + 2{q^2} - r = 0\) has always

If x and y are positive real numbers and m, n are any positive integers, then \({{{x^n}\,{y^m}} \over {(1 + {x^{2n}})\,(1 + {y^{2m}})}} > {1 \over 4}\)

Let \(ABC\) be a triangle with \(AB = AC\). If \(D\) is the midpoint of \(BC, E\) is the foot of the perpendicular drawn from \(D\) to \(AC\) and \(F\) the mid-point of \(DE\), prove that \(AF\) is perpendicular to \(BE\).

The general solutions of \(\,\sin x - 3\sin 2x + \sin 3x = \cos x - 3\cos 2x + \cos 3x\) is

In a triangle \(OAB,E\) is the midpoint of \(BO\) and \(D\) is a point on \(AB\) such that \(AD:DB=2:1.\) If \(OD\) and \(AE\) intersect at \(P,\) determine the ratio \(OP:PD\) using vector methods.

If vectors \(\overrightarrow A ,\overrightarrow B ,\overrightarrow C\) are coplanar, show that
$$$\left| {\matrix{
{} & {\overrightarrow {a.} } & {} & {\overrightarrow {b.} } & {} & {\overrightarrow {c.} } \cr
{\overrightarrow {a....

For any three vectors \({\overrightarrow a ,\,\overrightarrow b ,}\) and \({\overrightarrow c ,}\)
$$\left( {\overrightarrow a - \overrightarrow b } \right)\,.\,\left( {\overrightarrow b - \overrightarrow c } \right)\, \times \,\left( {\...