IIT-JEE 1992

What normal to the curve \(y = {x^2}\) forms the shortest chord?

A cubic \(f(x)\) vanishes at \(x=2\) and has relative minimum / maximum at \(x=-1\) and \(x = {1 \over 3}\) if \(\int\limits_{ - 1}^1 {f\,\,dx = {{14} \over 3}}\), find the cubic \(f(x)\).

In this questions there are entries in columns \(I\) and \(II\). Each entry in column \(I\) is related to exactly one entry in column \(II\). Write the correct letter from column \(II\) against the entry number in column \(I\) in your answe...

Sketch the region bounded by the curves \(y = {x^2}\) and
\(y = {2 \over {1 + {x^2}}}.\) Find the area.

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

Let a circle be given by 2x (x - a) + y (2y - b) = 0, \((a\, \ne \,0,\,\,b\, \ne 0)\). Find the condition on a abd b if two chords, each bisected by the x-axis, can be drawn to the circle from \(\left( {a,\,\,{b \over 2}} \right)\).

\({\rm{z }} \ne {\rm{0}}\) is a complex number
Column I
(A) Re z = 0
(B) Arg \(z = {\pi \over 4}\)
Column II
(p) Re\({z^2}\) = 0
(q) Im\({z^2}\) = 0
(r) Re\({z^2}\) = Im\({z^2}\)

Determine a positive integer \(n \le 5,\) such that
\($\int\limits_0^1 {{e^x}{{\left( {x - 1} \right)}^n}dx = 16 - 6e}\)$

Find the indefinite integral \(\int {\left( {{1 \over {\root 3 \of x + \root 4 \of 4 }} + {{In\left( {1 + \root 6 \of x } \right)} \over {\root 3 \of x + \root \, \of x }}} \right)} dx\)

Let \(p \ge 3\) be an integer and \(\alpha\), \(\beta\) be the roots of \({x^2} - \left( {p + 1} \right)x + 1 = 0\) using mathematical induction show that \({\alpha ^n} + {\beta ^n}.\)
(i) is an integer and (ii) is no...

The expansion \({\left( {x + {{\left( {{x^3} - 1} \right)}^{{1 \over 2}}}} \right)^5} + {\left( {x - {{\left( {{x^3} - 1} \right)}^{{1 \over 2}}}} \right)^5}\) is a polynomial of degree

If \(\sum\limits_{r = 0}^{2n} {{a_r}{{\left( {x - 2} \right)}^r}\,\, = \sum\limits_{r = 0}^{2n} {{b_r}{{\left( {x - 3} \right)}^r}} }\) and \({a_k} = 1\) for all \(k \ge n,\) then show that \({b_n} = {}^{2n + 1}{C_{n + 1}}\)

Three faces of a fair die are yellow, two faces red and one blue. The die is tossed three times. The probability that the colours, yellow, red and blue, appear in the first, second and the third tosses respectively is ...............

A lot contains \(50\) defective and \(50\) non defective bulbs. Two bulbs are drawn at random, one at a time, with replacement. The events \(A, B, C\) are defined as
\(A=\) (the first bulbs is defective)
\(B=\) (the second bulbs is non-def...

India plays two matches each with West Indies and Australia. In any match the probabilities of India getting, points \(0,\) \(1\) and \(2\) are \(0.45, 0.05\) and \(0.50\) respectively. Assuming that the outcomes are independent, the proba...

Three circles touch the one another externally. The tangent at their point of contact meet at a point whose distance from a point of contact is \(4\). Find the ratio of the product of the radii to the sum of the radii of the circles.

Let \(\alpha \,,\,\beta\) be the roots of the equation (x - a) (x - b) = c, \(c \ne 0\). Then the roots of the equation \((x - \alpha \,)\,(x - \beta ) + c = 0\) are

Let the harmonic mean and geometric mean of two positive numbers be the ratio 4 : 5. Then the two number are in the ratio .........

Determine all values of \(\alpha\) for which the point \(\left( {\alpha ,\,{\alpha ^2}} \right)\) lies insides the triangle formed by the lines
$$$\matrix{
{2x + 3y - 1 = 0} \cr
{x + 2y - 3 = 0} \cr
{5x - 6y - 1 = 0} \cr

...

If the sum of the distances of a point from two perpendicular lines in a plane is 1, then its locus is

In this questions there are entries in columns 1 and 2. Each entry in column 1 is related to exactly one entry in column 2. Write the correct letter from column 2 against the entry number in column 1 in your answer book.
$${{\sin \,3\alpha ...

Show that the value of \({{\tan x} \over {\tan 3x}},\) wherever defined never lies between \({1 \over 3}\) and 3.

A unit vector coplanar with \(\overrightarrow i + \overrightarrow j + 2\overrightarrow k\) and \(\overrightarrow i + 2\overrightarrow j + \overrightarrow k\) and perpendicular to $$\overrightarrow i + \overrightarrow j + \overrighta...