IIT-JEE 1993

Let \(f\left( x \right) = \left\{ {\matrix{ { - {x^3} + {{\left( {{b^3} - {b^2} + b - 1} \right)} \over {\left( {{b^2} + 3b + 2} \right)}},} & {0 \le x < 1} \cr {2x - 3} & {1 \le x \le 3} \cr } } \right.\)
Find all possible re...

Find the equation of the normal to the curve
\(y = {\left( {1 + x} \right)^y} + {\sin ^{ - 1}}\left( {{{\sin }^2}x} \right)\) at \(x=0\)

If \(f\left( x \right) = \left\{ {\matrix{ {3{x^2} + 12x - 1,} & { - 1 \le x \le 2} \cr {37 - x} & {2 < x \le 3} \cr } } \right.\) then:

Find the coordinates of the point at which the circles \({x^2}\, + \,{y^2} - \,4x - \,2y = - 4\,\,and\,\,{x^2}\, + \,{y^2} - \,12x - \,8y = - 36\) touch each other. Also find equations common tangests touching the circles in the distinct ...

The locus of the centre of a circle, which touches externally the circle \({x^2} + {y^2} - 6x - 6y + 14 = 0\) and also touches the y-axis, is given by the equation:

Consider a family of circles passing through two fixed points A (3, 7) and B (6, 5). Show that the chords on which the circle \({x^2}\, + \,{y^2} - \,4x - \,6y - 3 = 0\) cuts the members of the family are concurrent at a point. Find the coo...

The equation of the locus of the mid-points of the circle \(4{x^2} + 4{y^2} - 12x + 4y + 1 = 0\) that subtend an angle of \(2\pi /3\) at its centre is.................................

\(ABCD\) is a rhombus. Its diagonals \(AC\) and \(BD\) intersect at the point \(M\) and satisfy \(BD\) = 2\(AC\). If the points \(D\) and \(M\) represent the complex numbers \(1 + i\) and \(2 - i\) respectively, then A represents the comp[l...

Evaluate \(\int_2^3 {{{2{x^5} + {x^4} - 2{x^3} + 2{x^2} + 1} \over {\left( {{x^2} + 1} \right)\left( {{x^4} - 1} \right)}}} dx.\)

The value of \(\int\limits_0^{\pi /2} {{{dx} \over {1 + {{\tan }^3}\,x}}}\) is

The value of \(\int\limits_{\pi /4}^{3\pi /4} {{\phi \over {1 + \sin \phi }}d\phi }\) is ..............

Prove that \(\sum\limits_{r = 1}^k {{{\left( { - 3} \right)}^{r - 1}}\,\,{}^{3n}{C_{2r - 1}} = 0,}\) where \(k = \left( {3n} \right)/2\) and \(n\) is an even positive integer.

Using mathematical induction, prove that
$${\tan ^{ - 1}}\left( {1/3} \right) + {\tan ^{ - 1}}\left( {1/7} \right) + ........{\tan ^{ - 1}}\left\{ {1/\left( {{n^2} + n + 1} \right)} \right\} = {\tan ^{ - 1}}\left\{ {n/\left( {n + 2} \right...

\(E\) and \(F\) are two independent events. The probability that both \(E\) and \(F\) happen is \(1/12\) and the probability that neither \(E\) nor \(F\) happens is \(1/2.\) Then,

Numbers are selected at random, one at a time, from the two- digit numbers \(00, 01, 02 ......, 99\) with replacement. An event \(E\) occurs if only if the product of the two digits of a selected number is \(18\). If four numbers are select...

An unbiased die with faces marked \(1,2,3,4,5\) and \(6\) is rolled four times. Out of four face values obtained, the probability that the minimum face value is not less than \(2\) and the maximum face value is not greater than \(5,\) is th...

If in a triangle \(ABC\), \({{2\cos A} \over a} + {{\cos B} \over b} + {{2\cos C} \over c} = {a \over {bc}} + {b \over {ca}},\) then the value of the angle \(A\) is .................... degrees.

An observer at \(O\) notices that the angle of elevation of the top of a tower is \({30^ \circ }\). The line joining \(O\) to the base of the tower makes an angle of \({\tan ^{ - 1}}\left( {1/\sqrt 2 } \right)\) with the North and is inclin...

For \(0 < \phi < \pi /2,\) if
\(x =\)\(\sum\limits_{n = 0}^\infty {{{\cos }^{2n}}\phi ,y = \sum\limits_{n = 0}^\infty {{{\sin }^{2n}}\phi ,\,\,\,\,z = \sum\limits_{n = 0}^{} {{{\cos }^{2n}}\phi {{\sin }^{2n}}\phi } } } \infty\) then

A line through \(A (-5, -4)\) meets the line \(x + 3y + 2 = 0,\) \(2x + y + 4 = 0\) and \(x - y - 5 = 0\) at the points \(B, C\) and \(D\) respectively. If $${\left( {15/AB} \right)^2} + {\left( {10/AC} \right)^2} = {\left( {6/AD} \right)...

The vertices of a triangle are \(A\left( { - 1, - 7} \right)B\left( {5,\,1} \right)\) and \(C\left( {1,\,4} \right).\) The equation of the bisector of the angle \(\angle ABC\) is ............... .

Tagent at a point \({P_1}\) {other than \((0, 0)\)} on the curve \(y = {x^3}\) meets the curve again at \({P_2}\). The tangent at \({P_2}\) meets the curve at \({P_3}\), and so on. Show that the abscissae of $${P_1},\,{P_2},{P_3}......{P_n}...

Number of solutions of the equation \(\tan x + \sec x = 2\cos x\,\) lying in the interval \(\left[ {0,2\pi } \right]\) is:

If \(A > 0,B > 0\,\) and \(A + B = \pi /3,\) then the maximum value of tan A tan B is _______.

Determine the smallest positive value of number \(x\) (in degrees) for which
\($\tan \left( {x + {{100}^ \circ }} \right) = \tan \left( {x + {{50}^ \circ }} \right)\,\tan \left( x \right)\tan \left( {x - {{50}^ \circ }} \right).\)$

If \(K = \sin \left( {\pi /18} \right)\sin \left( {5\pi /18} \right)\sin \left( {7\pi /18} \right),\) then the numerical value of K is ______.

In a triangle \(ABC, D\) and \(E\) are points on \(BC\) and \(AC\) respectively, such that \(BD=2DC\) and \(AE=3EC.\) Let \(P\) be the point of intersection of \(AD\) and \(BE.\) Find \(BP/PE\) using vector methods.

Let \(\vec a = 2\hat i - \hat j + \hat k,\vec b = \hat i + 2\hat j - \hat k\) and \(\overrightarrow c = \widehat i + \widehat j - 2\widehat k - 2\widehat k\) be three vectors. A vector in the plane of \({\overrightarrow b }\) and $${\ove...

Let \(a, b, c\) 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