IIT-JEE 1999

The function \(f\left( x \right) = \int\limits_{ - 1}^x {t\left( {{e^t} - 1} \right)\left( {t - 1} \right){{\left( {t - 2} \right)}^3}\,\,\,{{\left( {t - 3} \right)}^5}}\) \(dt\) has a local minimum at \(x=\)

The function \(f(x)=\) \({\sin ^4}x + {\cos ^4}x\) increases if

For which of the following values of \(m\), is the area of the region bounded by the curve \(y = x - {x^2}\) and the line \(y=mx\) equals \(9/2\)?

Let \(f(x)\) be a continuous function given by
\($f\left( x \right) = \left\{ {\matrix{ {2x,} & {\left| x \right| \le 1} \cr {{x^2} + ax + b,} & {\left| x \right| > 1} \cr } } \right\}\)$
Find the area of the region in the thir...

If two distinct chords, drawn from the point (p, q) on the circle \({x^2}\, + \,{y^2} = \,px\, + \,qy\,\,(\,where\,pq\, \ne \,0)\) are bisected by the x - axis, then

Let \({T_1}\), \({T_2}\) be two tangents drawn from (- 2, 0) onto the circle \(C:{x^2}\,\, + \,{y^2} = 1\). Determine the circles touching C and having \({T_1}\), \({T_2}\) as their pair of tangents. Further, find the equations of all possi...

For complex numbers z and w, prove that \({\left| z \right|^2}w - {\left| w \right|^2}z = z - w\) if and only if \(z = w\,or\,z\overline {\,w} = 1\).

\(If\,i = \sqrt { - 1} ,\,\,then\,\,4 + 5{\left( { - {1 \over 2} + {{i\sqrt 3 } \over 2}} \right)^{334}} + 3{\left( { - {1 \over 2} + {{i\sqrt 3 } \over 2}} \right)^{365}}\) is equal to

If for a real number \(y\), \(\left[ y \right]\) is the greatest integer less than or
equal to \(y\), then the value of the integral \(\int\limits_{\pi /2}^{3\pi /2} {\left[ {2\sin x} \right]dx}\) is

Integrate \(\int\limits_0^\pi {{{{e^{\cos x}}} \over {{e^{\cos x}} + {e^{ - \cos x}}}}\,dx.}\)

\(\int\limits_{\pi /4}^{3\pi /4} {{{dx} \over {1 + \cos x}}}\) is equal to

A solution of the differential equation
\({\left( {{{dy} \over {dx}}} \right)^2} - x{{dy} \over {dx}} + y = 0\) is

The differential equation representing the family of curves
\({y^2} = 2c\left( {x + \sqrt c } \right),\) where \(c\) is a positive parameter, is of

Consider the family of circles \({x^2} + {y^2} = {r^2},\,\,2 < r < 5\). If in the first quadrant, the common taingent to a circle of this family and the ellipse \(4{x^2} + 25{y^2} = 100\) meets the co-ordinate axes at \(A\) and \(B\), then ...

On the ellipse \(4{x^2} + 9{y^2} = 1,\) the points at which the tangents are parallel to the line \(8x = 9y\) are

Find the co-ordinates of all the points \(P\) on the ellipse \({{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1\), for which the area of the triangle \(PON\) is maximum, where \(O\) denotes the origin and \(N\), the foot of the perpend...

Let \(P\) \(\left( {a\,\sec \,\theta ,\,\,b\,\tan \theta } \right)\) and \(Q\) \(\left( {a\,\sec \,\,\phi ,\,\,b\,\tan \,\phi } \right)\), where \(\theta + \phi = \pi /2,\), be two points on the hyperbola $${{{x^2}} \over {{a^2}}} - {{{y...

If \(x\) \(=\) \(9\) is the chord of contact of the hyperbola \({x^2} - {y^2} = 9,\) then the equation of the vcorresponding pair of tangents is

Integrate \(\int {{{{x^3} + 3x + 2} \over {{{\left( {{x^2} + 1} \right)}^2}\left( {x + 1} \right)}}dx.}\)

The number of real solutions of
\({\tan ^{ - 1}}\,\,\sqrt {x\left( {x + 1} \right)} + {\sin ^{ - 1}}\,\,\sqrt {{x^2} + x + 1} = \pi /2\) is

If in the expansion of \({\left( {1 + x} \right)^m}{\left( {1 - x} \right)^n},\) the coefficients of \(x\) and \({x^2}\) are \(3\) and \(-6\) respectively, then \(m\) is

Let \(n\) be any positive integer. Prove that
$$$\sum\limits_{k = 0}^m {{{\left( {\matrix{
{2n - k} \cr
k \cr

} } \right)} \over {\left( {\matrix{
{2n - k} \cr
n \cr

} } \right)}}.{{\left( {2n - 4k + 1} \right)} \ov...

The curve described parametrically by \(x = {t^2} + t + 1,\) \(y = {t^2} - t + 1\) represents

The probabilities that a student passes in Mathematics, Physics and Chemistry are \(m, p\) and \(c,\) respectively. Of these subjects, the student has a \(75%\) chance of passing in at least one, a \(50\)% chance of passing in at least two,...

Eight players \({P_1},{P_2},.....{P_8}\) play a knock-out tournament. It is known that whenever the players \({P_i}\) and \({P_j}\) play, the player \({P_i}\) will win if \(i < j.\) Assuming that the players are paired at random in each rou...

If the integers \(m\) and \(n\) are chosen at random from \(1\) to \(100\), then the probability that a number of the form \({7^m} + {7^n}\) is divisible by \(5\) equals

Let \(ABC\) be a triangle having \(O\) and \(I\) as its circumcenter and in centre respectively. If \(R\) and \(r\) are the circumradius and the inradius, respectively, then prove that $${\left( {IO} \right)^2} = {R^2} - 2{\mathop{\rm Rr}\n...

If the roots of the equation \({x^2} - 2ax + {a^2} + a - 3 = 0\) are real and less than 3, then

Let a, b, c, d be real numbers in G.P. If u, v, w, satisfy the system of equations
u + 2v + 3w = 6
4u + 5v + 6w = 12
6u + 9v = 4
then show that the roots of the equation \(\left( {{1 \over u} + {1 \over v} + {1 \over w}} \right){x^2}\)...

For a positive integer \(n\), let
\(a\left( n \right) = 1 + {1 \over 2} + {1 \over 3} + {1 \over 4} + .....\,{1 \over {\left( {{2^n}} \right) - 1}}\). Then

Let \({a_1},{a_2},......{a_{10}}\) be in \(A,\,P,\) and \({h_1},{h_2},......{h_{10}}\) be in H.P. If \({a_1} = {h_1} = 2\) and \({a_{10}} = {h_{10}} = 3,\) then \({a_4}{h_7}\) is

The harmonic mean of the roots of the equation \(\left( {5 + \sqrt 2 } \right){x^2} - \left( {4 + \sqrt 5 } \right)x + 8 + 2\sqrt 5 = 0\) is

If \({x_1},\,{x_2},\,{x_3}\) as well as \({y_1},\,{y_2},\,{y_3}\), are in G.P. with the same common ratio, then the points \(\left( {{x_1},\,{y_1}} \right),\left( {{x_2},\,{y_2}} \right)\) and \(\left( {{x_3},\,{y_3}} \right).\)

Let \({L_1}\) be a straight line passing through the origin and \({L_2}\) be the straight line \(x + y = 1\). If the intercepts made by the circle \({x^2} + {y^2} - x + 3y = 0\) on \({L_1}\) and \({L_2}\) are equal, then which of the follow...

Lt \(PQR\) be a right angled isosceles triangle, right angled at \(P(2, 1)\). If the equation of the line \(QR\) is \(2x + y = 3,\) then the equation representing the pair of lines \(PQ\) and \(PR\) is

In a triangle \(PQR,\angle R = \pi /2\). If \(\,\,\tan \left( {P/2} \right)\) and \(\tan \left( {Q/2} \right)\) are the roots of the equation \(a{x^2} + bx + c = 0\left( {a \ne 0} \right)\) then.

For a positive integer \(\,n\), let
$${f_n}\left( \theta \right) = \left( {\tan {\theta \over 2}} \right)\,\left( {1 + \sec \theta } \right)\,\left( {1 + \sec 2\theta } \right)\,\left( {1 + \sec 4\theta } \right).....\left( {1 + \sec {...

Let \(u\) and \(v\) be units vectors. If \(w\) is a vector such that \(w + \left( {w \times u} \right) = v,\) then prove that \(\left| {\left( {u \times v} \right) \cdot w} \right| \le 1/2\) and that the equality holds if and only if \(u\) ...

Let \(a=2i+j+k, b=i+2j-k\) and a unit vector \(c\) be coplanar. If \(c\) is perpendicular to \(a,\) then \(c =\)

Let \(a=2i+j-2k\) and \(b=i+j.\) If \(c\) is a vector such that \(a.\) \(c = \left| c \right|,\left| {c - a} \right| = 2\sqrt 2\) and the angle between \(\left( {a \times b} \right)\) and \(c\) is \({30^ \circ },\) then $$\left| {\left( {...

Let \(a\) and \(b\) two non-collinear unit vectors. If \(u = a - \left( {a\,.\,b} \right)\,b\) and \(v = a \times b,\) then \(\left| v \right|\) is