IIT-JEE 1994

Let \(p,q \in \left\{ {1,2,3,4} \right\}\,\). The number of equations of the form \(p{x^2} + qx + 1 = 0\) having real roots is

The number of points of intersection of two curves y = 2 sin x and y \(= 5{x^2} + 2x + 3\) is

If p, q, r are + ve and are on A.P., the roots of quadratic equation \(p{x^2} + qx + r = 0\) are all real for

If \(In\left( {a + c} \right),In\left( {a - c} \right),In\left( {a - 2b + c} \right)\) are in A.P., then

The equations to a pair of opposites sides of parallelogram are \({x^2} - 5x + 6 = 0\) and \({y^2} - 6y + 5 = 0,\) the equations to its diagonals are

The locus of a variable point whose distance from \(\left( { - 2,\,0} \right)\) is \(2/3\) times its distance from the line \(x = - {9 \over 2}\) is

Let \(n\) be a positive integer such that \(\sin {\pi \over {2n}} + \cos {\pi \over {2n}} = {{\sqrt n } \over 2}.\) Then

Let \(0 < x < {\pi \over 4}\) then \(\left( {\sec 2x - \tan 2x} \right)\) equals

If \(\omega \,\) is an imaginary cube root of unity then the value of \(\sin \left\{ {\left( {{\omega ^{10}} + {\omega ^{23}}} \right)\pi - {\pi \over 4}} \right\}\) is

Let \(2{\sin ^2}x + 3\sin x - 2 > 0\) and \({x^2} - x - 2 < 0\) (\(x\) is measured in radians). Then \(x\) lies in the interval

If the vectors \(\overrightarrow b ,\overrightarrow c ,\overrightarrow d ,\) are not coplanar, then prove that the vector
$$\left( {\overrightarrow a \times \overrightarrow b } \right) \times \left( {\overrightarrow c \times \overrightar...

The vector \(\,{1 \over 3}\left( {2\widehat i - 2\widehat j + \widehat k} \right)\) is

A block of mass 0.1 is held against a wall applying a horizontal force of 5 N on the block. If the coefficient of friction between the block and the wall is 0.5, the magnitude of the frictional force acting on the block is:

An object of mass 0.2 kg executes simple harmonic oscillation along the x-axis with a frequency of \(\left( {{{25} \over \pi }} \right)\) Hz. At the position x = 0.04, the object has kinetic energy of 0.5 J and potential energy 0.4 J. The a...

A particle of mass m is moving in a circular path of constant radius r such that its centripetal acceleration \({a_c}\) is varying with time t as \({a_c}\) = k2rt2 where k is a constant. The power delivered to the perticles by the force act...