IIT-JEE 1985

If \({n_1}\), \({n_2}\),.......\({n_p}\) are p positive integers, whose sum is an even number, then the number of odd integers among them is odd.

If \(P(x) = a{x^2} + bx + c\,\,and\,\,Q(x) = - a{x^2} + dx + c\), where \(ac \ne \,0\), then P(x) Q(x) = 0 has at least two real roots.

If \(a,\,b,\,c\) are in GP., then the equations \(\,\,\alpha {x^2} + 2bx + c = 0\) and \(d{x^2} + 2ex + f = 0\) have a common root if \({d \over a},\,{e \over b},{f \over c}\) are in ________.

Find the sum of the series :
$$$\sum\limits_{r = 0}^n {{{\left( { - 1} \right)}^r}\,{}^n{C_r}\left[ {{1 \over {{2^r}}} + {{{3^r}} \over {{2^{2r}}}} + {{{7^r}} \over {{2^{3r}}}} + {{{{15}^r}} \over {{2^{4r}}}}..........up\,\,to\,\,m\,\,terms...

One of the diameters of the circle circumscribing the rectangle \(ABCD\) is \(4y = x + 7\). If \(A\) and \(B\) are the points \((-3, 4)\) and \((5, 4)\) respectively, then find the area of rectangle.

Three lines \(px + qy + r = 0\), \(qx + ry + p = 0\) and \(rx + py + q = 0\) are concurrent if

Two sides of rhombus \(ABCD\) are parallel to the lines \(y = x + 2\) and \(y = 7x + 3\). If the diagonals of the rhombus intersect at the point \((1, 2)\) and the vertex \(A\) is on the \(y\)-axis, find possible co-ordinates of \(A\).

The orthocentre of the triangle formed by the lines \(x + y = 1,\,2x + 3y = 6\) and \(4x - y + 4 = 0\) lies in quadrant number .............

If \(\overrightarrow A \overrightarrow {\,B} \overrightarrow {\,C}\) are three non-coplannar vectors, then -
$${{\overrightarrow A .\overrightarrow B \times \overrightarrow C } \over {\overrightarrow C \times \overrightarrow A .\overrig...

If \(\overrightarrow A = \left( {1,1,1} \right),\,\,\overrightarrow C = \left( {0,1, - 1} \right)\) are given vectors, then a vector \(B\) satifying the equations \(\overrightarrow A \times \overrightarrow B = \overrightarrow {\,C}\) a...

Two identical trains are moving on rails along the equator on the earth in opposite directions with the same speed. They will exert the same pressure on the rails.

Plank' constant has dimension __________.

A uniform chain of length L and mass M is lying on a smooth table and one third of its length is hanging vertically down over the edge of the table. If g is acceleration due to gravity, the work required to pull the hanging part on to the t...