IIT-JEE 1980

Two circles \({x^2} + {y^2} = 6\) and \({x^2} + {y^2} - 6x + 8 = 0\) are given. Then the equation of the circle through their points of intersection and the point (1, 1) is

A square is inscribed in the circle \({x^2} + {y^2} - 2x + 4y + 3 = 0\). Its sides are parallel to the coordinate axes. The one vertex of the square is

The smallest positive integer n for which \({\left( {{{1 + i} \over {1 - i}}} \right)^n} = 1\) is

Find the real values of x and y for which the following equation is satisfied \(\,{{(1 + i)x - 2i} \over {3 + i}} + {{(2 + 3i)y + i} \over {3 - i}} = i\)

Given \(y = {{5x} \over {3\sqrt {{{\left( {1 - x} \right)}^2}} }} + {\cos ^2}\left( {2x + 1} \right)\); Find \({{dy} \over {dx}}\).

Two events \(A\) and \(B\) have probabilities \(0.25\) and \(0.50\) respectively. The probability that both \(A\) and \(B\) occur simultaneously is \(0.14\). Then the probability that neither \(A\) nor \(B\) occurs is

The probability that an event \(A\) happens in one trial of an experiment is \(0.4.\) Three independent trials of the experiment are performed. The probability that the event \(A\) happens at least once is

(i) \(PQ\) is a vertical tower. \(P\) is the foot and \(Q\) is the top of the tower. \(A, B, C\) are three points in the horizontal plane through \(P\). The angles of elevation of \(Q\) from \(A\), \(B\), \(C\) are equal, and each is equal ...

In a \(\Delta ABC,\,\angle A = {90^ \circ }\) and \(AD\) is an altitude. Complete the relation \({{BD} \over {BA}} = {{AB} \over {\left( {....} \right)}}\).

\(ABC\) is a triangle with \(\angle B\) greater than \(\angle C.\,D\) and \(E\) are points on \(BC\) such that \(AD\) is perpendicular to \(BC\) and \(AE\) is the bisector of angle \(A\). Complete the relation
$$$\angle DAE = {1 \over 2}\l...

\(ABC\) is a triangle. \(D\) is the middle point of \(BC\). If \(AD\) is perpendicular to \(AC\), then prove that
\($\cos A\,\cos C = {{2\left( {{c^2} - {a^2}} \right)} \over {3ac}}\)$

\(ABC\) is a triangle, \(P\) is a point on \(AB\), and \(Q\) is point on \(AC\) such that \(\angle AQP = \angle ABC\). Complete the relation
$$${{area\,\,of\,\,\Delta APQ} \over {area\,\,of\,\,\Delta ABC}} = {{\left( {...} \right)} \over {...

\(ABC\) is a triangle with \(AB=AC\). \(D\) is any point on the side \(BC\). \(E\) and \(F\) are points on the side \(AB\) and \(AC\), respectively, such that \(DE\) is parallel to \(AC\), and \(DF\) is parallel to \(AB\). Prove that
$$$DF...

Find the solution set of the system
\($\matrix{ {x + 2y + z = 1;} \cr {2x - 3y - w = 2;} \cr {x \ge 0;\,y \ge 0;\,z \ge 0;\,w \ge 0.} \cr }\)$

Both the roots of the equation (x - b) (x - c) + (x - a) (x - c) + (x - a) (x - b) = 0 are always

For what values of \(m,\) does the system of equations
\($\matrix{ {3x + my = m} \cr {2x - 5y = 20} \cr }\)$
has solution satisfying the conditions \(x > 0,\,y > 0.\)

Let \(y = \sqrt {{{\left( {x + 1} \right)\left( {x - 3} \right)} \over {\left( {x - 2} \right)}}}\)
Find all the real values of \(x,\) for which \(y\) takes real values.

Given \({n^4} < {10^n}\) for a fixed positive integer \(n \ge 2,\) prove that \({\left( {n + 1} \right)^4} < {10^{n + 1}}.\)

The least value of the expression \(2\,\,{\log _{10}}\,x\, - \,{\log _x}(0.01)\) for x > 1, is

If \(\,({x^2} + px + 1)\,\) is a factor of \((a{x^3} + bx + c)\), then

The interior angles of a polygon are in arithmetic progression. The smallest angle is \({120^ \circ }\), and the common difference is \({5^ \circ }\), Find the number of sides of the polygon.

The point \(\,\left( {4,\,1} \right)\) undergoes the following three transformations successively.
Reflection about the line \(y=x\).
Translation through a distance 2 units along the positive direction of x-axis.
Rotation through an angle ...

A straight line \(L\) is perpendicular to the line \(5x - y = 1.\) The area of the triangle formed by the line \(L\) and the coordinate axes is \(5\). Find the equation of the Line \(L\).

Given \(\alpha + \beta - \gamma = \pi ,\) prove that
\(\,{\sin ^2}\alpha + {\sin ^2}\beta - {\sin ^2}\gamma = 2\sin \alpha {\mkern 1mu} \sin \beta {\mkern 1mu} \cos y\)

The equation \(\,2{\cos ^2}{x \over 2}{\sin ^2}x = {x^2} + {x^{ - 2}};\,0 < x \le {\pi \over 2}\) has

Given \(A = {\sin ^2}\theta + {\cos ^4}\theta\) then for all real values of \(\theta\)

Given \(A = \left\{ {x:{\pi \over 6} \le x \le {\pi \over 3}} \right\}\) and
\(f\left( x \right) = \cos x - x\left( {1 + x} \right);\) find \(f\left( A \right).\)

For all \(\theta\) in \(\left[ {0,\,\pi /2} \right]\) show that, \(\cos \left( {\sin \theta } \right) \ge \,\sin \,\left( {\cos \theta } \right).\)