IIT-JEE 1978

From a point \(O\) inside a triangle \(ABC,\) perpendiculars \(OD\), \(OE, OF\) are drawn to the sides \(BC, CA, AB\) respectively. Prove that the perpendiculars from \(A, B, C\) to the sides \(EF, FD, DE\) are concurrent.

Find the equation of the circle whose radius is 5 and which touches the circle \({x^2}\, + \,{y^2}\, - \,2x\,\, - 4y\, - 20 = 0\,\) at the point (5, 5).

If x = a + b, y = a\(\gamma\) + b\(\beta\) and z = a\(\beta\) +b\(\gamma\) where \(\gamma\) and \(\beta\) are the complex cube roots of unity, show that xyz = \({a^3} + {b^3}\).

Express \({1 \over {1 - \cos \,\theta + 2i\sin \theta }}\) in the form x + iy.

Find the derivative of \(\sin \left( {{x^2} + 1} \right)\) with respect to \(x\) first principle.

Evaluate \(\int {{{\sin x} \over {\sin x - \cos x}}dx}\)

Six X' s have to be placed in the squares of figure below in such a way that each row contains at least one X. In how many different ways can this be done.

Balls are drawn one-by-one without replacement from a box containing \(2\) black, \(4\) white and \(3\) red balls till all the balls are drawn. Find the probability that the balls drawn are in the order \(2\) black, \(4\) white and \(3\) re...

A triangle \(ABC\) has sides \(AB=AC=5\) cm and \(BC=6\) cm Triangle \(A'B'C'\) is the reflection of the triangle \(ABC\) in a line parallel to \(AB\) placed at a distance \(2\) cm from \(AB\), outside the triangle \(ABC\). Triangle $$A''B'...

Show that the square of \(\,{{\sqrt {26 - 15\sqrt 3 } } \over {5\sqrt 2 - \sqrt {38 + 5\sqrt 3 } }}\) is a rational number.

Solve for \(x:\,\sqrt {x + 1} - \sqrt {x - 1} = 1.\)

Solve the following equation for \(x:\,\,2\,{\log _x}a + {\log _{ax}}a + 3\,\,{\log _{{a^2}x}}\,a = 0,a > 0\)

Solve for \(x:{4^x} - {3^{^{x - {1 \over 2}}}}\, = {3^{^{x + {1 \over 2}}}}\, - {2^{2x - 1}}\)

If \(\left( {m\,,\,n} \right) = {{\left( {1 - {x^m}} \right)\left( {1 - {x^{m - 1}}} \right).......\left( {1 - {x^{m - n + 1}}} \right)} \over {\left( {1 - x} \right)\left( {1 - {x^2}} \right).........\left( {1 - {x^n}} \right)}}\)
where $...

Sketch the solution set of the following system of inequalities:
\(${x^2} + {y^2} - 2x \ge 0;\,\,3x - y - 12 \le 0;\,\,y - x \le 0;\,\,y \ge 0.\)$

Find all integers \(x\) for which \(\left( {5x - 1} \right) < {\left( {x + 1} \right)^2} < \left( {7x - 3} \right).\)

The area of a triangle is \(5\). Two of its vertices are \(A\left( {2,1} \right)\) and \(B\left( {3, - 2} \right)\). The third vertex \(C\) lies on \(y = x + 3\). Find \(C\).

One side of rectangle lies along the line \(4x + 7y + 5 = 0.\) Two of its vertices are \((-3, 1)\) and \((1, 1).\) Find the equations of the other three sides.

A straight line segment of length \(\ell\) moves with its ends on two mutually perpendicular lines. Find the locus of the point which divides the line segment in the ratio \(1 : 2\)

If \(\tan \alpha = {m \over {m + 1}}\,\) and \(\tan \beta = {2 \over {2m + 1}},\) find the possible values of \(\left( {\alpha + \beta } \right).\)