IIT-JEE 1979

Prove that the minimum value of \({{\left( {a + x} \right)\left( {b + x} \right)} \over {\left( {c + x} \right)}},\)
\(a,b > c,x > - c\) is \({\left( {\sqrt {a - c} + \sqrt {b - c} } \right)^2}\).

If x + iy = \(\sqrt {{{a + ib} \over {c + id}}}\), prove that \({({x^2} + {y^2})^2} = {{{a^2} + {b^2}} \over {{c^2} + {d^2}}}\).

If the cube roots of unity are \(1,\,\omega ,\,{\omega ^2},\) then the roots of the equation \({\left( {x - 1} \right)^3} + 8 = 0\) are

Find the derivative of
\($f\left( x \right) = \left\{ {\matrix{ {{{x - 1} \over {2{x^2} - 7x + 5}}} & {when\,\,x \ne 1} \cr { - {1 \over 3}} & {when\,\,x = 1} \cr } } \right.\)$
at \(x=1\)

Evaluate \(\int {{{{x^2}dx} \over {{{\left( {a + bx} \right)}^2}}}}\)

Given that \({C_1} + 2{C_2}x + 3{C_3}{x^2} + ......... + 2n{C_{2n}}{x^{2n - 1}} = 2n{\left( {1 + x} \right)^{2n - 1}}\)
where \({C_r} = {{\left( {2n} \right)\,!} \over {r!\left( {2n - r} \right)!}}\,\,\,\,\,r = 0,1,2,\,............,2n\)
P...

\({}^n{C_{r - 1}} = 36,{}^n{C_r} = 84\,\,and\,\,{}^n{C_{r + 1}} = 126\), then r is :

Two fair dice are tossed. Let \(x\) be the event that the first die shows an even number and \(y\) be the event that the second die shows an odd number. The two events \(x\) and \(y\) are:

Six boys and six girls sit in a row randomly. Find the probability that
(i) the six girls sit together
(ii) the boys and girls sit alternately.

(a) A balloon is observed simultaneously from three points \(A, B\) and \(C\) on a straight road directly beneath it. The angular elevation at \(B\) is twice that at \(A\) and the angular elevation at \(C\) is thrice that at \(A\). If the d...

(a) If a circle is inscribed in a right angled triangle \(ABC\) with the right angle at \(B\), show that the diameter of the circle is equal to \(AB+BC-AC\).
(b) If a triangle is inscribed in a circle, then the product of any two sides of ...

If the bisector of the angle \(P\) of a triangle \(PQR\) meets \(QR\) in \(S\), then

The equation x + 2y + 2z = 1 and 2x + 4y + 4z = 9 have

If \(\ell\), m, n are real, \(\ell \ne m\), then the roots by the equation :
\((\ell - m)\,{x^2} - 5\,(\ell + m)\,x - 2\,(\ell - m) = 0\) are

If x, y and z are real and different and \(\,u = {x^2} + 4{y^2} + 9{z^2} - 6yz - 3zx - 2xy\), then u is always.

Let a > 0, b > 0 and c > 0. Then the roots of the equation \(a{x^2} + bx + c = 0\)

If \(\alpha ,\,\beta\) are the roots of \({x^2} + px + q = 0\) and \(\gamma ,\,\delta\) are the roots of \({x^2} + rx + s = 0,\) evaluate $$\left( {\alpha - \gamma } \right)\left( {\alpha - \delta } \right)\left( {\beta - \gamma } \rig...

The harmonic mean of two numbers is 4.Their arithmetic mean \(A\) and the geometric mean \(G\) satisfy the relation. \(2A + {G^2} = 27\)

The points \(\left( { - a,\, - b} \right),\,\left( {0,\,0} \right),\,\left( {a,\,b} \right)\) and \(\left( {{a^2},\,ab} \right)\) are :

(a) Two vertices of a triangle are \((5, -1)\) and \((-2, 3).\) If the orthocentre of the triangle is the origin, find the coordinates of the third point.
(b) Find the equation of the line which bisects the obtuse angle between the lines $...

If \(\alpha + \beta + \gamma = 2\pi ,\) then

If \(\tan \theta = - {4 \over 3},then\sin \theta \,is\,\)

(a) Draw the graph of \(y = {1 \over {\sqrt 2 }}\left( {cinx + \cos x} \right)\) from \(x = - {\pi \over 2}\) to \(x = {\pi \over 2}\).
(b) If $$\cos \left( {\alpha + \beta } \right) = {4 \over 5},\,\,\sin \,\left( {\alpha - \beta } \r...