IIT-JEE 1986

If the line \(ax+by+c=0\) is a normal to the curve \(xy=1\), then

Let \(P\left( x \right) = {a_0} + {a_1}{x^2} + {a_2}{x^4} + ...... + {a_n}{x^{2n}}\) be a polynomial in a real variable \(x\) with
\(0 < {a_0} < {a_1} < {a_2} < ..... < {a_n}.\) The function \(P(x)\) has

From the point A(0, 3) on the circle \({x^2} + 4x + {(y - 3)^2} = 0\), a chord AB is drawn and extended to a point M such that AM = 2AB. The equation of the locus of M is..........................

Lines 5x + 12y - 10 = 0 and 5x - 12y - 40 = 0 touch a circle \(C_1\) of diameter 6. If the centre of \(C_1\) lies in the first quadrant, find the equation of the circle \(C_2\) which is concentric with \(C_1\) and cuts intercepts of length ...

The equation of the line passing through the points of intersection of the circles \(3{x^2} + 3{y^2} - 2x + 12y - 9 = 0\) and \({x^2} + {y^2} - 6x + 2y - 15 = 0\) is..............................

Show that the area of the triangle on the Argand diagram formed by the complex numbers z, iz and z + iz is \({1 \over 2}\,{\left| z \right|^2}\) .

Let \({z_1}\) and \({z_2}\) be complex numbers such that \({z_1}\) \(\ne\) \({z_2}\) and \(\left| {{z_1}} \right| =\,\left| {{z_2}} \right|\). If \({z_1}\) has positive real and \({z_2}\) has negative imaginary part, then $${{{z_1}\, ...

Evaluate : \(\int\limits_0^\pi {{{x\,dx} \over {1 + \cos \,\alpha \,\sin x}},0 < \alpha < \pi }\)

The derivative of \({\sec ^{ - 1}}\left( {{1 \over {2{x^2} - 1}}} \right)\) with respect to \(\sqrt {1 - {x^2}}\) at \(x = {1 \over 2}\) is ...............

The principal value of \({\sin ^{ - 1}}\left( {\sin {{2\pi } \over 3}} \right)\) is

If \({C_r}\) stands for \({}^n{C_r},\) then the sum of the series
$${{2\left( {{n \over 2}} \right){\mkern 1mu} !{\mkern 1mu} \left( {{n \over 2}} \right){\mkern 1mu} !} \over {n!}}\left[ {C_0^2 - 2C_1^2 + 3C_2^2 - } \right......... + {\l...

A box contains two white balls, three black balls and four red balls. In how many ways can three balls be drawn from the box if at least one black ball is to be included in the draw?

A student appears for tests, \(I\), \(II\) and \(III\). The student is successful if he passes either in tests \(I\) and \(II\) or tests \(I\) and \(III\). The probabilities of student passing in tests \(I\), \(II\) and \(III\) are \(p, q\)...

A lot contains \(20\) articles. The probability that the lot contains exactly \(2\) defective articles is \(0.4\) and the probability that the lot contains exactly \(3\) defective articles is \(0.6\). Articles are drawn from the lot at rand...

The probability that at least one of the events \(A\) and \(B\) occurs is \(0.6\). If \(A\) and \(B\) occur simultaneously with probability \(0.2,\) then \(P\left( {\overline A } \right) + P\left( {\overline B } \right)\) is

If \({{1 + 3p} \over 3},\,\,\,{{1 - p} \over 4}\) and \(\,{{1 - 2p} \over 2}\) are the probabilities of three mutually exclusive events, then the set of all values of \(p\) is ..............

If in a triangle \(ABC\), \(\cos A\cos B + \sin A\sin B\sin C = 1,\) Show that \(a:b:c = 1:1:\sqrt 2\)

There exists a triangle \(ABC\) satisfying the conditions

For \(a \le 0,\) determine all real roots of the equation \(${x^2} - 2a\left| {x - a} \right| - 3{a^2} = 0\)$

If the quadratic equations \({x^2} + ax + b = 0\) and \({x^2} + bx + a = 0\) \((a \ne b)\) have a common root, then the numerical value of a + b is..........................

If \(S\) is the set of all real \(x\) such that \({{2x - 1} \over {2{x^3} + 3{x^2} + x}}\) is positive, then \(S\) contains

If \(a,\,b\) and \(c\) are distinct positive numbers, then the expression
\(\left( {b + c - a} \right)\left( {c + a - b} \right)\left( {a + b - c} \right) - abc\) is

The solution of equation \({\log _7}\,{\log _5}\,\left( {\sqrt {x + 5} + \sqrt x } \right) = 0\) is ........................

The solution of the equation \(lo{g_7}\) \(lo{g_5}\) \(\left( {\sqrt {x + 5} + \sqrt x } \right) = 0\) is .............

All points lying inside the triangle formed by the points \(\left( {1,\,3} \right),\,\left( {5,\,0} \right)\) and \(\left( { - 1,\,2} \right)\) satisfy

The points \(\left( {0,{8 \over 3}} \right),\,\,\left( {1,\,3} \right)\) and \(\left( {82,\,30} \right)\) are vertices of

A vector \(\overline a\) has components \(2p\) and \(1\) with respect to a rectangular cartesian system. This system is rotated through a certain angle about the origin in the counter clockwise sense. If, with respect to the new system, $$...

The expression \(2\left[ {{{\sin }^6}\left( {{\pi \over 2} + \alpha } \right) + {{\sin }^6}\left( {5\pi - \alpha } \right)} \right]\) is equal to

Let \(\overrightarrow a = {a_1}i + {a_2}j + {a_3}k,\,\,\,\overrightarrow b = {b_1}i + {b_2}j + {b_3}k\) and \(\overrightarrow c = {c_1}i + {c_2}j + {c_3}k\) be three non-zero vectors such that \(\overrightarrow c\) is a unit vector perp...

The position vectors of the points \(A, B, C\) and \(D\) are \(3\widehat i - 2\widehat j - \widehat k,\,2\widehat i + 3\widehat j - 4\widehat k,\, - \widehat i + \widehat j + 2\widehat k\) and $$4\widehat i + 5\widehat j + \lambda \widehat ...