IIT-JEE 1996

The position vectors of the vertices \(A, B\) and \(C\) of a tetrahedron \(ABCD\) are \(\widehat i + \widehat j + \widehat k,\,\widehat i\) and \(3\widehat i\,,\) respectively. The altitude from vertex \(D\) to the opposite face \(ABC\) mee...

Let \(f\left( x \right) = \left\{ {\matrix{ {x{e^{ax}},\,\,\,\,\,\,\,x \le 0} \cr {x + a{x^2} - {x^3},\,x > 0} \cr } } \right.\)
Where a is a positive constant. Find the interval in which \(f'(x)\) is increasing.

A curve \(y=f(x)\) passes through the point \(P(1, 1)\). The normal to the curve at \(P\) is \(a(y-1)+(x-1)=0\). If the slope of the tangent at any point on the curve is proportional to the ordinate of the point, determine the equation of t...

Determine the points of maxima and minima of the function
\(f\left( x \right) = {1 \over 8}\ell n\,x - bx + {x^2},x > 0,\) where \(b \ge 0\) is a constant.

Let \({A_n}\) be the area bounded by the curve \(y = {\left( {\tan x} \right)^n}\) and the
lines \(x=0,\) \(y=0,\) and \(x = {\pi \over 4}.\) Prove that for \(n > 2,\)
\({A_n} + {A_{n - 2}} = {1 \over {n - 1}}\) and deduce $${1 \over {2n ...

Find the intervals of value of a for which the line y + x = 0 bisects two chords drawn from a point \(\left( {{{1\, + \,\sqrt 2 a} \over 2},\,{{1\, - \,\sqrt 2 a} \over 2}} \right)\) to the circle $$\,\,2{x^2}\, + \,2{y^2} - (\,1\, + \sqrt ...

A circle passes through three points A, B and C with the line segment AC as its diameter. A line passing through A angles DAB and CAB are \(\,\alpha \,\,and\,\,\beta\) respectively and the distance between the point A and the mid point of ...

The intercept on the line y = x by the circle \({x^2} + {y^2} - 2x = 0\) is AB. Equation of the circle with AB as a diameter is................................

The angle between a pair of tangents drawn from a point P to the circle \({x^2}\, + \,{y^2}\, + \,\,4x\, - \,6\,y\, + \,9\,{\sin ^2}\,\alpha \, + \,13\,{\cos ^2}\,\alpha \, = \,0\) is \(2\,\alpha\).
The equation of the locus of the point...

The value of the expression
$$1 \bullet \left( {2 - \omega } \right)\left( {2 - {\omega ^2}} \right) + 2 \bullet \left( {3 - \omega } \right)\left( {3 - {\omega ^2}} \right) + \,....... + \left( {n - 1} \right).\left( {n - \omega } \right)\...

Find all non-zero complex numbers Z satisfying \(\overline Z = i{Z^2}\).

For positive integers \({n_1},\,{n_2}\) the value of the expression \({\left( {1 + i} \right)^{^{{n_1}}}} + {\left( {1 + {i^3}} \right)^{{n_1}}} + {\left( {1 + {i^5}} \right)^{{n_2}}} + {\left( {1 + {i^7}} \right)^{{n_2}}},\)
where $$i = \...

For \(n>0,\) \(\int_0^{2\pi } {{{x{{\sin }^{2n}}x} \over {{{\sin }^{2n}}x + {{\cos }^{2n}}x}}} dx =\)

If for nonzero \(x\), \(af(x)+\) \(bf\left( {{1 \over x}} \right) = {1 \over x} - 5\) where \(a \ne b,\) then
\(\int_1^2 {f\left( x \right)dx} = .......\)

Determine the equation of the curve passing through the origin, in the form \(y=f(x),\) which satisfies the differential equation \({{dy} \over {dx}} = \sin \left( {10x + 6y} \right).\,\)

If \(x{e^{xy}} = y + {\sin ^2}x,\) then at \(x = 0,{{dy} \over {dx}} = ..............\)

An ellipse has eccentricity \({1 \over 2}\) and one focus at the point \(P\left( {{1 \over 2},1} \right)\). Its one directrix is the common tangent, nearer to the point \(P\), to the circle \({x^2} + {y^2} = 1\) and the hyperbol;a $${x^2} -...

Evaluate \(\int {{{\left( {x + 1} \right)} \over {x{{\left( {1 + x{e^x}} \right)}^2}}}dx}\).

Using mathematical induction prove that for every integer \(n \ge 1,\,\,\left( {{3^{2n}} - 1} \right)\) is divisible by \({2^{n + 2}}\) but not by \({2^{n + 3}}\).

From a point \(A\) common tangents are drawn to the circle \({x^2} + {y^2} = {a^2}/2\) and parabola \({y^2} = 4ax\). Find the area of the quadrilateral formed by the common tangents, the chord of contact of the circle and the chord of cont...

Points \(A, B\) and \(C\) lie on the parabola \({y^2} = 4ax\). The tangents to the parabola at \(A, B\) and \(C\), taken in pairs, intersect at points \(P, Q\) and \(R\). Determine the ratio of the areas of the triangles \(ABC\) and \(PQR\)...

For the three events \(A, B,\) and \(C,P\) (exactly one of the events \(A\) or \(B\) occurs) \(=P\) (exactly one of the two events \(B\) or \(C\) occurs)\(=P\) (exactly one of the events \(C\) or \(A\) occurs)\(=p\) and \(P\) (all the three...

In how many ways three girls and nine boys can be seated in two vans, each having numbered seats, \(3\) in the front and \(4\) at the back? How many seating arrangements are possible if \(3\) girls should sit together in a back row on adja...

In a triangle \(ABC\), \(a:b:c=4:5:6\). The ratio of the radius of the circumcircle to that of the incircle is ............

Let n and k be positive such that \(n \ge {{k(k + 1)} \over 2}\) . The number of solutions \(\,({x_1},\,{x_2},\,.....{x_k}),\,{x_1}\,\, \ge \,1,\,{x_2}\, \ge \,2,.......,{x_k} \ge k\), all integers, satisfying $${x_1} + {x_2} + \,..... + {x...

The real numbers \({x_1}\), \({x_2}\), \({x_3}\) satisfying the equation \({x^3} - {x^2} + \beta x + \gamma = 0\) are in AP. Find the intervals in which \(\beta \,\,and\,\gamma\) lie.

For any odd integer \(n\) \(\ge 1,\,\,{n^3} - {\left( {n - 1} \right)^3} + .... + {\left( { - 1} \right)^{n - 1}}\,{1^3} = ........\)

A rectangle \(PQRS\) has its side \(PQ\) parallel to the line \(y = mx\) and vertices \(P, Q\) and \(S\) on the lines \(y = a, x = b\) and \(x = -b,\) respectively. Find the locus of the vertex \(R\).

General value of \(\theta\) satisfying the equation \({\tan ^2}\theta + \sec \,2\,\theta = 1\) is _________.

Find all values of \(\theta\) in the interval \(\left( { - {\pi \over 2},{\pi \over 2}} \right)\) satisfying the equation $$\left( {1 - \tan \,\theta } \right)\left( {1 + \tan \,\theta } \right)\,\,{\sec ^2}\theta + \,\,{2^{{{\tan }^2}...

\({\sec ^2}\theta = {{4xy} \over {{{\left( {x + y} \right)}^2}}}\,\) is true if and only if

If \(\overrightarrow b \,\) and \(\overrightarrow c \,\) are two non-collinear unit vectors and \(\overrightarrow a \,\) is any vector, then $$\left( {\overrightarrow a .\overrightarrow b } \right)\overrightarrow b + \left( {\overrightarro...

A nonzero vector \(\overrightarrow a\) is parallel to the line of intersection of the plane determined by the vectors \(\widehat i,\widehat i + \widehat j\) and the plane determined by the vectors $$\widehat i - \widehat j,\widehat i + \wi...