IIT-JEE 1994

Let \(\alpha ,\beta ,\gamma\) be distinct real numbers. The points with position
vectors $$\alpha \widehat i + \beta \widehat j + \gamma \widehat k,\,\,\beta \widehat i + \gamma \widehat j + \alpha \widehat k,\,\,\gamma \widehat i + \alph...

Let \(\overrightarrow p\) and \(\overrightarrow q\) be the position vectors of \(P\) and \(Q\) respectively, with respect to \(O\) and \(\left| {\overrightarrow p } \right| = p,\left| {\overrightarrow q } \right| = q.\) The points \(R\) a...

A unit vector perpendicular to the plane determined by the points \(P\left( {1, - 1,2} \right)Q\left( {2,0, - 1} \right)\) and \(R\left( {0,2,1} \right)\) is ............

The circle \({x^2} + {y^2} = 1\) cuts the \(x\)-axis at \(P\) and \(Q\). Another circle with centre at \(Q\) and variable radius intersects the first circle at \(R\) above the \(x\)-axis and the line segment \(PQ\) at \(S\). Find the maximu...

The function defined by \(f\left( x \right) = \left( {x + 2} \right){e^{ - x}}\)

Let \(P\) be a variable point on the ellipse \({{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1\) with foci \({F_1}\) and \({F_2}\). If \(A\) is the area of the triangle \(P{F_1}{F_2}\) then the maximum value of \(A\) is ..........

Which one of the following curves cut the parabola \({y^2} = 4ax\) at right angles?

The curve \(y = a{x^3} + b{x^2} + cx + 5\), touches the \(x\)-axis at \(P(-2, 0)\) and cuts the \(y\) axis at a point \(Q\), where its gradient is \(3\). Find \(a, b, c\).

Let \(C\) be the curve \({y^3} - 3xy + 2 = 0\). If \(H\) is the set of points on the curve \(C\) where the tangent is horizontal and \(V\) is the set of the point on the curve \(C\) where the tangent is vertical then \(H=\).............. an...

In what ratio does the \(x\)-axis divide the area of the region
bounded by the parabolas \(y = 4x - {x^2}\) and \(y = {x^2} - x?\)

The circles \({x^2} + {y^2} - 10x + 16 = 0\) and \({x^2} + {y^2} = {r^2}\) intersect each other in two distinct points if

Suppose Z1, Z2, Z3 are the vertices of an equilateral triangle inscribed in the circle \(\left| Z \right| = 2.\) If Z1 = \(1 + i\sqrt 3\) then Z2 = ......., Z3 =..............

Show that \(\int\limits_0^{n\pi + v} {\left| {\sin x} \right|dx = 2n + 1 - \cos \,v}\) where \(n\) is a positive integer and \(\,0 \le v < \pi .\)

The value of \(\int\limits_2^3 {{{\sqrt x } \over {\sqrt {3 - x} + \sqrt x }}} dx\) is ...........

A normal is drawn at a point \(P(x,y)\) of a curve. It meets the \(x\)-axis at \(Q.\) If \(PQ\) is of constant length \(k,\) then show that the differential equation describing such curves is $$y = {{dy} \over {dx}} = \pm \sqrt {{k^2} - {y...

If \(y = {\left( {\sin x} \right)^{\tan x}},\) then \({{dy} \over {dx}}\) is equal to

Let \(E\) be the ellipse \({{{x^2}} \over 9} + {{{y^2}} \over 4} = 1\) and \(C\) be the circle \({x^2} + {y^2} = 9\). Let \(P\) and \(Q\) be the points \((1, 2)\) and \((2, 1)\) respectively. Then

The equation \(2{x^2} + 3{y^2} - 8x - 18y + 35 = k\) represents

Find the indefinite integral \(\,\int {\cos 2\theta {\mkern 1mu} ln\left( {{{\cos \theta + \sin \theta } \over {\cos \theta - \sin \theta }}} \right)} {\mkern 1mu} d\theta\)

If we consider only the principle values of the inverse trigonometric functions then the value of
\(\tan \left( {{{\cos }^{ - 1}}{1 \over {5\sqrt 2 }} - {{\sin }^{ - 1}}{4 \over {\sqrt {17} }}} \right)\) is

If \(x\) is not an integral multiple of \(2\pi\) use mathematical induction to prove that :
\($\cos x + \cos 2x + .......... + \cos nx = \cos {{n + 1} \over 2}x\sin {{nx} \over 2}\cos ec{x \over 2}\)$

Let \(n\) be a positive integer and \({\left( {1 + x + {x^2}} \right)^n} = {a_0} + {a_1}x + ............ + {a_{2n}}{x^{2n}}\)
Show that \(a_0^2 - a_1^2 + a_2^2...... + {a_{2n}}{}^2 = {a_n}\)

Let \(n\) be positive integer. If the coefficients of 2nd, 3rd, and 4th terms in the expansion of \({\left( {1 + x} \right)^n}\) are in A.P., then the value of \(n\) is ................

The point of intersection of the tangents at the ends of the latus rectum of the parabola \({y^2} = 4x\) is ...... .

Through the vertex \(O\) of parabola \({y^2} = 4x\), chords \(OP\) and \(OQ\) are drawn at right angles to one another . Show that for all positions of \(P\), \(PQ\) cuts the axis of the parabola at a fixed point. Also find the locus of the...

A committee of 12 is to be formed from 9 women and 8 men. In how many ways this can be done if at least five women have to included in a committee? In how many of these committees? In how may of these committees
(a) The women are in majo...

If two events \(A\) and \(B\) are such that \(P\,\,\left( {{A^c}} \right)\,\, = \,\,0.3,\,\,P\left( B \right) = 0.4\) and \(P\left( {A \cap {B^c}} \right) = 0.5,\) then \(P\left( {B/\left( {A \cup {B^c}} \right)} \right.\)$$\left. \, \righ...

An unbiased coin is tossed. If the result is a head, a pair of unbiased dice is rolled and the number obtained by adding the numbers on the two faces is noted. If the result is a tail, a card from a well shuffled pack of eleven cards number...

Let \(A, B, C\) be three mutually independent events. Consider the two statements \({S_1}\) and \({S_2}\)
\({S_1}\,:\,A\) and \(B \cup C\) are independent
\({S_2}\,:\,A\) and \(B \cap C\) are independent
Then,

In a triangle \(ABC\), \(AD\) is the altitude from \(A\). Given \(b>c\), \(\angle C = {23^ \circ }\) and \(AD = {{abc} \over {{b^2} - {c^2}}}\) then \(\angle B =\).................

Let \({A_1},{A_2},........,{A_n}\) be the vertices of an \(n\)-sided regular polygon such that \({1 \over {{A_1}{A_2}}} = {1 \over {{A_1}{A_3}}} + {1 \over {{A_1}{A_4}}}\), Find the value of \(n\).

Consider the following statements connecting a triangle \(ABC\)
(i) The sides \(a, b, c\) and area \(\Delta\) are rational.
(ii) \(a,\tan {B \over 2},\tan {c \over 2}\) are rational.
(iii) \(a,\sin A,\sin B,\sin C\) are rational.
Prove th...

A circle is inscribed in an equilateral triangle of side \(a\). The area of any square inscribed in this circle is ..............

A tower \(AB\) leans towards west making an angle \(\alpha\) with the vertical. The angular elevation of \(B\), the topmost point of the tower is \(\beta\) as observed from a point \(C\) due west of \(A\) at a distance \(d\) from \(A\). I...

If the lengths of the sides of triangle are \(3, 5, 7\) then the largest angle of the triangle is

Let \(p,q \in \left\{ {1,2,3,4} \right\}\,\). The number of equations of the form \(p{x^2} + qx + 1 = 0\) having real roots is

The number of points of intersection of two curves y = 2 sin x and y \(= 5{x^2} + 2x + 3\) is

If p, q, r are + ve and are on A.P., the roots of quadratic equation \(p{x^2} + qx + r = 0\) are all real for

If \(In\left( {a + c} \right),In\left( {a - c} \right),In\left( {a - 2b + c} \right)\) are in A.P., then

The equations to a pair of opposites sides of parallelogram are \({x^2} - 5x + 6 = 0\) and \({y^2} - 6y + 5 = 0,\) the equations to its diagonals are

The locus of a variable point whose distance from \(\left( { - 2,\,0} \right)\) is \(2/3\) times its distance from the line \(x = - {9 \over 2}\) is

Let \(n\) be a positive integer such that \(\sin {\pi \over {2n}} + \cos {\pi \over {2n}} = {{\sqrt n } \over 2}.\) Then

Let \(0 < x < {\pi \over 4}\) then \(\left( {\sec 2x - \tan 2x} \right)\) equals

If \(\omega \,\) is an imaginary cube root of unity then the value of \(\sin \left\{ {\left( {{\omega ^{10}} + {\omega ^{23}}} \right)\pi - {\pi \over 4}} \right\}\) is

Let \(2{\sin ^2}x + 3\sin x - 2 > 0\) and \({x^2} - x - 2 < 0\) (\(x\) is measured in radians). Then \(x\) lies in the interval

If the vectors \(\overrightarrow b ,\overrightarrow c ,\overrightarrow d ,\) are not coplanar, then prove that the vector
$$\left( {\overrightarrow a \times \overrightarrow b } \right) \times \left( {\overrightarrow c \times \overrightar...

The vector \(\,{1 \over 3}\left( {2\widehat i - 2\widehat j + \widehat k} \right)\) is