IIT-JEE 1985

Let \(f\left( x \right) = {\sin ^3}x + \lambda {\sin ^2}x, - {\pi \over 2} < x < {\pi \over 2}.\) Find the intervals in which \(\lambda\) should lie in order that \(f(x)\) has exactly one minimum and exactly one maximum.

Find all the tangents to the curve
\(y = \cos \left( {x + y} \right),\,\, - 2\pi \le x \le 2\pi ,\) that are parallel to the line \(x+2y=0\).

Sketch the region bounded by the curves \(y = \sqrt {5 - {x^2}}\) and \(y = \left| {x - 1} \right|\) and find its area.

From the origin chords are drawn to the circle \({(x - 1)^2} + {y^2} = 1\). The equation of the locus of the mid-points of these chords is.............

Let \({x^2} + {y^2} - 4x - 2y - 11 = 0\) be a circle. A pair of tangentas from the point (4, 5) with a pair of radi from a quadrilateral of area............................

No tangent can be drawn from the point (5/2, 1) to the circumcircle of the triangle with vertices \(\left( {1,\sqrt 3 } \right)\,\,\left( {1, - \sqrt 3 } \right),\,\,\left( {3,\sqrt 3 } \right)\).

If three complex numbers are in A.P. then they lie on a circle in the complex plane.

If \(a,\,b,\,c\) and \(u,\,v,\,w\) are complex numbers representing the vertics of two triangles such that \(c = \left( {1 - r} \right)a + rb\) and \(w = \left( {1 - r} \right)u + rv,\) where \(w = \left( {1 - r} \right)u + rv,\) is a compl...

If \({z_1}\) = a + ib and \({z_2}\) = c + id are complex numbers such that \(\left| {{z_1}} \right| = \left| {{z_2}} \right| = 1\) and \({\mathop{\rm Re}\nolimits} ({z_1}\,{\overline z _2}) = 0\), then the pair of complex numbers \({w_1}\)...

Evaluate the following : \(\,\,\int\limits_0^{\pi /2} {{{x\sin x\cos x} \over {{{\cos }^4}x + {{\sin }^4}x}}} dx\)

For any integer \(n\) the integral ...........
\(\int\limits_0^\pi {{e^{{{\cos }^2}x}}{{\cos }^3}\left( {2n + 1} \right)xdx}\) has the value

If \(f\left( x \right) = {\log _x}\left( {In\,x} \right),\) then \(f'\left( x \right)\) at \(x=e\) is ................

If \({f_r}\left( x \right),{g_r}\left( x \right),{h_r}\left( x \right),r = 1,2,3\) are polynomials in \(x\) such that \({f_r}\left( a \right) = {g_r}\left( a \right) = {h_r}\left( a \right),r = 1,2,3\)
and $$F\left( x \right) = \left| {\ma...

Evaluate the following \(\int {\sqrt {{{1 - \sqrt x } \over {1 + \sqrt x }}dx} }\)

Use method of mathematical induction \({2.7^n} + {3.5^n} - 5\) is divisible by \(24\) for all \(n > 0\)

If \(\left| {\matrix{ a & {{a^2}} & {1 + {a^3}} \cr b & {{b^2}} & {1 + {b^3}} \cr c & {{c^2}} & {1 + {c^3}} \cr } } \right| = 0\) and the vectors
$$\overrightarrow A = \left( {1,a,{a^2}} \right),\,\,\overrightarrow B = \...

7 relatives of a man comprises 4 ladies and 3 gentlemen ; his wife has also 7 relatives ; 3 of them are ladies and 4 gentlemen. In how many ways can they invite a dinner party of 3 ladies and 3 gentlemen so that there are 3 of man's relativ...

The product of any r consecutive natural numbers is always divisible by r!

In a multiple-choice question there are four alternative answers, of which one or more are correct. A candidate will get marks in the question only if he ticks the correct answers. The candidate decides to tick the answers at random, If he ...

\(P\left( {A \cup B} \right) = P\left( {A \cap B} \right)\) if and only if the relation between \(P(A)\) and \(P(B)\) is .............

A box contains \(100\) tickets numbered \(1, 2, ....., 100.\) Two tickets are chosen at random. It is given that the maximum number on the two chosen tickets is not more than \(10.\) The minimum number on them is \(5\) with probability .......

In a triangle \(ABC\), if cot \(A\), cot \(B\), cot \(C\) are in A.P., then \({a^2},{b^2},{c^2}\), are in ............... progression.

In a triangle \(ABC\), the median to the side \(BC\) is of length
\(${1 \over {\sqrt {11 - 6\sqrt 3 } }}\)$ and it divides the angle \(A\) into angles \({30^ \circ }\) and \({45^ \circ }\). Find the length of the side \(BC\).

The set of all real numbers \(a\) such that \({a^2} + 2a,2a + 3\) and \({a^2} + 3a + 8\) are the sides of a triangle is ...........

A ladder rests against a wall at an angle \(\alpha\) to the horizintal. Its foot is pulled away from the wall through a distance \(a\), so that it slides \(a\) distance \(b\) down the wall making an angle \(\beta\) with the horizontal. Sh...

Solve for \(x\) ; \({\left( {5 + 2\sqrt 6 } \right)^{{x^2} - 3}} + {\left( {5 - 2\sqrt 6 } \right)^{{x^2} - 3}} = 10\)

If \({\log _{0.3}}\,(x\, - \,1) < {\log _{0.09}}(x - 1)\), then x lies in the interval-

If \({n_1}\), \({n_2}\),.......\({n_p}\) are p positive integers, whose sum is an even number, then the number of odd integers among them is odd.

If \(P(x) = a{x^2} + bx + c\,\,and\,\,Q(x) = - a{x^2} + dx + c\), where \(ac \ne \,0\), then P(x) Q(x) = 0 has at least two real roots.

If \(a,\,b,\,c\) are in GP., then the equations \(\,\,\alpha {x^2} + 2bx + c = 0\) and \(d{x^2} + 2ex + f = 0\) have a common root if \({d \over a},\,{e \over b},{f \over c}\) are in ________.

Find the sum of the series :
$$$\sum\limits_{r = 0}^n {{{\left( { - 1} \right)}^r}\,{}^n{C_r}\left[ {{1 \over {{2^r}}} + {{{3^r}} \over {{2^{2r}}}} + {{{7^r}} \over {{2^{3r}}}} + {{{{15}^r}} \over {{2^{4r}}}}..........up\,\,to\,\,m\,\,terms...

One of the diameters of the circle circumscribing the rectangle \(ABCD\) is \(4y = x + 7\). If \(A\) and \(B\) are the points \((-3, 4)\) and \((5, 4)\) respectively, then find the area of rectangle.

Three lines \(px + qy + r = 0\), \(qx + ry + p = 0\) and \(rx + py + q = 0\) are concurrent if

Two sides of rhombus \(ABCD\) are parallel to the lines \(y = x + 2\) and \(y = 7x + 3\). If the diagonals of the rhombus intersect at the point \((1, 2)\) and the vertex \(A\) is on the \(y\)-axis, find possible co-ordinates of \(A\).

The orthocentre of the triangle formed by the lines \(x + y = 1,\,2x + 3y = 6\) and \(4x - y + 4 = 0\) lies in quadrant number .............

If \(\overrightarrow A \overrightarrow {\,B} \overrightarrow {\,C}\) are three non-coplannar vectors, then -
$${{\overrightarrow A .\overrightarrow B \times \overrightarrow C } \over {\overrightarrow C \times \overrightarrow A .\overrig...

If \(\overrightarrow A = \left( {1,1,1} \right),\,\,\overrightarrow C = \left( {0,1, - 1} \right)\) are given vectors, then a vector \(B\) satifying the equations \(\overrightarrow A \times \overrightarrow B = \overrightarrow {\,C}\) a...