IIT-JEE 1984

For \(0 < a < x,\) the minimum value of the function \(lo{g_a}x + {\log _x}a\) is \(2\).

Find the area of the region bounded by the \(x\)-axis and the curves defined by
\($y = \tan x, - {\pi \over 3} \le x \le {\pi \over 3};\,\,y = \cot x,{\pi \over 6} \le x \le {{3\pi } \over 2}\)$

The lines 3x - 4y + 4 = 0 and 6x - 8y - 7 = 0 are tangents to the same circle. The radius of this circle is ........................................

The abscissa of the two points A and B are the roots of the equation \({x^2}\, + \,2ax\, - {b^2} = 0\) and their ordinates are the roots of the equation \({x^2}\, + \,2px\, - {q^2} = 0\). Find the equation and the radius of the circle with ...

The locus of the mid-point of a chord of the circle \({x^2} + {y^2} = 4\) which subtends a right angle at the origin is

If the complex numbers, \({Z_1},{Z_2}\) and \({Z_3}\) represent the vertics of an equilateral triangle such that
\(\left| {{Z_1}} \right| = \left| {{Z_2}} \right| = \left| {{Z_3}} \right|\) then \({Z_1} + {Z_2} + {Z_3} = 0.\)

If 1, \({{a_1}}\), \({{a_2}}\)......,\({a_{n - 1}}\) are the n roots of unity, then show that (1- \({{a_1}}\)) (1- \({{a_2}}\)) (1- \({{a_3}}\)) ....\((1 - \,a{ - _{n - 1}}) = n\)

Given a function \(f(x)\) such that
(i) it is integrable over every interval on the real line and
(ii) \(f(t+x)=f(x),\) for every \(x\) and a real \(t\), then show that
the integral \(\int\limits_a^{a + 1} {f\,\,\left( x \right)} \,dx\) ...

Evaluate the following \(\int\limits_0^{{1 \over 2}} {{{x{{\sin }^{ - 1}}x} \over {\sqrt {1 - {x^2}} }}dx}\)

If \(\alpha\) be a repeated root of a quadratic equation \(f(x)=0\) and \(A(x), B(x)\) and \(C(x)\) be polynomials of degree \(3\), \(4\) and \(5\) respectively,
then show that $$\left| {\matrix{
{A\left( x \right)} & {B\left( x \right...

Evaluate the following \(\int {{{dx} \over {{x^2}{{\left( {{x^4} + 1} \right)}^{3/4}}}}}\)

The numerical value of \(\tan \left\{ {2{{\tan }^{ - 1}}\left( {{1 \over 5}} \right) - {\pi \over 4}} \right\}\) is equal to __________

Given \({s_n} = 1 + q + {q^2} + ...... + {q^2};\)
\({S_n} = 1 + {{q + 1} \over 2} + {\left( {{{q + 1} \over 2}} \right)^2} + ........ + {\left( {{{q + 1} \over 2}} \right)^n}\,\,\,,q \ne 1\) Prove that $${}^{n + 1}{C_1} + {}^{n + 1}{C_2}{s...

If \(p\) be a natural number then prove that \({p^{n + 1}} + {\left( {p + 1} \right)^{2n - 1}}\) is divisible by \({p^2} + p + 1\) for every positive integer \(n\).

The side AB, BC and CA of a triangle ABC have 3, 4 and 5 interior points respectively on them. The number of triangles that can be constructed using these interior points as vertices is .......................

A box contains \(24\) identical balls of which \(12\) are white and \(12\) are black. The balls are drawn at random from the box one at a time with replacement. The probability that a white ball is drawn for the \(4\)th time on the \(7\)th ...

If \(M\) and \(N\) are any two events, the probability that exactly one of them occurs is

In a certain city only two newspapers \(A\) and \(B\) are published, it is known that \(25\)% of the city population reads \(A\) and \(20\)% reads \(B\) while \(8\)% reads both \(A\) and \(B\). It is also known that \(30\)% of those who rea...

Three identical dice are rolled. The probability that the same number will appear on each of them is

For a triangle \(ABC\) it is given that \(\cos A + \cos B + \cos C = {3 \over 2}\). Prove that the triangle is equilateral.

With usual notation, if in a triangle \(ABC\);
\({{b + c} \over {11}} = {{c + a} \over {12}} = {{a + b} \over {13}}\) then prove that \({{\cos A} \over 7} = {{\cos B} \over {19}} = {{\cos C} \over {25}}\).

If a < b < c < d, then the roots of the equation (x - a) (x - c) + 2 ( x - b) (x - d) = 0 are real and distinct.

For real \(x\), the function \(\,{{\left( {x - a} \right)\left( {x - b} \right)} \over {x - c}}\) will assume all real values provided

If \(\,{a^2} + {b^2} + {c^2} = 1\), then ab + bc + ca lies in the interval

The equation \(x - {2 \over {x - 1}} = 1 - {2 \over {x - 1}}\) has

If the product of the roots of the equation \(\,{x^2} - 3\,k\,x + 2\,{e^{2lnk}} - 1 = 0\,\,\,\,is\,7\), then the roots are real for k = .................................

If \(a > 0,\,b > 0\) and \(\,c > 0,\) prove that \(\,c > 0,\) prove that \(\left( {a + b + c} \right)\left( {{1 \over a} + {1 \over b} + {1 \over c}} \right) \ge 9\)

The sum of integers from 1 to 100 that are divisible by 2 or 5 is ............

If \(n\) is a natural number such that
\(n = {p_1}{}^{{\alpha _1}}{p_2}{}^{{\alpha _2}}.{p_3}{}^{{\alpha _3}}........{p_k}{}^{{\alpha _k}}\) and \({p_1},{p_2},\,\,......,\,{p_k}\) are distinct primes, then show that \(In\) \(n \ge k\) $$in...

If \(a,\,b\) and \(c\) are in A.P., then the straight line \(ax + by + c = 0\) will always pass through a fixed point whose coordinates are ...............

Two equal sides of an isosceles triangle are given by the equations \(7x - y + 3 = 0\) and \(x + y - 3 = 0\) and its thirds side passes through the point \((1, -10)\). Determine the equation of the third side.

Find the values of \(x \in \left( { - \pi , + \pi } \right)\) which satisfy the equation \({g^{(1 + \left| {\cos x} \right| + \left| {{{\cos }^2}x} \right| + \left| {{{\cos }^3}x} \right| + ...)}} = {4^3}\)

There exists a value of \(\theta\) between 0 and \(2\pi\) that satisfies the equation \(\,\,{\sin ^4}\theta - 2{\sin ^2}\theta - 1 = 0.\)

\(\left( {1 + \cos {\pi \over 8}} \right)\left( {1 + \cos {{3\pi } \over 8}} \right)\left( {1 + \cos {{5\pi } \over 8}} \right)\left( {1 + \cos {{7\pi } \over 8}} \right)\) is equal to

The points with position vectors \(a+b,\) \(a-b,\) and \(a+kb\) are collinear for all real values of \(k.\)

\(A, B, C\) and \(D,\) are four points in a plane with position vectors \(a, b, c\) and \(d\) respectively such that
$$$\left( {\overrightarrow a - \overrightarrow d } \right)\left( {\overrightarrow b - \overrightarrow c } \right) = \left...