IIT-JEE 1990

A point \(P\) is given on the circumference of a circle of radius \(r\). Chord \(QR\) is parallel to the tangent at \(P\). Determine the maximum possible area of the triangle \(PQR\).

Show that \(2\sin x + \tan x \ge 3x\) where \(0 \le x < {\pi \over 2}\).

Compute the area of the region bounded by the curves \(\,y = ex\,\ln x\) and \(y = {{\ln x} \over {ex}}\) where \(ln\) \(e=1.\)

A circle touches the line y = x at a point P such that OP = \({4\sqrt 2 \,}\), where O is the origin. The circle contains the point (- 10, 2) in its interior and the length of its chord on the line x + y = 0 is \({6\sqrt 2 \,}\). Determine ...

Let \({z_1}\) = 10 + 6i and \({z_2}\) = 4 + 6i. If Z is any complex number such that the argument of \({{(z - {z_1})} \over {(z - {z_2})}}\,is{\pi \over 4}\) , then prove that \(\left| {z - 7 - 9i} \right| = 3\sqrt 2\).

Let \(f:R \to R\) and \(\,\,g:R \to R\) be continuous functions. Then the value of the integral
$$\int\limits_{ - \pi /2}^{\pi /2} {\left[ {f\left( x \right) + f\left( { - x} \right)} \right]\left[ {g\left( x \right) - g\left( { - x} \ri...

Prove that for any positive integer \(k\),
\({{\sin 2kx} \over {\sin x}} = 2\left[ {\cos x + \cos 3x + ......... + \cos \left( {2k - 1} \right)x} \right]\)
Hence prove that $$\int\limits_0^{\pi /2} {\sin 2kx\,\cot \,x\,dx = {\pi \over 2}} ...

Show that \(\int\limits_0^{\pi /2} {f\left( {\sin 2x} \right)\sin x\,dx = \sqrt 2 } \int\limits_0^{\pi /4} {f\left( {\cos 2x} \right)\cos x\,dx}\)

Let \(f(x)\) be a quadratic expression which is positive for all the real values of \(x\). If \(g(x)=f(x)+f''(x)\), then for any real \(x\),

If \(f\left( x \right) = \left| {x - 2} \right|\) and \(g\left( x \right) = f\left[ {f\left( x \right)} \right]\), then \(g'\left( x \right) = ...............\) for \(x > 20\)

If \(\int {{{4{e^x} + 6{e^{ - x}}} \over {9{e^x} - 4{e^{ - x}}}}\,dx = Ax + B\,\,\log \left( {9{e^{2x}} - 4} \right) + C,}\) then
\(A = .....,B = .....\) and \(C = .....\)

Prove that \({{{n^7}} \over 7} + {{{n^5}} \over 5} + {{2{n^3}} \over 3} - {n \over {105}}\) is an integer for every positive integer \(n\)

A is a set containing \(n\) elements. \(A\) subset \(P\) of \(A\) is chosen at random. The set \(A\) is reconstructed by replacing the elements of \(P.\) \(A\) subset \(Q\) of \(A\) is again chosen at random. Find the probability that \(P\)...

Let \(A\) and \(B\) be two events such that \(P\,\,\left( A \right)\,\, = \,\,0.3\) and \(P\left( {A \cup B} \right) = 0.8.\) If \(A\) and \(B\) are independent events then \(P(B)=\) ................

In a triangle \(ABC\), angle \(A\) is greater than angle \(B\). If the measures of angles \(A\) and \(B\) satify the equation \(3{\mathop{\rm sinx}\nolimits} - 4si{n^3}x - k = 0,\) \(0 < k < 1\), then the measure of angle \(C\) is

A vertical tower \(PQ\) stands at a point \(P\). Points \(A\) and \(B\) are located to the South and East of \(P\) respectively. \(M\) is the mid point of \(AB\). \(PAM\) is an equilateral triangle; and \(N\) is the foot of the perpendicula...

If \(\,x < 0,\,\,y < 0,\,\,x + y + {x \over y} = {1 \over 2}\) and \((x + y)\,{x \over y} = - {1 \over 2}\), then x =..........and y =.........

The number of solutions of the equation sin\({(e)^x} = {5^x} + {5^{ - x}}\) is

The number \({\log _2}\,7\) is

If \({\log _3}\,2\,,\,\,{\log _3}\,({2^x} - 5)\,,\,and\,\,{\log _3}\,\left( {{2^x} - {7 \over 2}} \right)\) are in arithmetic progression, determine the value of x.

Straight lines \(3x + 4y = 5\) and \(4x - 3y = 15\) intersect at the point \(A\). Points \(B\) and \(C\) are choosen on these two lines such that \(AB = AC\). Determine the possible equations of the line \(BC\) passing through the point $$(...

Line \(L\) has intercepts \(a\) and \(b\) on the coordinate axes. When the axes are rotated through a given angle, keeping the origin fixed, the same line \(L\) has intercepts \(p\) and \(q\), then

A line cuts the \(x\)-axis at \(A (7, 0)\) and the \(y\)-axis at \(B (0, -5)\). A variable line \(PQ\) is drawn perpendicular to \(AB\) cutting the \(x\)axis in \(P\) and they \(Y\)-axis in \(Q\). If \(AQ\) and \(BP\) intersect at \(R\), fi...

\(ABC\) is a triangle such that
\($\sin \left( {2A + B} \right) = \sin \left( {C - A} \right) = \, - \sin \left( {B + 2C} \right) = {1 \over 2}.\)$
If \(A,\,B\) and \(C\) are in arithmetic progression, determine the values of \(A,\,B\) and...

The equation \(\left( {\cos p - 1} \right){x^2} + \left( {\cos p} \right)x + \sin p = 0\,\)
In the variable x, has real roots. Then p can take any value in the interval

Let \(\overrightarrow A = 2\overrightarrow i + \overrightarrow k ,\,\overrightarrow B = \overrightarrow i + \overrightarrow j + \overrightarrow k ,\) and $$\overrightarrow C = 4\overrightarrow i - 3\overrightarrow j + 7\overrightarr...