IIT-JEE 1988

Investigate for maxima and minimum the function
\($f\left( x \right) = \int\limits_1^x {\left[ {2\left( {t - 1} \right){{\left( {t - 2} \right)}^3} + 3{{\left( {t - 1} \right)}^2}{{\left( {t - 2} \right)}^2}} \right]} dt\)$

Find the area of the region bounded by the curve \(C:y=\)
\(\tan x,\) tangent drawn to \(C\) at \(x = {\pi \over 4}\) and the \(x\)-axis.

The equations of the tangents drawn from the origin to the circle \({x^2}\, + \,{y^2}\, - \,2rx\,\, - 2hy\, + {h^2} = 0\), are

If the circle \({C_1}:{x^2} + {y^2} = 16\) intersects another circle \({C_2}\) of radius 5 in such a manner that common chord is of maximum lenght and has a slope equal to 3/4, then the coordinates of the centre of \({C_2}\) are...............

If a circle passes through the point (a, b) and cuts the circle \({x^2}\, + \,{y^2}\, = \,{k^2}\) orthogonally, then the equation of the locus of its centre is

For any two complex numbers \({z_1},{z_2}\) and any real number a and b.
\(\,{\left| {a{z_1} - b{z_2}} \right|^2} + {\left| {b{z_1} + a{z_2}} \right|^2} = .........\)

The cube roots of unity when represented on Argand diagram form the vertices of an equilateral triangle.

The integral \(\int\limits_0^{1.5} {\left[ {{x^2}} \right]dx,}\)
Where [ ] denotes the greatest integer function, equals .............

The value of the integral \(\int\limits_0^{2a} {[{{f\left( x \right)} \over {\left\{ {f\left( x \right) + f\left( {2a - x} \right)} \right\}}}]\,dx}\) is equal to \(a\).

Evaluate \(\int\limits_0^1 {\log \left[ {\sqrt {1 - x} + \sqrt {1 + x} } \right]dx}\)

If \({y^2} = P\left( x \right)\), a polynomial of degree \(3\), then \(2{d \over {dx}}\left( {{y^3}{{{d^2}y} \over {d{x^2}}}} \right)\) equals

Let \(R\) \(= {\left( {5\sqrt 5 + 11} \right)^{2n + 1}}\) and \(f = R - \left[ R \right],\) where [ ] denotes the greatest integer function. Prove that \(Rf = {4^{2n + 4}}\)

Total number of ways in which six ' + ' and four ' - ' signs can be arranged in a line such that no two ' - ' signs occur together is.....................................

There are four balls of different colours and four boxes of colours, same as those of the balls. The number of ways in which the balls, one each in a box, could be placed such that a ball does not go to a box of its own colour is..............

Urn \(A\) contains \(6\) red and \(4\) black balls and urn \(B\) contains \(4\) red and \(6\) black balls. One ball is drawn at random from urn \(A\) and placed in urn \(B\). The one ball is drawn at random from urn \(B\) and placed in urn ...

One hundred identical coins, each with probability, \(p,\) of showing up heads are tossed once. If \(0 < p < 1\) and the probability of heads showing on \(50\) coins is equal to that of heads showing on \(51\) coins, then the value of \(p\)...

A box contains \(2\) fifty paise coins, \(5\) twenty five paise coins and a certain fixed number \(N\,\,\left( { \ge 2} \right)\) of ten and five paise coins. Five coins are taken out of the box at random. Find the probability that the tota...

For two given events \(A\) and \(B,\) \(P\left( {A \cap B} \right)\)

If the angles of a triangle are \({30^ \circ }\) and \({45^ \circ }\) and the included side is \(\left( {\sqrt 3 + 1} \right)\) cms, then the area of the triangle is ...............

A sign -post in the form of an isosceles triangle \(ABC\) is mounted on a pole of height \(h\) fixed to the ground. The base \(BC\) of the triangle is parallel to the ground. A man standing on the ground at a distance \(d\) from the sign-po...

Solve \(\left| {{x^2} + 4x + 3} \right| + 2x + 5 = 0\)

The sum of the first n terms of the series \({1^2} + {2.2^2} + {3^2} + {2.4^2} + {5^2} + {2.6^2} + .........\) is
\(n\,\,{\left( {n + 1} \right)^2}/2,\) when \(n\) is even. When \(n\) is odd, the sum is .............

Sum of the first n terms of the series \({1 \over 2} + {3 \over 4} + {7 \over 8} + {{15} \over {16}} + ............\) is equal to

If the first and the \((2n-1)\)st terms of an A.P., a G.P. and an H.P. are equal and their \(n\)-th terms are \(a,b\) and \(c\) respectively, then

The lines \(2x + 3y + 19 = 0\) and \(9x + 6y - 17 = 0\) cut the coordinates axes in concyclic points.

If \(P=(1, 0),\) \(Q=(-1, 0)\) and \(R=(2, 0)\) are three given points, then locus of the point \(S\) satisfying the relation \(S{Q^2} + S{R^2} = 2S{P^2},\) is

Lines\({L_1} = ax + by + c = 0\) and \({L_2} = lx + my + n = 0\) intersect at the point \(P\) and make an angle \(\theta\) with each other. Find the equation of a line \(L\) different from \({L_2}\) which passes through \(P\) and makes the...

The values of \(\theta\) lying between \(\theta = \theta\) and \(\theta = \pi /2\) and satisfying the equation
$$\left| {\matrix{
{1 + {{\sin }^2}\theta } & {{{\cos }^2}\theta } & {4\sin 4\theta } \cr
{{{\sin }^2}\theta } & {1 ...

The value of the expression \(\sqrt 3 \,\cos \,ec\,{20^0} - \sec \,{20^0}\) is equal to

Let \(\overrightarrow a ,\overrightarrow b ,\overrightarrow c ,\) be three non-coplanar vectors and \(\overrightarrow p ,\overrightarrow q ,\overrightarrow r,\) are vectors defined by the relations $$\overrightarrow p = {{\overrightarrow b...

Let \(OA\) \(CB\) be a parallelogram with \(O\) at the origin and \(OC\) a diagonal. Let \(D\) be the midpoint of \(OA.\) Using vector methods prove that \(BD\) and \(CO\) intersect in the same ratio. Determine this ratio.

The components of a vector \(\overrightarrow a\) along and perpendicular to a non-zero vector \(\overrightarrow b\) are ......and .....respectively.