IIT-JEE 1991

A window of perimeter \(P\) (including the base of the arch) is in the form of a rectangle surmounded by a semi circle. The semi-circular portion is fitted with coloured glass while the rectangular part is fitted with clear glass transmits ...

Sketch the curves and identify the region bounded by
\(x = {1 \over 2},x = 2,y = \ln \,x\) and \(y = {2^x}.\) Find the area of this region.

If \('f\) is a continuous function with \(\int\limits_0^x {f\left( t \right)dt \to \infty }\) as \(\left| x \right| \to \infty ,\) then show that every line \(y=mx\)

intersects the curve $${y^2} + \int\limits_0^x {f\left( t \right)dt = ...

Two circles, each of radius 5 units, touch each other at (1, 2). If the equation of their common tangent is 4x + 3y = 10, find the equation of the circles.

If a circle passes through the points of intersection of the coordinate axes with the lines \(\lambda \,x - y + 1 = 0\) and x - 2y + 3 = 0, then the value of \(\lambda\) = .........

Evaluate \(\,\int\limits_0^\pi {{{x\,\sin \,2x\,\sin \left( {{\pi \over 2}\cos x} \right)} \over {2x - \pi }}dx}\)

Find \({{{dy} \over {dx}}}\) at \(x=-1\), when
\({\left( {\sin y} \right)^{\sin \left( {{\pi \over 2}x} \right)}} + {{\sqrt 3 } \over 2}{\sec ^{ - 1}}\left( {2x} \right) + {2^x}\tan \left( {In\left( {x + 2} \right)} \right) = 0\)

Using induction or otherwise, prove that for any non-negative integers \(m\), \(n\), \(r\) and \(k\) ,
$$\sum\limits_{m = 0}^k {\left( {n - m} \right)} {{\left( {r + m} \right)!} \over {m!}} = {{\left( {r + k + 1} \right)!} \over {k!}}\left...

Three normals are drawn from the point \((c, 0)\) to the curve \({y^2} = x.\) Show that \(c\) must be greater than \(1/2\). One normal is always the \(x\)-axis. Find \(c\) for which the other two normals are perpendicular to each other.

Eighteen guests have to be seated, half on each side of a long table. Four particular guests desire to sit on one particular side and three others on the other side. Determine the number of ways in which the sitting arrangements can be made...

In a test an examine either guesses or copies or knows the answer to a multiple choice question with four choices. The probability that he make a guess is \(1/3\) and the probability that he copies the answer is \(1/6\). The probability th...

If the mean and the variance of binomial variate \(X\) are \(2\) and \(1\) respectively, then the probability that \(X\) takes a value greater than one is equal to ...............

For any two events \(A\) and \(B\) in a simple space

In a triangle of base a the ratio of the other two sides is \(r\left( { < 1} \right)\). Show that the altitude of the triangle is less than of equal to \({{ar} \over {1 - {r^2}}}\)

The sides of a triangle are three consecutive natural numbers and its largest angle is twice the smallest one. Determine the sides of the triangle.

A man notices two objects in a straight line due west. After walking a distance \(c\) due north he observes that the objects subtend an angle \(\alpha\) at his eye; and, after walking a further distance \(2c\) due north, an angle $$\beta ...

The product of \(n\) positive numbers is unity. Then their sum is

If \({S_1}\), \({S_2}\), \({S_3}\),.............,\({S_n}\) are the sums of infinite geometric series whose first terms are 1, 2, 3, ...................,n and whose common ratios are \({1 \over 2}\), \({1 \over 3}\), \({1 \over 4}\),...........

Let p be the first of the n arithmetic means between two numbers and q the first of n harmonic means between the same numbers. Show that q does not lie between p and \(\,{\left( {{{n + 1} \over {n - 1}}} \right)^2}\,p\).

Find the equation of the line passing through the point \((2, 3)\) and making intercept of length 2 units between the lines \(y + 2x = 3\) and \(y + 2x = 5\).

Show that all chords of the curve \(3{x^2} - {y^2} - 2x + 4y = 0,\) which subtend a right angle at the origin, pass through a fixed point. Find the coordinates of the point.

Let the algebraic sum of the perpendicular distances from the points \(\left( {2,0} \right),\,\left( {0,\,2} \right)\) \(\left( {1,\,1} \right)\) to a variable straight line be zero; then the line passes through a fixed point whose cordinat...

If \(\exp \,\,\,\left\{ {\left( {\left( {{{\sin }^2}x + {{\sin }^4}x + {{\sin }^6}x + \,\,\,..............\infty } \right)\,In\,\,2} \right)} \right\}\) satiesfies the equation \({x^2} - 9x + 8 = 0,\) find the value of $${{\cos x} \over {\...

The value of
\(\sin {\pi \over {14}}\sin {{3\pi } \over {14}}\sin {{5\pi } \over {14}}\sin {{7\pi } \over {14}}\sin {{9\pi } \over {14}}\sin {{11\pi } \over {14}}\sin {{13\pi } \over {14}}\) is equal to ______.

Given that \(\overrightarrow a = \left( {1,1,1} \right),\,\,\overrightarrow c = \left( {0,1, - 1} \right),\,\overrightarrow a .\overrightarrow b = 3\) and \(\overrightarrow a \times \overrightarrow b = \overrightarrow c ,\) then $$\ove...

Determine the value of \('c'\) so that for all real \(x,\) the vector
\(cx\widehat i - 6\widehat j - 3\widehat k\) and \(x\widehat i + 2\widehat j + 2cx\widehat k\) make an obtuse angle with each other.