IIT-JEE 1997

If \(f\left( x \right) = {x \over {\sin x}}\) and \(g\left( x \right) = {x \over {\tan x}}\), where \(0 < x \le 1\), then in this interval

Let \(a+b=4\), where \(a<2,\) and let \(g(x)\) be a differentiable function.
If \({{dg} \over {dx}} > 0\) for all \(x\), prove that \(\int_0^a {g\left( x \right)dx + \int_0^b {g\left( x \right)dx} }\)
increases as \((b-a)\) increases.

Let \(f(x)= Maximum\) \(\,\left\{ {{x^2},{{\left( {1 - x} \right)}^2},2x\left( {1 - x} \right)} \right\},\) where \(0 \le x \le 1.\)
Determine the area of the region bounded by the curves
\(y = f\left( x \right),\) \(x\)-axes, \(x=0\) a...

If \(g\left( x \right) = \int_0^x {{{\cos }^4}t\,dt,}\) then \(g\left( {x + \pi } \right)\) equals

Let C be any circle with centre \(\,\left( {0\, , \sqrt {2} } \right)\). Prove that at the most two rational points can to there on C. (A rational point is a point both of whose coordinates are rational numbers.)

The chords of contact of the pair of tangents drawn from each point on the line 2x + y = 4 to circle \({x^2} + {y^2} = 1\) pass through the point........................

For each natural number k, let \({C_k}\) denote the circle with radius k centimetres and centre at the origin. On the circle \({C_k}\), a-particle moves k centimetres in the counter-clockwise direction. After completing its motion on $${C_k...

Let \({z_1}\) and \({z_2}\) be roots of the equation \({z^2} + pz + q = 0\,\) , where the coefficients p and q may be complex numbers. Let A and B represent \({z_1}\) and \({z_2}\) in the complex plane. If \(\angle AOB = \alpha \ne 0\,\) a...

Let \({d \over {dx}}\,F\left( x \right) = {{{e^{\sin x}}} \over x},\,x > 0.\) If \(\int_1^4 {{{2{e^{\sin {x^2}}}} \over x}} \,\,dx = F\left( k \right) - F\left( 1 \right)\)
then one of the possible values of \(k\) is ............

Determine the value of \(\int_\pi ^\pi {{{2x\left( {1 + \sin x} \right)} \over {1 + {{\cos }^2}x}}} \,dx.\)

The value of \(\int_1^{{e^{37}}} {{{\pi \sin \left( {\pi In\,x} \right)} \over x}\,dx}\) is ...............

Let \(u(x)\) and \(v(x)\) satisfy the differential equation \({{du} \over {dx}} + p\left( x \right)u = f\left( x \right)\) and \({{dv} \over {dx}} + p\left( x \right)v = g\left( x \right),\) where \(p(x) f(x)\) and \(g(x)\) are continuous f...

A tangent to the ellipse x2 + 4y2 = 4 meets the ellipse x2 + 2y2 = 6 at P and Q. Prove that the tangents at P and Q of the ellipse x2 + 2y2 = 6 are at right angles.

The sum of the rational terms in the expansion of \({\left( {\sqrt 2 + {3^{1/5}}} \right)^{10}}\) is ...............

Let \(0 < {A_i} < n\) for \(i = 1,\,2....,\,n.\) Use mathematical induction to prove that
\($\sin {A_1} + \sin {A_2}....... + \sin {A_n} \le n\,\sin \,\,\left( {{{{A_1} + {A_2} + ...... + {A_n}} \over n}} \right)\)$
where \(\ge 1\) is a n...

If \(p\) and \(q\) are chosen randomly from the set \(\left\{ {1,2,3,4,5,6,7,8,9,10} \right\},\) with replacement, determine the probability that the roots of the equation \({x^2} + px + q = 0\) are real.

Let \(S\) be a square of unit area. Consider any quadrilateral which has one vertex on each side of \(S\). If \(a,\,b,\,c\) and \(d\) denote the lengths of the sides of the quadrilateral, prove that $$2 \le {a^2} + {b^2} + {c^2} + {d^2} \le...

The sum of all the real roots of the equation \({\left| {x - 2} \right|^2} + \left| {x - 2} \right| - 2 = 0\) is ............................

Let \(p\) and \(q\) be roots of the equation \({x^2} - 2x + A = 0\) and let \(r\) and \(s\) be the roots of the equation \({x^2} - 18x + B = 0.\) If \(p < q < r < s\) are in arithmetic progression, then \(A = \,..........\) and $$B = \,.......

The real roots of the equation \(\,{\cos ^7}x + {\sin ^4}x = 1\) in the interval \(\left( { - \pi ,\pi } \right)\) are ...., ...., and ______.

Prove that \(\sum\limits_{k = 1}^{n - 1} {\left( {n - k} \right)\,\cos \,{{2k\pi } \over n} = - {n \over 2},}\) where \(n \ge 3\) is an integer.

Prove that the values of the function \({{\sin x\cos 3x} \over {\sin 3x\cos x}}\) do not lie between \({1 \over 3}\) and 3 for any real \(x.\)

If \(A,B\) and \(C\) are vectors such that \(\left| B \right| = \left| C \right|.\) Prove that
\(\left[ {\left( {A + B} \right) \times \left( {A + C} \right)} \right] \times \left( {B \times C} \right)\left( {B + C} \right) = 0\,\,.\)

Let \(OA=a,\) \(OB=10a+2b\) and \(OC=b\) where \(O,A\) and \(C\) are non-collinear points. Let \(p\) denote the area of the quadrilateral \(OABC,\) and let \(q\) denote the area of the parallelogram with \(OA\) and \(OC\) as adjacent sides....