IIT-JEE 1998

Suppose \(f(x)\) is a function satisfying the following conditions
(a) \(f(0)=2,f(1)=1\),
(b) \(f\)has a minimum value at \(x=5/2\), and
(c) for all \(x\),
$$$f'\left( x \right) = \matrix{
{2ax} & {2ax - 1} & {2ax + b + 1} \cr
b ...

If \(f\left( x \right) = {{{x^2} - 1} \over {{x^2} + 1}},\) for every real number \(x\), then the minimum value of \(f\)

Let \(h\left( x \right) = f\left( x \right) - {\left( {f\left( x \right)} \right)^2} + {\left( {f\left( x \right)} \right)^3}\) for every real number \(x\). Then

A curve \(C\) has the property that if the tangent drawn at any point \(P\) on \(C\) meets the co-ordinate axes at \(A\) and \(B\), then \(P\) is the mid-point of \(AB\). The curve passes through the point \((1, 1)\). Determine the equation...

The number of values of \(x\) where the function
\(f\left( x \right) = \cos x + \cos \left( {\sqrt 2 x} \right)\) attains its maximum is

If the circle \({x^2}\, + \,{y^2} = \,{a^2}\) intersects the hyperbola \(xy = {c^2}\) in four points \(P\,({x_1},\,{y_1}),\,Q\,\,({x_2},\,{y_2}),\,\,R\,({x_3},\,{y_3}),\,S\,({x_4},\,{y_4}),\) then

The number of common tangents to the circles \({x^2}\, + \,{y^2} = 4\) and \({x^2}\, + \,{y^2}\, - 6x\, - 8y = 24\) is

\(C_1\) and \(C_2\) are two concentric circles, the radius of \(C_2\) being twice that of \(C_1\). From a point P on \(C_2\), tangents PA and PB are drawn to \(C_1\). Prove that the centroid of the triangle PAB lies on \(C_1\).

The value of the sum \(\,\,\sum\limits_{n = 1}^{13} {({i^n}} + {i^{n + 1}})\) , where i = \(\sqrt { - 1}\), equals

If \(\,\left| {\matrix{ {6i} & { - 3i} & 1 \cr 4 & {3i} & { - 1} \cr {20} & 3 & i \cr } } \right| = x + iy\) , then

If \({\omega}\) is an imaginary cube root of unity, then \({(1\, + \omega \, - {\omega ^2})^7}\) equals

If \(\int_0^x {f\left( t \right)dt = x + \int_x^1 {t\,\,f\left( t \right)\,\,dt,} }\) then the value of \(f(1)\) is

Prove that \(\int_0^1 {{{\tan }^{ - 1}}} \,\left( {{1 \over {1 - x + {x^2}}}} \right)dx = 2\int_0^1 {{{\tan }^{ - 1}}} \,x\,dx.\)
Hence or otherwise, evaluate the integral
\(\int_0^1 {{{\tan }^{ - 1}}\left( {1 - x + {x^2}} \right)dx.}\)

Let \(f\left( x \right) = x - \left[ x \right],\) for every real number \(x\), where \(\left[ x \right]\) is the integral part of \(x\). Then \(\int_{ - 1}^1 {f\left( x \right)\,dx}\) is

The order of the differential equation whose general solution is given by
\(y = \left( {{C_1} + {C_2}} \right)\cos \left( {x + {C_3}} \right) - {C_4}{e^{x + {C_5}}},\) where
\({C_1},{C_2},{C_3},{C_4},{C_5},\) are arbitrary constants, is

If\(\,\,\,\) \(y = {{a{x^2}} \over {\left( {x - a} \right)\left( {x - b} \right)\left( {x - c} \right)}} + {{bx} \over {\left( {x - b} \right)\left( {x - c} \right)}} + {c \over {x - c}} + 1\),
prove that $${{y'} \over y} = {1 \over x}\lef...

The number of values of \(c\) such that the straight line \(y=4x + c\) touches the curve \(\left( {{x^2}/4} \right) + {y^2} = 1\) is

If \(P=(x, y)\), \({F_1} = \left( {3,0} \right),\,{F_2} = \left( { - 3,0} \right)\) and \(16{x^2} + 25{y^2} = 400,\) then \(P{F_1} + P{F_2}\) equals

The angle between a pair of tangents drawn from a point \(P\) to the parabola \({y^2} = 4ax\) is \({45^ \circ }\). Show that the locus of the point \(P\) is a hyperbola.

If \({a_n} = \sum\limits_{r = 0}^n {{1 \over {{}^n{C_r}}},\,\,\,then\,\,\,\sum\limits_{r = 0}^n {{r \over {{}^n{C_r}}}} }\) equals

Let \(p\) be a prime and \(m\) a positive integer. By mathematical induction on \(m\), or otherwise, prove that whenever \(r\) is an integer such that \(p\) does not divide \(r\), \(p\) divides \({}^{np}{C_r},\)
[Hint: You may use the fact ...

An n-digit number is a positive number with exactly digits. Nine hundred distinct n-digit numbers are to be formed using only the three digits 2, 5 and 7. The smallest value of n for which this is possible is

If from each of the three boxes containing \(3\) white and \(1\) black, \(2\) white and \(2\) black, \(1\) white and \(3\) black balls, one ball is drawn at random, then the probability that \(2\) white and \(1\) black ball will be drawn is

If \(\overline E\) and \(\overline F\) are the complementary events of events \(E\) and \(F\) respectively and if \(0 < P\left( F \right) < 1,\) then

Seven white balls and three black balls are randomly placed in a row. The probability that no two black balls are placed adjacently equals

Three players, \(A,B\) and \(C,\) toss a coin cyclically in that order (that is \(A, B, C, A, B, C, A, B,...\)) till a head shows. Let \(p\) be the probability that the coin shows a head. Let \(\alpha ,\,\,\,\beta\) and \(\gamma\) be, res...

If \(E\) and \(F\) are events with \(P\left( E \right) \le P\left( F \right)\) and \(P\left( {E \cap F} \right) > 0,\) then

There are four machines and it is known that exactly two of them are faulty. They are tested, one by one, in a random order till both the faulty machines are identified. Then the probability that only two tests are needed is

Let \({C_1}\) and \({C_2}\) be the graphs of the functions \(y = {x^2}\) and \(y = 2x,\) \(0 \le x \le 1\) respectively. Let \({C_3}\) be the graph of a function \(y=f(x),\) \(0 \le x \le 1,\) \(f(0)=0.\) For a point \(P\) on \({C_1},\) let...

A fair coin is tossed repeatedly. If the tail appears on first four tosses, then the probability of the head appearing on the fifth toss equals

Let \({A_0}{A_1}{A_2}{A_3}{A_4}{A_5}\) be a regular hexagon inscribed in a circle of unit radius. Then the product of the lengths of the line segments \({A_0}{A_1},{A_0}{A_2}\) and \({A_0}{A_4}\) is

A bird flies in a circle on a horizontal plane. An observer stands at a point on the ground. Suppose \({60^ \circ }\) and \({30^ \circ }\) are the maximum and the minimum angles of elevation of the bird and that they occur when the bird is ...

Prove that a triangle \(ABC\) is equilateral if and only if \(\tan A + \tan B + \tan C = 3\sqrt 3\).

If in a triangle \(PQR\), \(\sin P,\sin Q,\sin R\) are in \(A.P.,\) then

Number of divisor of the form 4\(n\)\(+ 2\left( {n \ge 0} \right)\) of the integer 240 is

If \(x > 1,y > 1,z > 1\) are in G.P., then \({1 \over {1 + In\,x}},{1 \over {1 + In\,y}},{1 \over {1 + In\,z}}\) are in

Let \({T_r}\) be the \({r^{th}}\) term of an A.P., for \(r=1, 2, 3, ....\) If for some positive integers \(m\), \(n\) we have
\({T_m} = {1 \over n}\) and \({T_n} = {1 \over m},\) then \({T_n} = {1 \over m},\) equals

Let \(n\) be an odd integer. If \(\sin n\theta = \sum\limits_{r = 0}^n {{b_r}{{\sin }^r}\theta ,}\) for every value of \(\theta ,\) then

If \(\left( {P\left( {1,2} \right),\,Q\left( {4,6} \right),\,R\left( {5,7} \right)} \right)\) and \(S\left( {a,b} \right)\) are the vertices of a parrallelogram \(PQRS,\) then

Using co-ordinate geometry, prove that the three altitudes of any triangle are concurrent.

If the vertices \(P, Q, R\) of a triangle \(PQR\) are rational points, which of the following points of the triangle \(PQR\) is (are) always rational point(s)?

The diagonals of a parralleogram \(PQRS\) are along the lines \(x + 3y = 4\) and \(6x - 2y = 7\). Then \(PQRS\) must be a.

Which of the following number(s) is /are rational?

The number of values of \(x\,\,\) in the interval \(\left[ {0,\,5\pi } \right]\) satisfying the equation \(3\,{\sin ^2}x - 7\,\sin \,x + 2 = 0\) is

Prove that \(\tan \,\alpha + 2\tan 2\alpha + 4\tan 4\alpha + 8\cot 8\alpha = \cot \alpha\)

Prove, by vector methods or otherwise, that the point of intersection of the diagonals of a trapezium lies on the line passing through the mid-points of the parallel sides. (You may assume that the trapezium is not a parallelogram.)

Which of the following expressions are meaningful?

For any two vectors \(u\) and \(v,\) prove that
(a) \({\left( {u\,.\,v} \right)^2} + {\left| {u \times v} \right|^2} = {\left| u \right|^2}{\left| v \right|^2}\) and
(b) $$\left( {1 + {{\left| u \right|}^2}} \right)\left( {1 + {{\left| v...

For three vectors \(u,v,w\) which of the following expression is not equal to any of the remaining three?

If \(a = i + j + k,\overrightarrow b = 4i + 3j + 4k\) and \(c = i + \alpha j + \beta k\) are linearly dependent vectors and \(\left| c \right| = \sqrt 3 ,\) then