IIT-JEE 1981

For all \(x\) in \(\left[ {0,1} \right]\), let the second derivative \(f''(x)\) of a function \(f(x)\) exist and satisfy \(\left| {f''\left( x \right)} \right| < 1.\) If \(f(0)=f(1)\), then show that $$\left| {f\left( x \right)} \right| < 1...

Let \(x\) and \(y\) be two real variables such that \(x>0\) and \(xy=1\). Find the minimum value of \(x+y\).

Use the function \(f\left( x \right) = {x^{1/x}},x > 0\). to determine the bigger of the two numbers \({e^\pi }\) and \({\pi ^e}\)

Find the area bounded by the curve \({x^2} = 4y\) and the straight

Let A be the centre of the circle \({x^2}\, + \,{y^2}\, - \,2x\,\, - 4y\, - 20 = 0\,\). Suppose that the tangents at the points B (1, 7) and D (4. - 2) on the circle meet at the point C. Find the area of the quadrilateral ABCD.

Find the equations of the circle passing through (- 4, 3) and touching the lines x + y = 2 and x - y = 2.

Let the complex number \({{z_1}}\), \({{z_2}}\) and \({{z_3}}\) be the vertices of an equilateral triangle. Let \({{z_0}}\) be the circumcentre of the triangle. Then prove that \(z_1^2 + z_2^2 + z_3^2 = 3z_0^2\).

The complex numbers \(z = x + iy\) which satisfy the equation \(\,\left| {{{z - 5i} \over {z + 5i}}} \right| = 1\) lie on

For complex number \({z_1} = {x_1} + i{y_1}\) and \({z_2} = {x_2} + i{y_2},\) we write \({z_1} \cap {z_2},\,\,if\,\,{x_1} \le {x_2}\,\,and\,\,{y_1} \le {y_2}.\)
Then for all complex numbers \(z\,\,with\,\,1 \cap z,\) we have $${{1 - z} \ov...

Let \(a, b, c\) be non-zero real numbers such that
\(\int\limits_0^1 {\left( {1 + {{\cos }^8}x} \right)\left( {a{x^2} + bx + c} \right)dx = \int\limits_0^2 {\left( {1 + {{\cos }^8}x} \right)\left( {a{x^2} + bx + c} \right)dx.} }\)
Then th...

The value of the definite integral \(\int\limits_0^1 {\left( {1 + {e^{ - {x^2}}}} \right)} \,\,dx\)

Show that : \(\mathop {\lim }\limits_{n \to \infty } \left( {{1 \over {n + 1}} + {1 \over {n + 2}} + .... + {1 \over {6n}}} \right) = \log 6\)

Let \(y = {e^{x\,\sin \,{x^3}}} + {\left( {\tan x} \right)^x}\). Find \({{dy} \over {dx}}\)

Each of the four inequalties given below defines a region in the \(xy\) plane. One of these four regions does not have the following property. For any two points \(\left( {{x_1},{y_1}} \right)\) and \(\left( {{x_2},{y_2}} \right)\) in the r...

The equation \({{{x^2}} \over {1 - r}} - {{{y^2}} \over {1 + r}} = 1,\,\,\,\,r > 1\) represents

Evaluate \(\int {\left( {{e^{\log x}} + \sin x} \right)\cos x\,\,dx.}\)

Let \(a, b, c\) be positive real numbers Let
\(\theta = {\tan ^{ - 1}}\sqrt {{{a\left( {a + b + c} \right)} \over {bc}}} + {\tan ^{ - 1}}\sqrt {{{b\left( {a + b + c} \right)} \over {ca}}}\) $$ + {\,\,\tan ^{ - 1}}\sqrt {{{c\left( {a + b...

Find the value of : \(\cos \left( {2{{\cos }^{ - 1}}x + {{\sin }^{ - 1}}x} \right)\) at \(x = {1 \over 5}\), where
\(0 \le {\cos ^{ - 1}}x \le \pi\) and \(- \pi /2 \le {\sin ^{ - 1}}x \le \pi /2\).

Suppose that the normals drawn at three different points on the parabola \({y^2} = 4x\) pass through the point \((h, k)\). Show that \(h>2\).

Five balls of different colours are to be placed in there boxes of different size. Each box can hold all five. In how many different ways can be place the balls so that no box remains emply?

For a biased die the probabilities for the different faces to turn up are given below :

This die tossed and you are told that either face \(1\) or face \(2\) has turned up. Then the probability that it is face \(1\) is ...............

An anti-aircraft gun can take a maximum of four shots at an enemy plane moving away from it. The probabilities of hitting the plane at the first, second, third and fourth shot are \(0.4, 0.3, 0.2\) and \(0.1\) respectively. What is the pr...

Let the angles \(A, B, C\) of a triangle \(ABC\) be in A.P. and let \(b:c = \sqrt 3 :\sqrt 2\). Find the angle \(A\).

For every integer n > 1, the inequality \({(n!)^{1/n}} < {{n + 1} \over 2}\) holds.

The area enclosed within the curve \(\left| x \right| + \left| y \right| = 1\) is .................

Suppose \({\sin ^3}\,x\sin 3x = \sum\limits_{m = 0}^n {{C_m}\cos \,mx}\) is an identity in x, where C0, C1 ,....Cn are constants, and \({C_n} \ne 0\) , then the value of n is _____.

The general solution of the trigonometric equation sin x+cos x=1 is given by:

Let \(\overrightarrow A ,\overrightarrow B ,\overrightarrow C\) be vectors of length \(3, 4, 5\) respectively. Let \(\overrightarrow A\) be perpendicular to \(\overrightarrow B + \overrightarrow C ,\overrightarrow B\) to $$\overrightarr...

Let \(\overrightarrow A ,\overrightarrow B\) and \({\overrightarrow C }\) be unit vectors suppose that \(\overrightarrow A .\overrightarrow B = \overrightarrow A .\overrightarrow C = 0,\) and thatthe angle between $${\overrightarrow B }$...

The scalar \(\overrightarrow A .\left( {\overrightarrow B + \overrightarrow C } \right) \times \left( {\overrightarrow A + \overrightarrow B + \overrightarrow C } \right)\) equals :