IIT-JEE 2009 Paper 1 Offline

Let \(P(3,2,6)\) be a point in space and \(Q\) be a point on the line
\($\widehat r = \left( {\widehat i - \widehat j + 2\widehat k} \right) + \mu \left( { - 3\widehat i + \widehat j + 5\widehat k} \right)\)$
Then the value of \(\mu\) for...

Area of the region bounded by the curve \(y = {e^x}\) and lines \(x=0\) and \(y=e\) is

Let \(f\) be a non-negative function defined on the interval \([0,1]\).
If \(\int\limits_0^x {\sqrt {1 - {{(f'(t))}^2}dt} = \int\limits_0^x {f(t)dt,0 \le x \le 1} }\), and \(f(0) = 0\), then

Tangents drawn from the point P (1, 8) to the circle
\({x^2}\, + \,{y^2}\, - \,6x\, - 4y\, - 11 = 0\)
touch the circle at the points A and B. The equation of the cirumcircle of the triangle PAB is

Let \(z = x + iy\) be a complex number where x and y are integers. Then the area of the rectangle whose vertices are the roots of the equation \(\overline z {z^3} + z{\overline z ^3} = 350\) is

Let \(z = \,\cos \,\theta \, + i\,\sin \,\theta\) . Then the value of \(\sum\limits_{m = 1}^{15} {{\mathop{\rm Im}\nolimits} } ({z^{2m - 1}})\,at\,\theta \, = {2^ \circ }\) is

Match the statements/expressions in Column I with the open intervals in Column II :

.tg {border-collapse:collapse;border-spacing:0;}
.tg td{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14p...

In a triangle \(ABC\) with fixed base \(BC\), the vertex \(A\) moves such that
\($\cos \,B + \cos \,C = 4{\sin ^2}{A \over 2}.\)$
If \(a, b\) and \(c\) denote the lengths of the sides of the triangle opposite to the angles \(A, B\) and $$C...

The line passing through the extremity \(A\) of the major axis and extremity \(B\) of the minor axis of the ellipse \({x^2} + 9{y^2} = 9\) meets its auxiliary circle at the point \(M\). Then the area of the triangle with vertices at \(A\), ...

Match the conics in Column I with the statements/expressions in Column II :

.tg {border-collapse:collapse;border-spacing:0;}
.tg td{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px;
ove...

Let \(L = \mathop {\lim }\limits_{x \to 0} {{a - \sqrt {{a^2} - {x^2}} - {{{x^2}} \over 4}} \over {{x^4}}},a > 0\). If L is finite, then

The number of matrices in A is

The number of matrices A in A for which the system of linear equations \(A\left[ {\matrix{ x \cr y \cr z \cr } } \right] = \left[ {\matrix{ 1 \cr 0 \cr 0 \cr } } \right]\) has a unique solution, is

The number of matrices A in A for which the system of linear equations \(A\left[ {\matrix{ x \cr y \cr z \cr } } \right] = \left[ {\matrix{ 1 \cr 0 \cr 0 \cr } } \right]\) is inconsistent, is

The number of seven digit integers, with sum of the digits equal to 10 and formed by using the digits 1, 2 and 3 only, is

The probability that X = 3 equals

The probability that \(X\ge3\) equals :

The conditional probability that \(X\ge6\) given \(X>3\) equals :

If \({{{{\sin }^4}x} \over 2} + {{{{\cos }^4}x} \over 3} = {1 \over 5},\) then

If \(\overrightarrow a ,\overrightarrow b ,\overrightarrow c\) and \(\overrightarrow d\) are unit vectors such that \((\overrightarrow a \times \overrightarrow b )\,.\,(\overrightarrow c \times \overrightarrow d ) = 1\) and $$\overright...