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

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

Then the value of $$\mu $$ for which the vector $${\overrightarrow {PQ} }$$ is parallel to the plane $$x - 4y + 3z = 1$$ is :

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 $$\overrightarrow a \,.\,\overrightarrow c = {1 \over 2}$$, then

The probability that $$X\ge3$$ equals :

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

The probability that X = 3 equals

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

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

	Column I		Column II
(A)	Circle	(P)	The locus of the point ($$h,k$$) for which the line $$hx+ky=1$$ touches the circle $$x^2+y^2=4$$.
(B)	Parabola	(Q)	Points z in the complex plane satisfying $$|z+2|-|z-2|=\pm3$$.
(C)	Ellipse	(R)	Points of the conic have parametric representation $$x = \sqrt 3 \left( {{{1 - {t^2}} \over {1 + {t^2}}}} \right),y = {{2t} \over {1 + {t^2}}}$$
(D)	Hyperbola	(S)	The eccentricity of the conic lies in the interval $$1 \le x \le \infty $$.
		(T)	Points z in the complex plane satisfying $${\mathop{\rm Re}\nolimits} {(z + 1)^2} = |z{|^2} + 1$$.

If $$a, b$$ and $$c$$ denote the lengths of the sides of the triangle opposite to the angles $$A, B$$ and $$C$$, respectively, then

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 $$A$$, $$M$$ and the origin $$O$$ is

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

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

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

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

	Column I		Column II
(A)	Interval contained in the domain of definition of non-zero solutions of the differential equation $${(x - 3)^2}y' + y = 0$$	(P)	$$\left( { - {\pi \over 2},{\pi \over 2}} \right)$$
(B)	Interval containing the value of the integral $$\int\limits_1^5 {(x - 1)(x - 2)(x - 3)(x - 4)(x - 5)dx} $$	(Q)	$$\left( {0,{\pi \over 2}} \right)$$
(C)	Interval in which at least one of the points of local maximum of $${\cos ^2}x + \sin x$$ lies	(R)	$$\left( {{\pi \over 8},{{5\pi } \over 4}} \right)$$
(D)	Interval in which $${\tan ^{ - 1}}(\sin x + \cos x)$$ is increasing	(S)	$$\left( {0,{\pi \over 8}} \right)$$
		(T)	$$( - \pi ,\pi )$$

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