If $$C$$ and $$D$$ are two events such that $$C\subset D$$ and $$P\left(D\right)\ne 0,$$ then the correct statement among the following is :

{{{d^2}x} \over {d{y^2}}} equals:

Let $$I$$ be the purchase value of an equipment and $$V(t)$$ be the value after it has been used for $$t$$ years. The value $$V(t)$$ depreciates at a rate given by differential equation {{dv\left( t \right)} \over {dt}} = - k\left( {T - t} \right), where $$k>0$$ is a constant and $$T$$ is the total life in years of the equipment. Then the scrap value $$V(T)$$ of the equipment is

If {{dy} \over {dx}} = y + 3 > 0\,\, and $$y(0)=2,$$ then $$y\left(\ln 2\right)$$ is equal to :

The lines $$L_1:y-x=0$$ and $$L_2:2x+y=0$$ intersect the line $$L_3:y+2=0$$ at $$P$$ and $$Q$$ respectively. The bisector of the acute angle between $$L_1$$ and $$L_2$$ intersects $$L_3$$ at $$R$$. Statement-1: The ratio $$PR$$ : $$RQ$$ equals $$2\sqrt{2}:\sqrt{5}$$ Statement-2: In any triangle, bisector of an angle divide the triangle into two similar triangles.

The coefficient of $$x^7$$ in the expansion of $${\left(1-x-x^2+x^3\right)}^6$$ is

The domain of the function f(x) = {1 \over {\sqrt {\left| x \right| - x} }} is

The value of \int\limits_0^1 {{{8\log \left( {1 + x} \right)} \over {1 + {x^2}}}} dx is

Let $$\alpha ,\beta$$ be real and z be a complex number. If $$z^2+\alpha z+\beta =0$$ has two distinct roots on the line Re z = 1, then it is necessary that :

If $$\omega (\ne 1)$$ is a cube root of unity, and $$(1+\omega {)}^7=A+B\omega$$. Then $$(A,B)$$ equals :

The two circles x2 + y2 = ax, and x2 + y2 = c2 (c > 0) touch each other if :

Equation of the ellipse whose axes of coordinates and which passes through the point $$(-3,1)$$ and has eccentricity \sqrt {{2 \over 5}} is :

Let $R$ be the set of real numbers. Statement I : $A=\{(x,y)\in R\times R:y-x$ is an integer $\}$ is an equivalence relation on $R$. Statement II : $B=\{(x,y)\in R\times R:x=\alpha y$ for some rational number $\alpha \}$ is an equivalence relation on $R$.

The value of $$p$$ and $$q$$ for which the function f\left( x \right) = \left\{ {\matrix{ {{{\sin (p + 1)x + \sin x} \over x}} & {,x 0} \cr } } \right. is continuous for all $$x$$ in R, are

These are 10 points in a plane, out of these 6 are collinear, if N is the number of triangles formed by joining these points. then:

Statement - 1: The number of ways of distributing 10 identical balls in 4 distinct boxes such that no box is emply is $${}^9{C}_3$$. Statement - 2: The number of ways of choosing any 3 places from 9 different places is $${}^9{C}_3$$.

If the angle between the line x = {{y - 1} \over 2} = {{z - 3} \over \lambda } and the plane $$x+2y+3z=4$$ is {\cos ^{ - 1}}\left( {\sqrt {{5 \over {14}}} } \right), then $$\lambda$$ equals :

Statement - 1 : The point $$A(1,0,7)$$ is the mirror image of the point $$B(1,6,3)$$ in the line : {x \over 1} = {{y - 1} \over 2} = {{z - 2} \over 3} Statement - 2 : The line {x \over 1} = {{y - 1} \over 2} = {{z - 2} \over 3} bisects the line segment joining $$A(1,0,7)$$ and $$B(1,6,3)$$

If the mean deviation about the median of the numbers a, 2a,........., 50a is 50, then |a| equals