The differential equation which represents the family of curves $$y=c_1{e}^{c_{2}x},$$ where $$c_1$$ , and $$c_2$$ are arbitrary constants, is

The shortest distance between the line $$y-x=1$$ and the curve $$x=y^2$$ is :

The lines $$p\left(p^2+1\right)x-y+q=0$$ and $$\left(p^2+1\right){}^{2}x+\left(p^2+1\right)y+2q$$ $$=0$$ are perpendicular to a common line for :

The remainder left out when $$8^{2n}-{\left(62\right)}^{2n+1}$$ is divided by 9 is :

Let $$f\left(x\right)={\left(x+1\right)}^2-1,x\ge -1$$ Statement - 1 : The set $$\left\{x:f\left(x\right)=f^{-1}\left(x\right)\right\}=\left\{0,-1\right\}$$. Statement - 2 : $$f$$ is a bijection.

For real x, let f(x) = x3 + 5x + 1, then

$$\underset{0}{\overset{\pi }{\int }}\left[\cot x\right]dx,$$ where $$\left[.\right]$$ denotes the greatest integer function, is equal to:

If \,\left| {z - {4 \over z}} \right| = 2, then the maximum value of $$\left|z\right|$$ is equal to :

Three distinct points A, B and C are given in the 2 -dimensional coordinates plane such that the ratio of the distance of any one of them from the point $$(1,0)$$ to the distance from the point $$(-1,0)$$ is equal to {1 \over 3}. Then the circumcentre of the triangle ABC is at the point :

If $$P$$ and $$Q$$ are the points of intersection of the circles $$x^2+y^2+3x+7y+2p-5=0$$ and $$x^2+y^2+2x+2y-p^2=0$$ then there is a circle passing through $$P,Q$$ and $$(1,1)$$ for :

The ellipse $$x^2+4y^2=4$$ is inscribed in a rectangle aligned with the coordinate axex, which in turn is inscribed in another ellipse that passes through the point $$(4,0)$$. Then the equation of the ellipse is :

If $A, B$ and $C$ are three sets such that $A\cap B=A\cap C$ and $A\cup B=A\cup C$, then :

Let $$f\left(x\right)=x\left|x\right|$$ and $$g\left(x\right)=\sin x.$$ Statement-1: gof is differentiable at $$x=0$$ and its derivative is continuous at that point. Statement-2: gof is twice differentiable at $$x=0$$.

From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are to be selected and arranged in a row on a shelf so that the dictionary is always in the middle. Then the number of such arrangement is :

If the roots of the equation $$b{x}^2+cx+a=0$$ imaginary, then for all real values of $$x$$, the expression $$3b^2{x}^2+6bcx+2c^2$$ is :

The projections of a vector on the three coordinate axis are $$6,-3,2$$ respectively. The direction cosines of the vector are :

Let the line {{x - 2} \over 3} = {{y - 1} \over { - 5}} = {{z + 2} \over 2} lie in the plane $$x+3y-\alpha z+\beta =0.$$ Then $$\left(\alpha ,\beta \right)$$ equals

Statement - 1 : The variance of first n even natural numbers is {{{n^2} - 1} \over 4} Statement - 2 : The sum of first n natural numbers is {{n\left( {n + 1} \right)} \over 2} and the sum of squares of first n natural numbers is {{n\left( {n + 1} \right)\left( {2n + 1} \right)} \over 6}

If the mean deviation of number 1, 1 + d, 1 + 2d,........, 1 + 100d from their mean is 255, then the d is equal to

The sum to infinite term of the series 1 + {2 \over 3} + {6 \over {{3^2}}} + {{10} \over {{3^3}}} + {{14} \over {{3^4}}} + ..... is