Class 12th

Suppose a differentiable function $f\left(x\right)$ satisfies the(Functions)

identity $f(x+y)=f\left(x\right)+f\left(y\right)+x{y}^2+x^{2}y$ , for all real $x$ and $y$ . If ${\mathrm{l}\mathrm{i}\mathrm{m}}_{x\to 0} \frac{f\left(x\right)}{x}=1$ then $f^{\mathrm{\text{'}}}\left(3\right)$ is equal to

If the system of equations

$x-2y+3z=9$

$2x+y+z=b$

$x-7y+az=24$

has infinitely many solutions, then $a-b$ is equal to

If the system of equations

If the equation of a plane $P$ , passing through the intersection of the planes, $x+4y-z+7$ $=0$ and $3x+y+5z=8$ is $ax+by+6z=15$ for some $a,b\in R$ , then the distance of the point $(\mathrm{3,2},-1)$ from the plane $P$ is(

If the equation of a plane $P$ , passing through the intersection of the planes, $x+4y-z+7$ $=0$ and $3x+y+5z=8$ is $ax+by+6z=15$ for some $a,b\in R$ , then the distance of the point $(\mathrm{3,2},-1)$ from the plane $P$ is

Let ${\left(2x^2+3x+4\right)}^{10}={\sum }_{r=0}^{20}?a_r{x}^r$ . Then $\frac{a_7}{a_{13}}$ is equal to

Let $y=y\left(x\right)$ be the solution of the differential equation, $x{y}^{\mathrm{\text{'}}}-y=x^2(x\mathrm{c}\mathrm{o}\mathrm{s}x+\mathrm{s}\mathrm{i}\mathrm{n}x),x>0$ . If $y\left(\pi \right)=\pi$ , then $y^{\mathrm{\text{'}}\mathrm{\text{'}}}\left(\frac{\pi }{2}\right)+y\left(\frac{\pi }{2}\right)$ is equal to:

A survey shows that $63\mathrm{\%}$ of the people in a city read newspaper $A$ whereas $76\mathrm{\%}$ read newspaper $B$ . if $x\mathrm{\%}$ of the people read both the newspapers, then a possible value of $x$ can be:

Let $f\left(x\right)=\int \frac{\sqrt[]{x}}{(1+x{)}^2}dx(x\ge 0)$ . Then $f\left(3\right)-f\left(1\right)$ is equal to:

Let $\left[t\right]$ denote the greatest integer $\le t$ . Then the equation in $x,[x{]}^2+2[x+2]-7=0$ has: