Let $f:\mathbf{R}\to \mathbf{R}$ be a twice differentiable function such that $f(x+y)=f(x) f(y)$ for all $x,y\in \mathbf{R}$. If $f^{\prime }(0)=4\mathrm{a}$ and $f$ satisfies $f^{\prime \prime }(x)-3\mathrm{a}{f}^{\prime }(x)-f(x)=0,\mathrm{a}>0$, then the area of the region $\mathrm{R}=\{(x,y)\mid 0\le y\le f(ax),0\le x\le 2\}$ is :

If $x=f(y)$ is the solution of the differential equation $\left(1+y^2\right)+\left(x-2{\mathrm{e}}^{{\tan }^{-1}y}\right)\frac{\mathrm{d}y}{\mathrm{d}x}=0,y\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)$ with $f(0)=1$, then $f\left(\frac{1}{\sqrt{3}}\right)$ is equal to :

Let a curve $y=f(x)$ pass through the points $(0,5)$ and $\left({\log }_{e}2,k\right)$. If the curve satisfies the differential equation $2(3+y)e^{2x}dx-\left(7+e^{2x}\right)dy=0$, then $k$ is equal to

Let $x=x(y)$ be the solution of the differential equation $y=\left(x-y\frac{\mathrm{d}x}{\mathrm{d}y}\right)\sin \left(\frac{x}{y}\right),y>0$ and $x(1)=\frac{\pi }{2}$. Then $\cos (x(2))$ is equal to :

Let $\mathrm{y}=\mathrm{y}(\mathrm{x})$ be the solution of the differential equation $\left(xy-5x^2\sqrt{1+x^2}\right)dx+\left(1+x^2\right)dy=0,y(0)=0$. Then $y(\sqrt{3})$ is equal to

Let for some function $\mathrm{y}=f(x),{\int }_0^{x}tf(t)dt=x^{2}f(x),x>0$ and $f(2)=3$. Then $f(6)$ is equal to

Let $f(x) = x - 1$ and $g(x)=e^x$ for $x\in \mathbb{R}$. If $\frac{dy}{dx}=\left(e^{-2\sqrt{x}}g\left(f(f(x))\right)-\frac{y}{\sqrt{x}}\right)$, $y(0) = 0$, then $y(1)$ is

Let y = y(x) be the solution of the differential equation $(x^2+1)y^{\prime }-2xy=(x^4+2x^2+1)\cos x$, $y(0) = 1$. Then $\underset{-3}{\overset{3}{\int }}y(x)dx$ is :

Let $y=y(x)$ be the solution curve of the differential equation $x\left(x^2+e^x\right)dy+\left({\mathrm{e}}^x(x-2)y-x^3\right)\mathrm{d}x=0,x>0$, passing through the point $(1,0)$. Then $y(2)$ is equal to :

Let $g$ be a differentiable function such that ${\int }_0^{x}g(t)dt=x-{\int }_0^x\mathrm{tg}(t)dt,x\ge 0$ and let $y=y(x)$ satisfy the differential equation $\frac{dy}{dx}-y\tan x=2(x+1)\sec xg(x),x\in \left[0,\frac{\pi }{2}\right)$. If $y(0)=0$, then $y\left(\frac{\pi }{3}\right)$ is equal to

If a curve $y=y(x)$ passes through the point $\left(1,\frac{\pi }{2}\right)$ and satisfies the differential equation $\left(7x^4\cot y-{\mathrm{e}}^x\mathrm{cosec}y\right)\frac{\mathrm{d}x}{\mathrm{d}y}=x^5,x\ge 1$, then at $x=2$, the value of $\cos y$ is :

Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}+3\left({\tan }^{2}x\right)y+3y={\sec }^{2}x,y(0)=\frac{1}{3}+e^3$. Then $y\left(\frac{\pi }{4}\right)$ is equal to :

Let ΔABC be a triangle formed by the lines 7x – 6y + 3 = 0, x + 2y – 31 = 0 and 9x – 2y – 19 = 0. Let the point (h, k) be the image of the centroid of ΔABC in the line 3x + 6y – 53 = 0. Then h2 + k2 + hk is equal to :

Let the line x + y = 1 meet the axes of x and y at A and B, respectively. A right angled triangle AMN is inscribed in the triangle OAB, where O is the origin and the points M and N lie on the lines OB and AB, respectively. If the area of the triangle AMN is $\frac{4}{9}$ of the area of the triangle OAB and AN : NB = $\lambda :1$, then the sum of all possible value(s) of $\lambda$ is:

Let the triangle PQR be the image of the triangle with vertices $(1,3),(3,1)$ and $(2,4)$ in the line $x+2 y=2$. If the centroid of $△\mathrm{PQR}$ is the point $(\alpha ,\beta )$, then $15(\alpha -\beta )$ is equal to :

Two equal sides of an isosceles triangle are along $ -x + 2y = 4 $ and $ x + y = 4 $. If $ m $ is the slope of its third side, then the sum, of all possible distinct values of $ m $, is:

If A and B are the points of intersection of the circle $x^2+y^2-8x=0$ and the hyperbola $\frac{x^2}{9}-\frac{y^2}{4}=1$ and a point P moves on the line $2x - 3y + 4 = 0$, then the centroid of $\Delta PAB$ lies on the line :

A rod of length eight units moves such that its ends $A$ and $B$ always lie on the lines $x-y+2=0$ and $y+2=0$, respectively. If the locus of the point $P$, that divides the rod $A B$ internally in the ratio $2: 1$ is $9\left(x^2+\alpha y^2+\beta xy+\gamma x+28y\right)-76=0$, then $\alpha -\beta -\gamma$ is equal to :

Let the lines $3x-4y-\alpha =0,8x-11y-33=0$, and $2x-3y+\lambda =0$ be concurrent. If the image of the point $(1,2)$ in the line $2x-3y+\lambda =0$ is $\left(\frac{57}{13},\frac{-40}{13}\right)$, then $|\alpha \lambda |$ is equal to

Let the points $\left(\frac{11}{2},\alpha \right)$ lie on or inside the triangle with sides $x+y=11, x+2 y=16$ and $2 x+3 y=29$. Then the product of the smallest and the largest values of $\alpha$ is equal to :