Q 11 :

Consider the triangles with vertices A(2, 1), B(0, 0) and C(t, 4), $t\in$[0, 4]. If the maximum and the minimum perimeters of such triangles are obtained at $t=\alpha$ and $t=\beta$ respectively, then $6\alpha +21\beta$ is equal to _____ .             [2023]

Q 12 :

Consider the triangles with vertices A(2, 1), B(0, 0) and C(t, 4), $t\in$[0, 4]. If the maximum and the minimum perimeters of such triangles are obtained at $t=\alpha$ and $t=\beta$ respectively, then $6\alpha +21\beta$ is equal to _____ .             [2023]

Q 13 :

Let $(2\alpha ,\alpha )$ be the largest interval in which the function $f\left(t\right)=\frac{|t+1|}{t^2},t<0,$ is strictly decreasing. Then the local maximum value of the function $g\left(x\right)=2{\log }_e(x-2)+\alpha x^2+4x-\alpha ,x>2,$ is ________ .              [2026]

Q 14 :

Let $f\left(x\right)=x^{2025}-x^{2000}$, $x\in [0,1]$, and the minimum value of the function $f\left(x\right)$ in the interval [0,1] be ${\left(80\right)}^{80}{\left(n\right)}^{-81}$. Then $n$ is equal to         [2026]

• $-80$

   

• $-81$

   

• $-41$

   

• $-40$

   

Q 15 :

The least value of $({\cos }^2\theta -6\sin \theta \cos \theta +3{\sin }^2\theta +2)$ is  [2026]

• $4-\sqrt{10}$

   

• $4+\sqrt{10}$

   

• 1

   

• -1

   

Q 16 :

 If the solution curve $y=f\left(x\right)$ of the differential equation

$(x^2-4)y\text{'}-2xy+2x{(4-x^2)}^2=0,x>2,$

passes through the point (3,15) then the local maximum value of $f$ is _____.   [2026]