Q 1 :

Let $a,b,c$ and $d$ be positive real numbers such that $a+b+c+d=11.$ If the maximum value of $a^5{b}^3{c}^{2}d$ is $3750\beta$, then the value of $\beta$ is        [2023]

• 55 

   

• 108

   

• 90 

   

• 110

   

Q 2 :

Let $A_1$ and $A_2$ be two arithmetic means and $G_1,G_2,G_3$ be three geometric means of two distinct positive numbers. Then $G_1^4+G_2^4+G_3^4+G_1^2{G}_3^2$ is equal to      [2023]

• $2(A_1+A_2)G_1^2{G}_3^2$

   

• ${(A_1+A_2)}^2{G}_1{G}_3$

   

• $(A_1+A_2)G_1^2{G}_3^2$

   

• $2(A_1+A_2)G_1{G}_3$

   

Q 3 :

Let three real numbers $a,b,c$ be in arithmetic progression and $a+1,b,c+3$ be in geometric progression. If $a>10$ and the arithmetic mean of $a,b$ and $c$ is 8, then the cube of the geometric mean of $a,b$ and $c$ is             [2024]

• 316

   

• 128

   

• 120

   

• 312

   

Q 4 :

Let the range of the function $f\left(x\right)=\frac{1}{2+\sin 3x+\cos 3x},$$x\in \mathrm{ℝ}$ be $[a,b]$. If $\alpha$ and $\beta$ are respectively the A.M. and the G.M. of $a$ and $b,$ then $\frac{\alpha }{\beta }$ is equal to          [2024]

• $\pi$

   

• $2$

   

• $\sqrt{\pi }$

   

• $\sqrt{2}$

   

Q 5 :

Let $p,q,r$ are positive real numbers, such that $27pqr\ge {(p+q+r)}^3$ and $3p+4q+5r=12$, then $p^3+q^4+r^5=$

• 3

   

• 6

   

• 2

   

• 4

   

Q 6 :

For the function $f\left(\theta \right)=\alpha {\tan }^2\theta +\beta {\cot }^2\theta$ and $g\left(\theta \right)= \alpha {\sin }^2\theta +\beta {\cos }^2\theta$, $\alpha >\beta >0$, let $\underset{0<\theta <\frac{\pi }{2}}{\min }f\left(\theta \right)=\underset{0<\theta <\pi }{\max }g\left(\theta \right)$. If the first term of a G.P. is $\left(\frac{\alpha }{2\beta }\right)$, its common ratio is $\left(\frac{2\beta }{a}\right)$ and the sum of its first 10 terms is $\frac{m}{n}$, gcd (m, n) = 1, then m + n is equal to:          [2026]