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If $\left(x^2+\frac{1}{x^2}\right)=25$ and $x>0$, then the value of $\left(x^7+\frac{1}{x^7}\right)$ is

If $(a + b\sqrt{3})^2 = 52 + 30\sqrt{3}$, where $a$ and $b$ are natural numbers, then $a + b$ equals

Let x, y, and z be real numbers satisfying

$4(x^2 + y^2 + z^2) = a$

$4(x - y - z) = 3+ a$

Then a equals

If $(a+b \sqrt{n})$ is the positive square root of $(29-12 \sqrt{5})$, where $a$ and $b$ are integers, and $n$ is a natural number, then the maximum possible value of $(a+b+n)$ is

If $x$ is a positive real number such that $x^8 + \frac{1}{x^8} = 47$, then the value of $x^9 + \frac{1}{x^9}$ is

If $p^{2}+q^{2}-29=2 p q-20=52-2 p q$, then the difference between the maximum and minimum possible value of $\left(p^{3}-q^{3}\right)$ is

If $x$ and $y$ are real numbers such that $x^2 + (x - 2y − 1)^2 = - 4y(x + y)$, then the value $x - 2y$ is

Consider the pair of equations: $x^{2}-x y-x=22$ and $y^{2}-x y+y=34$. If $x>y$, then $x-y$ equals

The product of two positive numbers is $616$. If the ratio of the difference of their cubes to the cube of their difference is $157: 3$, then the sum of the two numbers is

$u² + (u − 2v − 1)² = −4v (u + v)$. Then the value of $u + 3v$ is

If $x + 1 = x^2$ and $x > 0$, then $2x^4$ is