I see the image contains multiple math questions covering a variety of topics like number theory, algebra, polynomials, linear equations, and irrational numbers. Let me know which specific problem you'd like to start with, or if you'd like help with solving all of them, one by one.

Here’s a list of the questions visible from the image:

1. Check whether $6^n$ can end with the digit 0 for any natural number $n$.
2. Show that $2\sqrt{2}$ is an irrational number.
3. Show that the reciprocal of $3 + 2\sqrt{2}$ is an irrational number.
4. The LCM of two numbers is 14 times their HCF. The sum of LCM and HCF is 600. If one number is 280, find the other number.
5. In a seminar, the number of participants in Hindi, English, and Mathematics are 60, 84, and 108, respectively. Find the minimum number of rooms required.
6. Find the smallest number which when increased by 17 is exactly divisible by both 520 and 468.
7. Find the smallest number which leaves remainders 8 and 12 when divided by 28 and 32 respectively.
8. If $\alpha$ and $\beta$ are the zeroes of the polynomial $ax^2 + bx + c$, find the value of $\alpha^2 + \beta^2$.
9. Find the condition that zeroes of $p(x) = ax^2 + bx + c$ are reciprocal of each other.
10. Form a quadratic polynomial whose zeroes are $3 + \sqrt{2}$ and $3 - \sqrt{2}$.
11. If the zeroes of the polynomial $x^2 + px + q$ are double the zeroes of $2x^2 - 5x - 3$, find the value of $p$ and $q$.
12. Find the zeroes of the quadratic polynomial $3x^2 - 75$ and verify the relationship between the zeroes and coefficients.
13. If $\alpha$ and $\beta$ are the zeroes of the polynomial $6y^2 - 7y + 2$, find a quadratic polynomial whose zeroes are $\frac{1}{\alpha}$ and $\frac{1}{\beta}$.
14. If $\alpha$ and $\beta$ are the zeroes of $p(x) = 2x^2 + 5x + k$, satisfying the relation $\alpha^2 + \beta^2 + \alpha\beta = 214$, find the value of $k$.
15. Find the value of $k$ for which the following pair of linear equations has infinitely many solutions:
    • $2x + 3y = 7$
    • $(k + 1)x + (2k - 1)y = 4k + 1$
16. If $51x + 23y = 116$ and $23x + 51y = 106$, find the value of $x - y$.
17. Solve $2x + 3y = 11$ and $x - 2y = -12$ algebraically, and hence find the value of $m$ for which $y = mx + 3$.
18. Represent the system of linear equations $3x + y = 5$ and $2x + y = 5$ graphically. Find the points where the lines intersect the y-axis.

Let me know which question you'd like to work on first!