Mathematics and Social Sciences Projects

Assume √2 + √3 is rational, then let √2 + √3 = p/q, where p and q are integers. Show that this leads to a contradiction by deriving that both √2 and √3 must also be rational, which is not true.

First, express each number as a product of prime factors: 108 = $2^2 imes 3^3$, 120 = $2^3 imes 3^1 imes 5^1$, 252 = $2^2 imes 3^2 imes 7^1$. HCF = $2^2 imes 3^1 = 12$; LCM = $2^3 imes 3^3 imes 5^1 imes 7^1 = 1260.

Using the quadratic formula, the zeros are given by $x = \frac{8 \pm \sqrt{(-8)^2 - 4(3)(453)}}{2(3)}$. Verify that the sum of the zeros equals $rac{8}{3}$ and the product equals $rac{453}{3}$.

<p>The product of the zeros must equal 1, which leads to the equation $\frac{6a}{a^2 + 9} = 1$. Solving gives the value of a as 3, $a = 3$.</p> Signup and view all the answers

<p>To determine the vertices, find the intersections: for the first line, set x=0 and find y, and do the same for the second line. Solve the equations simultaneously to find their intersection as well.</p> Signup and view all the answers

<p>Assume √15 is rational, then it can be expressed as p/q. Since 15 is not a perfect square, this leads to a contradiction upon squaring both sides, confirming that √15 is irrational.</p> Signup and view all the answers

<p>Calculate 5! + 2 = 120 + 2 = 122, which can be factored as $2 \times 61$, showing it has divisors other than 1 and itself, thus it is composite.</p> Signup and view all the answers

<p>A voltmeter is connected in parallel with the component across which potential difference is to be measured, ensuring it reads the voltage accurately.</p> Signup and view all the answers

<p>Use Ohm's law: Resistance (R) = Voltage (V) / Current (I). From the table, calculate using any pair of V and I values.</p> Signup and view all the answers

<p>A balanced chemical equation has the same number of atoms of each element on both sides, maintaining the law of conservation of mass. It is crucial for accurate stoichiometric calculations.</p> Signup and view all the answers