Number System and Simplification

To find the fraction with the lowest denominator, compare the bottom numbers: 6, 7, 9, 4, and 3. The smallest is 3, so 7/3 (Option E) has the lowest denominator. This is a straightforward comparison task.

A proper fraction is a fraction where the numerator is less than the denominator (e.g., 3/4, 2/5). An improper fraction has a numerator greater than or equal to the denominator. A mixed fraction is a whole number plus a proper fraction. Equivalent fractions are different fractions representing the same value.

It is the reciprocal as it is written in - b/a for -a/b. Hence, this option is correct.

If two or more fractions will have the same value, then it is called as an equivalent fraction. The value of the fraction does not change, if the numerator and denominator are multiplied by a same natural number. Here 12/36 = 1/3  ..........[After dividing by 12 on both numerator and denominator]

$\frac{7x^2 + 2x + 5}{x+5}$ = $\frac{7(-1)^2 + 2(-1) + 5}{(-1)+5} = \frac{7-2 + 5}{4}$= $\frac{5}{2}$.

$\frac{7x^2y + 2xy^2 + 3}{(x+4)^3}$ = $\frac{7(1)^2 2 + 2(1)(2)^2 + 3}{(1+4)^3}$= $\frac{14+8+3}{125} = \frac{1}{5}$ 

$\frac{x+4}{x+2} = \frac{1}{5}$ 5 (x + 4) = x + 2 5x + 20 = x + 2 5x – x = 2 – 20 4x = –18 x = $\frac{-18}{4}$ x =$\frac{-9}{2}$

Denominator is 2 more than thrice its numerator, y = 2 + 3x

$\frac{8(x)^3 + 3(x)^2y + 2y}{x+y}$ = $\frac{8(-1)^3 + 3(-1)^2y + 2y}{-1+y}$ = $\frac{8(-1)^3 + 3(-1)^2 2 + 2(2)}{-1+2}$ = $\frac{8(-1) + 6 + 4}{1}$ = – 8 + 6 + 4 = 2

Denominator is 5 more than twice its numerator, y = 5 + 2x

Given y = x^3 + 40. The first derivative dy/dx = 3x^2. The second derivative d^2y/dx^2 = 6x.