Algebra and Arithmetic

The SIGN function returns: 1 for positive numbers, 0 for zero, and -1 for negative numbers. Option D correctly states it returns 1 for positive values. Options A and C are wrong because positive values return 1, not >0 or 0.

The phrase “not negative” means any number that is zero or greater. The condition “>=0” includes zero and all positive numbers, satisfying the definition. In contrast, “=0” only covers zero, “>=1” omits zero, and “<=0” incorrectly includes negative values, so they are not correct.

Not negative means a value is greater than or equal to zero (>=0). This includes zero and all positive numbers. It doesn't mean equal to 0 only, or >=1 (which excludes zero), or <=0 (which would include negative numbers).

Not negative means the value is either zero or positive. In mathematical notation, this is x >= 0. A value greater than or equal to 1 would exclude zero, while less than or equal to 0 would only include negative numbers and zero.

A number that is 'not negative' cannot be less than 0. The number 0 itself is not negative, and all positive numbers are also not negative. Therefore, 'not negative' means the number is greater than or equal to 0 (x ≥ 0).

R = E[X²] - (E[X])² is the definition of variance, which is always non-negative because variance measures spread and cannot be negative. R = 0 only when the random variable is constant. R > 0 when the variable has any variation. Therefore R >= 0 is always true.

We know that, If the four tense operators, P, F, H and G and following axions are $G (Q\rightarrow R)\rightarrow (GQ \rightarrow GR)$ $H (Q\rightarrow R) \rightarrow (HQ\rightarrow HR)$ $G - HQ \rightarrow Q$ $- H - GQ\rightarrow Q$

$cothx = \frac{cosh x}{sinh x} \\ \text{as} |x| < < 1, coshx \approx 1 \hspace{0.2cm} and \hspace{0.2cm} sinh x \approx x \\ \text{Thus} \hspace{0.2cm} coth x \approx \frac{1}{x}$

$\text{If} \hspace{0.5cm} x > y \\ \text{Then} \hspace{0.5cm} e^x > e^y \\ \text{Similarly, for x > y > 1 log x is increasing function. Hence,} \\ \text{In} x > In y$

When NPV is positive at a given discount rate, the project's internal rate of return (R) exceeds the cost of capital (K). The relationship K < R indicates that the project generates returns higher than the required rate, making it acceptable. Positive NPV and IRR > cost of capital are equivalent decision criteria.

Let the successive numbers be n and n + 1. Given: n * (n + 1) = 4556 n² + n = 4556 n² + n - 4556 = 0 On solving this, we will get 67 and 68. So, the larger number is 68.

Overall sum will be original sum - n times the constant quantity. Hence it decreases by same quantity.

The expression {(30 + 45) / 5} + {(18 - 3) x 20} can be solved by simplifying the brackets. {(30 + 45) / 5} + {(18 - 3) x 20} = {75 / 5} + {15 x 20} = 15 + 300 = 315  Hence, the value of the given expression is 315. The correct answer is option (1).