Determine whether the following statements are true or false, and provide justification: The expression 0/0 is indeterminate, 100/0 is undefined, and 0/100 is neither. The equation x2 + 1 = 0 has no s

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SECTION B (10 Marks)

21. The following expressions are either undefined or indeterminate: $\frac{0}{0}$, $\frac{100}{0}$, $\frac{0}{100}$, $\frac{0}{0}$

    • $\frac{0}{0}$ is indeterminate.
    • $\frac{100}{0}$ is undefined.
    • $\frac{0}{100} = 0$, which is neither undefined nor indeterminate.
    • Therefore, the statement is false.
    • Answer: False

22. The equation $x^2 + 1 = 0$ have no solution in the set of real numbers, $\mathbb{R}$

    • $x^2 = -1$.
    • The square of any real number cannot be negative.
    • Therefore, there are no real solutions.
    • Answer: True

23. $(\frac{2}{5})^{-3} = (\frac{5}{2})^3$

    • Using the property $(\frac{a}{b})^{-n} = (\frac{b}{a})^n$.
    • $(\frac{2}{5})^{-3} = (\frac{5}{2})^3$.
    • Answer: True

24. If $a$ is an even number, then $3a^3 + 3$ is an odd number.

    • If $a$ is even, then $a^3$ is even.
    • $3a^3$ is (odd $\times$ even) = even.
    • $3a^3 + 3$ is (even + odd) = odd.
    • Answer: True

25. The quadratic equation $2 + 6\sqrt{2}y + 9y^2$ has two equal roots.

    • Rearrange the equation: $9y^2 + 6\sqrt{2}y + 2 = 0$.
    • For two equal roots, the discriminant $B^2 - 4AC$ must be 0.
    • Here, $A=9$, $B=6\sqrt{2}$, $C=2$.
    • $B^2 - 4AC = (6\sqrt{2})^2 - 4(9)(2) = (36 \times 2) - 72 = 72 - 72 = 0$
    • Since the discriminant is 0, the equation has two equal roots.
    • Answer: True

26. There are infinitely many integers between 5.3 and 5.37.

    • Integers are whole numbers. There are no integers between 5.3 and 5.37.
    • Answer: False

27. $\log_b (M + N) = \log_b M + \log_b N$

    • The correct logarithm property is $\log_b (MN) = \log_b M + \log_b N$.
    • The given statement is incorrect.
    • Answer: False

28. $\frac{1}{2}x - 3 = 0$ is a linear equation in one variable.

    • This equation is of the form $ax + b = 0$, where $a = \frac{1}{2}$ and $b = -3$. It has one variable ($x$) raised to the power of 1.
    • Answer: True

29. We use the determinant of a quadratic equation to determine its nature of roots.

    • The discriminant ($b^2 - 4ac$) is used to determine the nature of roots for a quadratic equation. The term "determinant" is used for matrices.
    • Answer: False

30. $5y - 4x = 0$ is neither a quadratic equation nor a linear equation?

    • The equation $5y - 4x = 0$ is a linear equation in two variables ($x$ and $y$).
    • Therefore, the statement that it is neither is false.
    • Answer: False